AMERICAN MATHEMATICAL SOCIETY
COLLOQUIUM PUBLICATIONS
VOLUME XXIII
ORTHOGONAL POLYNOMIALS
BY
GABOR SZEGO
PROFESSOR OF MATHEMATICS
STANFORD tJNIVERSITT
010214
Published by the
American Mathematical Society
Providence, Rhode Island
1939

First edition, 1939
Revised edition, 1958
Third edition, 1967
Third edition, second printing, 1974
Fourth edition, 1975
Library of Congress Catalog Number 39-33497
International Standard Book Number 0-8218-1023-5
Printed in the United States of America

TO MY WIFE

PREFACE
Recent years have seen a great deal of progress in the field of orthogonal
polynomials, a subject closely related to many important branches of analysis.
Orthogonal polynomials are connected with trigonometric, hypergeometric,
Bessel, and elliptic functions, are related to the theory of continued fractions
and to important problems of interpolation and mechanical quadrature, and
are of occasional occurrence in the theories of differential and integral equations.
In addition, they furnish comparatively general and instructive illustrations of
certain situations in the theory of orthogonal systems. Recently, some of these
polynomials have been shown to be of significance in quantum mechanics and
in mathematical statistics.
The origins of the subject are to be found in the investigation of a certain
type of continued fractions, bearing the name of Stieltjes. Special cases of these
fractions were studied by Gauss, Jacobi, Christoffel, and Mehler, among others,
while more general aspects of their theory were given by Tchebichef, Heine,
Stieltjes, and A. Markoff.
Despite the close relationship between continued fractions and the problem
of moments, and notwithstanding recent important advances in this latter
subject, continued fractions have been gradually abandoned as a starting point
for the theory of orthogonal polynomials. In their place, the orthogonal
property itself has been taken as basic, and it is this point of view which has been
adopted in the following exposition of the subject. Choosing this same basic
property, we discuss certain special orthogonal polynomials, which have been
treated in great detail independently of the general theory, and indeed, even
before this theory existed at all. In this connection we add the names of La-
Laplace, Legendre, Fourier, Abel, Laguerre, and Hermite to those previously
mentioned.
As regards treatises on the subject, we note that the only systematic treat-
treatment thus far given is found in J. Shohat's monograph, Theorie Generate des
Polynomes Orthogonaux de Tchebichef, Memorial des Sciences Mathematiques,
Paris, 1934. Limitations of space have compelled that work to be brief, and
consequently, it does not enter into a detailed treatment of many problems
which have been especially advanced in recent years. It has therefore seemed
desirable to attempt a new and detailed development of the main ideas of this
field, devoting, in particular, some space to recent investigations of the distribu-
distribution of the zeros, of asymptotic representations, of expansion problems, and of
certain questions of interpolation and mechanical quadrature.
In what follows, we are concerned partly with the general theory of orthogonal
polynomials, and partly with the study of special classes of these polynomials.
As might be expected, we have more exhaustive results for these special classes,
and we cite as an instance the classical polynomials satisfying linear differential

vi PREFACE
equations of the second order. Also, when the primary importance of these
special classes in applications is taken into account, it should not be at all sur-
surprising that the present book is mainly devoted to their study. The general
theory, however, as developed in Chapters XII and XIII, doubtless represents
the most important progress made in recent years.
In the present work, no claim is made for completeness of treatment. On the
contrary, the aim has purposely been to make the material suggestive rather
than exhaustive. An attempt has been made to indicate the main and charac-
characteristic methods and to point out the relation of these to some general ideas in
modern analysis. As a rule, preference has been given to those topics to which
we were able to make some new, though modest, contributions, or which we
could present in a new setting. Thus the book contains a number of results not
previously published, some of which originated several years ago. For instance,
we have included a discussion of the Cesaro summability of the Jacobi series at
the end-points of the orthogonality interval (the method used here is of interest
even in the classical case of Legendre series). Further, a new and simpler ap-
approach has been given to S. Bernstein's asymptotic formula for orthogonal
polynomials. We also refer to certain details of minor importance, such as:
simplifications and additions in the asymptotic investigation of Jacobi and
Laguerre polynomials and in the discussion of the expansions in terms of these
polynomials; the discussion of the cases in which the Jacobi differential equation
has only polynomial solutions; the evaluation of the number of zeros of general
Jacobi polynomials in the intervals [— », — 1], [— 1, + 1], [+ 1, + °°]; a new
proof of the Heine-Stieltjes theorem on linear differential equations of the second
order with polynomial coefficients and polynomial solutions, and so on.
In general, we have preferred to discuss problems which may be stated and
treated simply, and which could be presented in a more or less complete form.
This was the main reason for devoting no space to the extremely interesting
arithmetic and algebraic properties of orthogonal polynomials, such as, for
instance, the recent important investigations of I. Schur concerning the irre-
ducibility and related properties of Laguerre and Hermite polynomials. Fur-
Furthermore, we have attached great importance to the idea of replacing incomplete
and overlapping theorems, scattered in the literature, by complete results
involving only intrinsic or necessary restrictions. We have also tried to exploit,
as far as seemed to be at all possible, definite methods, such as, for instance,
Sturm's methods in differential equations (see §§6.3, 6.31, 6.32, 6.83).
A complete treatment of Legendre polynomials was not feasible, and probably
not desirable, in the framework of the general theory. Besides, there are al-
already complete treatises on spherical and other harmonics.1 We have selected
and considered only those properties of Legendre polynomials which are the
starting points of generalizations to ultraspherical, Jacobi, or to more general
polynomials. Another subject which could not be included was Stieltjes'
1 For instance, E. W. Hobson 1 (see bibliography).

PREFACE vii
problem of moments, which has been omitted in spite of its great interest;
for this subject would have necessitated the development of a complicated
apparatus of results and methods. Orthogonal polynomials of more than one
variable also have not been treated.2
The book is based on a course given at Washington University during the
academic year 1935-1936. Acquaintance with the general ideas and methods
of the theory of functions of real and complex variables is naturally required.
Occasionally, Stieltjes-Lebesgue and Lebesgue integrals are considered. In the
greater part of the book, however, these integrals have been avoided, and, except
in a very few places, no detailed properties of them were used.
The problems at the end of this book are, with few exceptions, not new, and
they are not interconnected as are, for instance, those in Polya-Szego's Aufgabcn
und Lehrsdtze. They are more or less supplementary in character and serve as
illustrations and exercises; they sometimes differ widely from one another both
as to subject and method.
The list of references is not complete; it contains only original memoirs, a few
text books of primary importance, and monographs to which references are made
in the text.
For the suggestion of preparing a book on orthogonal polynomials for the
Colloquium Publications, I am indebted to Professor J. D. Tamarkin, who has
also participated in the present work by offering a great number of valuable
suggestions. It is with the greatest gratitude that I mention his friendly
interest.
1 have also received valuable advice from my friends and teachers L. Fejer
(Budapest), and G. Polya (Zurich). My colleagues P. Erdos (Manchester),
G. Grtinwald (Budapest), W. H. Roever (St. Louis), A. Ross (St. Louis), J.
Shohat (Philadelphia), and P. Turan (Budapest) gave generously and unstint-
ingly of their time. F. A. Butter, Jr. (at present in Los Angeles) collaborated
with me in the preparation of the manuscript. This last aid was made possible
through a grant from the Rockefeller Research Fund of Washington University
A936-1937). My student L. H. Kanter also rendered valuable assistance in
the preparation of the manuscript.
My gratitude for the encouragement and help of these friends, colleagues, and
institutions can hardly be measured by any formal acknowledgment. Lastly,
I wish to express to the American Mathematical Society my great appreciation
for the inclusion of the present book in its Colloquium Series.
G. Szego
Washington University, 1938.
2 Cf. the bibliography in Jackson 8, p. 423.

PREFACE TO THE REVISED EDITION
The first printing of this book published in 1939 was about exhausted in 1948.
Reprinting was arranged then but for various reasons no change in the text was
made. During the past twenty years since the preparation of the original
edition was completed, considerable progess was made in this field. A glance
at the pertinent section of the Mathematical Reviews suggests that the interest
in this topic is still very much alive. Systematic treatment of orthogonal
polynomials has been incorporated in various modern texts published in the
meantime. We refer only to the Higher Transcendental Functions published by
the Bateman Manuscript Project Staff (cf. in particular, vol. 2, Chapter X,
edited by Professor A. Erdelyi), and to the book of F. Tricomi, Vorlesungen
iiber Orthogonalreihen (Chapters IV—VI).
Recently the council of the American Mathematical Society has authorized
the author to prepare a revised edition of the book, adding a moderate amount
of material in order to bring it up to date. Naturally, limitations of space and
time did not allow including all new results (or, for that matter, the old ones
which were missing from the original edition). Only a few particularly interest-
interesting new items have been added as well as some details which deserve attention
because of elegance of the method or originality of ideas. We mention here in
particular the important Pollaczek polynomials; they are treated in an Appendix.
Further new material was incorporated in the form of Problems and Exercises.
New bibliographic items have been included, again in a rather selective way.
Finally, misprints have been corrected and numerous minor improvements and
additions made.
The author recollects again, as was stated in the Preface of 1938, that the prep-
preparation of this book was suggested to him by the late Professor J. D. Tamarkin.
Since his untimely death in 1945 his name is not too frequently mentioned. It is
justified and probably necessary to remind the younger mathematical gener-
generation, in the rush of modern developments, how much American mathematics
owes to his great energy and far-sighted intelligence.
Stanford University, 1958 G. Szego
IX

PREFACE TO THE THIRD EDITION
The interest of the mathematical community for orthogonal polynomials,
classical and non-classical, is still not entirely exhausted. During the past
years I lectured about this subject several times at Stanford. The attendants
of the course were upper division and graduate students, specializing in
mathematics, mathematical statistics, calculus of probability, etc.
Only minor changes have been made in the text. I owe numerous im-
improvements and corrections to various friends and colleagues. I mention
particularly Professor Paul Turan (Budapest, Hungary) and Professor
Lee Lorch (Edmonton, Canada). New references, published in the time
interval 1958-1966, have been included.
Stanford University, 1966 G. Szego
PREFACE TO THE FOURTH EDITION
Again the American Mathematical Society has taken the initiative to
reprint the present book, allowing some minor changes and new material.
Among the persons interested in the field of orthogonal polynomials who
have contributed to these changes and additions, I mention with particular
indebtedness my friend and colleague Professor Richard Askey (Madison,
Wisconsin) and the very active and original group of mathematicians, around
him. A very important set of lectures by Askey entitled, "Orthogonal
Polynomials and Special Functions," reached me too late to be incorporated
in the present edition.
Further material has been furnished by Professor Paul Turan (Budapest,
Hungary) and Professor Lee Lorch (Toronto, Canada). New problems and
exercises have also been included. Peter Szego (Redwood City, California)
gave me valuable assistance in preparing the present manuscript.
My gratitude goes to all these friends and colleagues.
Stanford University, 1975 G. Szego
XI

TABLE OF CONTENTS
PAOK
Preface v
Preface to the revised edition ix
Preface to the third edition xi
Preface to the fourth edition xi
Chapter I. Preliminaries 1
Chapter II. Definition of orthogonal polynomials; principal examples ... 23
Chapter III. General properties of orthogonal polynomials 38
Chapter IV. Jacobi polynomials 58
Chapter V. Laguerre and Hermite polynomials 100
Chapter VI. Zeros of orthogonal polynomials Ill
Chapter VII. Inequalities . 159
Chapter VIII. Asymptotic properties of the classical polynomials 191
Chapter IX. Expansion problems associated with the classical polynomials . 244
Chapter X. Representation of positive functions 274
Chapter XI. Polynomials orthogonal on the unit circle 287
Chapter XII. Asymptotic properties of general orthogonal polynomials . . . 296
Chapter XIII. Expansion problems associated with general orthogonal
POLYNOMIALS 313
Chapter XIV. Interpolation 329
Chapter XV. Mechanical quadrature 349
Chapter XVI. Polynomials orthogonal on an arbitrary curve 364
Problems and exercises 377
Further problems and exercises 387
Appendix 393
List of references 401
Further references 416
Index 425
xin

CHAPTER I
PRELIMINARIES
1.1. Notation
Numbers in bold face type, like 1, refer to the bibliography at the end of the
book. The system of section numbering used is Peano's decimal system, and
the numeration of formulas starts anew in each section. Thus, reference to §9.5
and (9.5.2) means section 9.5 in Chapter IX and formula (9.5.2) in the same
section, respectively. A similar numeration has been used for the theorems.
We use the symbol: 8nm = 0 or 1, according as n ^ m, or n = m.
The closed real interval a ^ x ^ b (a and b finite) will be denoted by [a, b].
The same symbol is used if either a or b is infinite or if both are; in this case the
equality sign is excluded.
We often write for a real x
A.1.1) sgnz = - 1,0, + 1,
according as -x is negative, zero, or positive; more generally, for arbitrary com-
complex x, x t6 0, we write
A.1.2) sgnz = \x pxx.
The symbol x denotes the conjugate complex value, 9?(z) the real part, and
3(x) the imaginary part of the complex number x.
If two sequences zn and wn of complex numbers have the property that wn ^ 0
and zn/wn —> 1 as n —> °°, we write zn ^ wn . If zn and wn are complex, wn ^ 0,
and the sequence | zn |/| wn | has finite positive limits of indetermination, we
write zn ~ wn .
Occasionally we make use of the notation
A.1.3) zn = O(an), zn = o(an)
if an > 0, to state that zn/an is bounded, or tends to 0, respectively, as»^ ».
A similar notation is used for a passage of limit other than n —> °c.
A function f(x) is called increasing (strictly increasing) if xi < x2 implies
f(xi) < f(x2); it is called non-decreasing if x\ < x2 implies f(xO S f(?-i) ¦ An
analogous terminology will be used for decreasing functions.
Let p ^ 1, and let a(x) be a non-decreasing function in [a, b] which is not
constant. The class of functions f{x) which are measurable with respect to
a(x) and for which the Stieltjes-Lebesgue integral J* |/(x) |p da(x) exists (see
§1.4) is called Lpa(a, b). In case a(x) = x we use the notation Lp(a, b); in case
p = 1, a(x) arbitrary, the notation La(a, b) is used. If f(x) and g(x) belong to
the class Lpa(a, b), the same is true for/(x) + g(x). (Cf. Kaczmarz-Steinhaus 1,
pp. 10-11.)
1

PRELIMINARIES
I]
1.11. Inequalities
A) Cauchy's inequality. Let {a,}, {b,\, v = 1, 2, • • • , n, be two systems of
complex numbers. Then
A.11.1)
»—i
The equality sign holds if and only if two numbers X, n, not both zero, exist
such that \ar + /J>p = 0,v = 1,2, • • • ,n.
B) Schwarz's inequality. Let/(a;) and g(x) be two functions of class L2a(a, b).
Then f{x)g{x) is of class La(a, b), and
A.11.2)
(bf(x)g(x) da(x) 2 ^ P | /(x) |2 d«(x) P | g(x) |2 rf«(x).
y« J" Ja
C) Inequality for the arithmetic and geometric mean. If f(x) > 0, we have
f{x)da(x)
log/(z) da(x)
A.11.3)
provided all integrals exist, and jbada(x) > 0. (Cf. Hardy-Littlewood-P61ya
1, pp. 137-138.)
D) Abel's transformation and Abel's inequality. From
A.11.4)
where
A.11.5)
= (/0 - /l)G0 + (/l - /2)Gl + • • • + (/„-! - /„)(?„-! + /»(?» ,
v = 0,1,2, ¦•-, n,
, = g0 + gi + ¦ ¦ ¦ + gt,
we obtain, assuming/o ^ /i ^ • • • ^ fn =? 0, and | (?, | ^ G, v = 0, 1, • • • , n,
the inequality
A.11.6) l/otfo + figi + ¦ • ¦ + fngn I S fo G.
E) Second mean-value theorem of the integral calculus. Let f(x) ^ 0 be a
non-increasing function, and let g(x) be continuous, a ^ x ^ b, a and b finite.
Then
A.11.7)
Ja
)g(x) dx =» /(a + 0)
Ja
)dx,
a ^ f ^ 6.
1.12. Polynomials and trigonometric polynomials
We shall consider polynomials in z of the form
A.12.1) p(x) = co + Cix + c2x2 + • • • + cmxm,

[1.2] TRIGONOMETRIC POLYNOMIALS 3
with arbitrary complex coefficients Co, d, c2, ¦ ¦ ¦ , cm. Here m is called the
degree; and if cm ^ 0, the precise degree of P(x). In what follows an arbitrary
polynomial of degree m will be denoted by wm. If po(z), pi(z), • • • , pn(z) are
arbitrary polynomials such that pm(x) has the precise degree m, every wn can be
represented as a linear combination of these polynomials with coefficients which
are uniquely determined.
A trigonometric polynomial in 0 of degree m has the form
A.12.2) gF) = ao + ai cos 0 + bi sin 0 + • ¦ • + am cos md + bm sin md,
with arbitrary complex coefficients. Here m is again called the degree of gF);
m is the precise degree if | am | + | bm \ > 0. According as all the b^ or all the a,
vanish, gF) is referred to as a cosine or a sine polynomial.
The functions cos md and sin (m + lH/sin 0 are polynomials in cos 0 = x of
the precise degree m and are called Tchebichef -polynomials of the first and second
kind, respectively. These polynomials play a fundamental role in subsequent
considerations. Setting
A.12.3) cos me = Tm (cos 0) = Tm(x), sin (m + Ve = Um (cos B) = Um(x)
sin 0
we see that any cosine polynomial of degree m is a polynomial of the same degree
in cos 6 = x, and conversely. Any sine polynomial of degree m, divided by
sin 6, furnishes a cosine polynomial of degree m — 1. Thus, a sine polynomial
can be represented as the product of sin 6 = A — x2I12 by a polynomial in
cos 6 = x.
The polynomials A.12.3) are special cases of the so-called Jacobi polynomials
(cf. Chapter IV). They contain only even or only odd powers of x according
as m is even or odd. Thus cos (m + §H/cos (#/2) and sin (m + ?H/sin@/2)
are cosine polynomials in 0 of degree m; they are also connected with the
Jacobi polynomials (see D.1.8)).
We define the "reciprocal" polynomial of A.12.1) by
A.12.4) p*(x) = x^ix'1) = cm + cm_xx + cm_2x2 + • • • + coxm.
If the zeros of p(x) are xi, x2, • • • , xm , those of p*(x) are x?, x?, • • • , x* , where
x* = x^1 is the point which is obtained from x^ by inversion with respect to the
unit circle | x | = 1 in the complex x-plane. The zeros must be counted accord-
according to their multiplicity, and 0*= oo,oo* = 0;a>asa zero of order k means
that the coefficients of the k highest powers vanish.
1.2. Representation of non-negative trigonometric polynomials
Theorem 1.2.1. Let gF) be a trigonometric polynomial with real coefficients
which is non-negative for all real values of 0. Then there exists a polynomial p{z)
of the same degree as gF) such that gF) = | p(z) |2, where z = e' . Conversely, if
z = elB, the expression \ p(z) \2 always represents a non-negative trigonometric poly-
polynomial in 0 of the same degree as the polynomial p(z).

4 PRELIMINARIES [ I ]
See FejeY 5. The second part of the statement is obvious. The first part is
easily derived from A.12.2) by introducing z* + z~k for 2 cos k6 and z* - z~k
for 2i sin he. We then find g(d) = z~mG(z), where G{z) is a *-2m for which G*(z) =
G(z). Now those zeros of G(z) which are different from 0 and a>, and which
do not have the absolute value 1, can be combined in pairs of the form z^, z*,
0 < | Zp | < 1, where z* has a meaning similar to that in §1.12. Furthermore,
every real zero 0O of g(d) is of even multiplicity, and e'e° is a zero of G(z) of the
same multiplicity. Thus
A.2.1) G(z) = cz' tl(z- z,)(z - z*) U(z- f,J,
0 < | z,, | < 1, | f, | = 1; k + a + t = m.
Since gF) = \ g(d) | = | G(z) |, z = eie, and | z - z, | = | z, | | z - z* \, z = e'fl,
the theorem is established.
The representation in question is, however, not unique. Indeed, if a denotes
an arbitrary zero of p(z), the polynomial p(z) A — az)/(z — a) furnishes another
representation. Hence assuming gF) ^ 0, we can gradually remove all the
zeros from | z | < 1 and obtain the following theorem :
Theorem 1.2.2. Let g(8) satisfy the condition of Theorem 1.2.1 and g(d) f? 0.
Then a representation gF) = | h(el6) |2 exists such that h{z) is a -polynomial of
the same degree as gF), with h(z) y^ 0 in | z | < 1, and h@) > 0. This polynomial
is uniquely determined. If gF) is a cosine polynomial, h(z) is a polynomial with
real coefficients.
A generalization of this normalized representation (its extension to a certain
class of non-negative functions gF)) is of great importance in the discussion of
the asymptotic behavior of orthogonal polynomials. (See Chapters X-XIII.)
1.21. Theorem of Lukacs concerning non-negative polynomials
A) Theorem 1.21.1 (Theorem of Lukacs). Let p(x) be a irm non-negative in
[ —1, +1]. Then p(x) can be represented in the form
\\A(x)J + A - x2){B(x)}2 if mis even,
1A + x)\C(x)}2 + A - x)\D(x)}2 if mis odd.
A.21.1) P(x) = , _
Here A{x), B(x), C(x), and D(x) are real polynomials such that the degrees of the
single terms on the right-hand side do not exceed m.
The proof can be based on Theorem 1.2.2. We have
/ -\ It/ i0\ |2 I —imO/2 -it i$\ 2
p(cos 6) = | h{e ) I = I e h{e ) ,
where h(z) is a wm with real coefficients. Now the expressions
sin (m + 1N cos (m + \N sin (m + \N
A.21.2) cos me,
cos (g/2)

[ 1.3 ] APPROXIMATION BY POLYNOMIALS
are all irm in cos 6 (see §1.12), so that
) + i sin 6 5(cos d) if m is ,
[2* cos @/2) C(cos 6) + i2* sin @/2)Z)(cos 6) if m is odd,.
where the degrees of A(x), B(x), C(x), D(x) are, respectively, m/2, m/2 - 1,
(w - l)/2, (m - l)/2.
B) The following theorem has a simpler character:
Theorem 1.21.2. Every polynomial in x, which is non-negative for all real
values of x, can be represented in the form \A{x)J + \B(x)}2. Every poly-
polynomial which is non-negative for x ^ 0, can be represented in the form
\A(x)}2 + \B(x)}2 + x[{C(x)}2 + \D(x)}2}. Here A(x), B(x), C(x), D(x) are
all real polynomials, and the degree of each term does not exceed the degree of the
given polynomial.
These representations can also be written in the form | P{x) \2 and
| P(x) \2 + x | Q(x) |2, respectively, where P(x) and Q(x) are polynomials with
complex coefficients; for the degrees the same remark holds as before. In the
case when x ^ 0, B(x) and D(x) can be chosen to vanish identically. See
Achieser 4, [2.54], and Karlin-Studden 1, Chapter V, Corollary 8.1.
In connection with this section see Polya-Szego 1, vol. 2, pp. 82, 275, 276,
problems 44, 45, 47.
1.22. Theorems of S. Bernstein
Theorem 1.22.1. // gF) is a trigonometric polynomial of degree m satisfying
the condition \ g@) \ ^ I, 0 arbitrary and real, then \ g'F) \ ^ m.
This theorem is due to S. Bernstein. (Cf. M. Biesz 1.) The upper bound m
cannot be replaced by a smaller one as is readily seen by taking gF) = cos m6.
The following special case is worthy of notice:
Theorem 1.22.2. Let p(z) be an arbitrary irm satisfying the condition
| p(z) | ^ 1, where z is complex, and \ z \ ^ 1; then \ p'(z) \ ^ m, \ z \ ^ 1.
With regard to this theorem see also Szasz 1, pp. 516-517. Finally we
mention the following consequence of Theorem 1.22.1:
Theorem 1.22.3. Let p(x) be a irm satisfying the condition \ p{x) \ ^ 1 in
-1 S x ^ +1. Then
| P'(x) | ^ A - x2yW
This follows by applying Theorem 1.22.1 to gF) = p(cos 6).
1.3. Approximation by polynomials
A) Theorem 1.3.1 (Theorem of Weierstrass). A function, continuous in a
finite closed interval, can be approximated with a preassigned accuracy by poly-
polynomials. A function of a real variable which is continuous and has the period 2tt,
can be approximated by trigonometric polynomials.

6 PRELIMINARIES f: j
For information concerning this theorem we refer to Jackson 4. In the second
part of the theorem let the function in question be even (odd); then the approxi-
approximating trigonometric polynomials can be chosen as cosine (sine) polynomials.
Theorem 1.3.2. Let w(<5) be the modulus of continuity of a given function
f(x), continuous in the finite interval [a, b],
A 3.1) «(«) = max \f(x') - f(x") \ if \ x' - x"\ S 5.
Then for each m we can find a polynomial p(x) of degree m, such that in the given
interval of length I we have
A.3.2) |/(x) - P(x)\ <Au(l/m).
In the case of a periodic function fF) with period 2tt, a trigonometric polynomial
gF) of degree m can be found such that
A.3.3) |/@) - gF) | <
Here A and B are absolute constants.
In this connection see Jackson 4, pp. 7, 15.
Theorem 1.3.3. Let f(x) have a continuous derivative of order n in the finite
interval \a, b], n ^ 1, and let wME) be the modulus of continuity offw(x). Then
a polynomial p(x) of degree m + n exists such that
\f{x) - p{x) | < C(l/mY^(l/m),
(L3'4) \f'{x) - P'(x) | < CWmy-W/m), I = b - a.
Here C is a constant depending only on n.
Analogous inequalities can be obtained for all the derivatives f(x), f'(x),
••¦ ,/w(x).
For the first inequality see Jackson 4 (p. 18, Theorem VIII). To prove the
second inequality we first establish the following lemma:
Lemma. Let fF) be a function of period 2ir satisfying the Lipschitz condition
A.3.5) |M) -M)| < Xl»i - M,
where X is a positive constant. Then there exist for each m trigonometric poly-
polynomials g{B) of degree m such that
A.3.6) \M -gF)\<~, \g'(8)\ <D"\,
fit
where D' and D" are absolute constants.
For the first inequality A.3.6) see Jackson 4, pp. 2-6. When we use his
notation and argument, it suffices to show that | X ll'm{8) I is less than an abso-
absolute constant. But

1.3
APPROXIMATION BY POLYNOMIALS
A.3.7)
and
lL(e) = - ^ f+Tl2 {f(e + 2u) - f(e)}F'm(u) du
A J—r/2
A.3.8)
/x/2 rT/2
u I F'm(u) \du = ± u
Jo
fr/2
= 0A) / U
Jo
+ 0A)
sin mu
msin u
d sin mu
du m sin u
du
sin mu
mu
d sin mu
r/2
U
sin mu
mu
du mu
sin mu
du
mu
du,
since u/sin u is analytic in the closed interval [0, w/2]. On writing mu = x,
d sin x
0{m
Jo
sin x
dx
dx + 0(m~2)
/•oo
/ x
Jo
sin x
dx = 0(m x)-
Now we use (cf. loc. cit.) hm = 0(m).
The analogue of the lemma for polynomials can be derived in the usual way.
Then in the upper bound of the first inequality of A.3.6) the factor b — a = I
appears. It is convenient to transform the interval a S x Hkb into — h ^ y ^ ?
(instead of — 1 ^ y ^ 1, cf. Jackson, loc. cit., p. 14), defining the function in
[—1, — ?] and [\, 1] by a constant.
In order to prove Theorem 1.3.3, we apply Theorem VIII of Jackson (loc.
cit., p. 18) to/'(x). (For this argument cf. loc. cit., p. 16.) Thus
\f'(x) -q(x)\ < K(l/my-\(l/m),
where q(x) is a proper ?r„,+„_! . Applying the lemma to f(x) — jxa q{t)dt,
which satisfies a Lipschitz condition with
X = K{l/mY~\{l/m),
we obtain a irm , say a(x), such that
/(¦
Ja
q(t) dt - a(x)
a'{x)
If we write /* q(t)-dt + a(x) — p(x), the statement is established.
The constants K, K', K" in the last three inequalities depend only on n.
B) Theorem 1.3.4 (Theorem of Runge-Walsh). Let f(x) be an analytic
function regular in the interior of a Jordan curve C and continuous in the closed
domain bounded by C. Then f(x) can be approximated with an arbitrary accuracy
by polynomials.
See Walsh 1, p. 36. This theorem has been proved by Runge in case f(x)
is analytic on C; the general case is due to Walsh.
We need also a supplement to the former theorem, due to Walsh A,

8 PRELIMINARIES [ I ]
pp. 75-76). Let C be again a Jordan curve in the complex x-plane. Let
x = 4>{z) be the map function carrying over the exterior of C into | z \ > 1
and preserving x = z = w. Then the circles \z\ = R, R > 1, correspond to
certain curves CR , called level curves. We have
Theorem 1.3.5. Let f(x) be analytic within and on C, and let CR be the largest
level curve in the interior of which f(x) is regular. Then to an arbitrary r,
0 < r < R, there corresponds a constant M > 0 such that, for each m, a -polynomial
Pm(x) of degree m exists satisfying the inequality
A.3.9) \f(x) - Pm(x) | < Mr'"', x on C.
This holds also if C is a Jordan arc, for example, the interval — 1 S x ^ + 1.
In the latter case CK is an ellipse with foci at ±1, and R is the sum of the semi-
axes (§1.9).
1.4. Orthogonality; weight function; vectors in function spaces
A) Let a(x) be a non-decreasing function in [a, b] which is not constant. If
a = — co (or b = + =o), we require that a(— <») = limI_>_Oo a(x) (a(+ °°) =
limI_>+Oo a(x)) should be finite. The scalar product of two real functions f(x)
and g(x), where x ranges over the real interval [a, b], is defined by the Stieltjes-
Lebesgue integral
A.4.1) (f,g) = ff(x)g(x)da(x),
where we assume that f(x)g(x) is of the class La(a, b). This is certainly the
case if f(x) and g{x) are both continuous, or both of bounded variation, and
[a, b] is a finite interval. For a fixed function a(x) the orthogonality with
respect to the "distribution" da(x) may be defined by the relation
A.4.2) (f,g) =0.
We shall also use the expression "fix) is orthogonal to g(x)."
If we permit f(x) and g(x) to be complex functions in general, definition
A.4.1) must be modified to read
A.4.3) U,g) = f"f(x)jG) da(x).
With this change in the definition of (/, g), we retain A.4.2) as the definition of
orthogonality.
[For the definition of Stieltjes-Lebesgue integrals see, for instance, Hildebrandt
1, pp. 185-194. This definition, given originally for a monotonic a(x), can
easily be extended to the case where a(x) is of bounded variation. Hildebrandt
1, pp. 177-178, may also be consulted for the definition of Riemann-Stieltjes
integrals.

[ 1.4 ] ORTHOGONALITY; WEIGHT FUNCTION; VECTORS 9
In what follows we sometimes need the formula for integration by parts:
A.4.4) jf f(x) da(x) + jf a(x) df(x) = f(b)a(b) - f(a)a(a),
where a and b are finite, a(x) is of bounded variation, and f(x) is continuous.
The integrals are taken as Biemann-Stieltjes integrals.
The expression "distribution" used above arises from the classical inter-
interpretation of da(x) as a continuous or discontinuous mass distribution in the
interval [a, b], the mass contributed by the interval [xi , x2] of [a, b] being
a(x-2) - a(xi).]
B) If a(x) is absolutely continuous, the scalar product A.4.1) reduces to
A.4.5) (f,g) = ?f(x)g(x)w(x) dx,
where the integral is assumed to exist in Lebesgue's sense. Here w(x) is a non-
negative function -measurable in Lebesgue's sense for which ft w{x) dx > 0.
We shall call w(x) the weight function, referring to a weight function of, or on,
the given interval. Instead of "weight function" the term "norm function"
is sometimes used in the literature.3 In the case of a distribution w(x) dx the
total mass corresponding to the interval [xi, ?2] is obviously Jl\ w(x) dx.
In what follows we refer to distributions of the form da(x) as distributions of
StieUjes type.
We use the same concept of distribution and weight function on a curve or on
an arc in the complex plane, for example, on the unit circle. Then we replace
the variable x by the real parameter which is used for the definition of the curve
or arc in question. (See Chapters XI and XVI.)
C) Let da(x), or w(x) dx, a ^ x ^ b, be a fixed distribution, and consider a
space of "vectors" defined by the set of real functions/(x) which belong to the
class h\{a, b). The scalar product of two vectors (functions) fix) and g{x) is
defined by A.4.1) and the length (magnitude, norm) of a vector/(x) by ||/ || =
(/, /)*. Vectors (functions) with ||/|| = 0 are called zero-vectors (zero-func-
(zero-functions); vectors (functions) with ||/|| = 1 are said to be normalized. When
f{x) is not a zero-function, X/(x) will be normalized provided X ?± 0 is a proper
constant, uniquely determined save possibly for sign. If the functions a(x) and
w{x) satisfy the conditions mentioned in A) and B), there exist functions of
positive length for both cases. In the second case/(z) is a zero-function if and
only if \f(x)}2w(x), or what amounts to the same thing, f(x)w(x), vanishes
everywhere in [a, b] except on a set of measure zero. If w{x) and f{x) are
integrable in Biemann's sense,/(x) is a zero-function provided f(x)w(x) vanishes
at every point of continuity.
We note the inequality of Schwarz (cf. A.11.2))
A.4-6) \\fg\\ ^ ||/|| || 9 II ,
3 Some corresponding German and French terms are: Belegungsfunktion, Gewichts-
funktion, fonction caracteristique (Stekloff), poids (S. Bernstein).

10 PRELIMINARIES [I]
the equality sign holding if and only if X/(z) + ng(x) is a zero-function with
X and n proper constants not both zero.
A finite set of functions fo(x), fi(x), ¦ ¦ ¦ , /,(x) is said to be linearly inde-
independent if the equation
|| Xo/o(x) + Xjxlx) + • • • + X,/,(x) ||=0
can be true only for
Xo = Xi = • • • = X; = 0.
Evidently no zero-function can be contained in such a system. An enumerable
set of functions (I = °°) is called linearly independent if the preceding condi-
condition is satisfied for every finite subset of the given set.
The extension of these considerations to complex vector spaces is not difficult.
The scalar product is then defined as in A.4.3).
Concerning the axiomatic foundation of these concepts see Stone 1, Chapter I.
1.5. Closure; integral approximations
(l) Definition. Let p ^ 1, and let a(x) be a non-decreasing function in [a, b]
which is not constant. Let the functions
A.5.1) Mx),f1(x),Mx)) ... ,fn(x), ¦¦¦
be of the class L^{a, b). The system A.5.1) is called closed in Lpa{a, b) if for every
f(x) of La (a, b) and for every e > 0 a function of the form
A.5.2) k(x) = co/o(x) + Ci/i(x) + • • • + cB/B(x)
A.5.3) / |/(x) - k(x) \pda(x) < e.
Ja
With regard to this definition see Kaczmarz-Steinhaus 1, p. 49. These authors
use the term "Abgeschlossenheit" for "closure."
B) Theorem 1.5.1. Let p and a(x) have the same meaning as in the previous
definition, and let the function f(x) be of the class Lpa{a, b), a and b finite. Then
for every e > 0 a continuous function F(x) can be determined such that
A.5.4) [ \f(x) - F(x) Tda(x) <e.
Ja
For a Riemann-integrable function with a(x) = x, this follows by a well-
known argument from the definition of the integral. In the general case, it is
convenient to use the method of W. H. Young of approximating Stieltjes-
Lebesgue integrals. (See Hildebrandt 1, p. 190.)
Applying Weierstrass' theorem, we obtain the following:
4 See the remark at the beginning of §1.4 A).

[1.6] LINEAR FUNCTIONAL OPERATIONS 11
Theorem 1.5.2. Let p, a, b, a(x), f(x) satisfy the conditions of Theorem 1.5.1.
For every e > 0 there exists a polynomial p(x) such that
A.5.5) ( |/(z) - p(x)\pda(x) < e.
This means the closure of the system
A.5.6) {xn\, n = 0,1,2,..-,
in the class L'aia, b). In what follows, we shall use in particular the cases p = 1
and p = 2.
An analogous statement holds for the "mean approximation" of f(x) by
trigonometric polynomials, which is equivalent to the property of closure of
the system
A.5.7) 1, cos x, sin x, cos 2x, sin 2x, ¦ ¦ ¦ , cos nx, sin nx, ¦ • ¦
in La( — ir, +ir).
C) A more precise form of Theorem 1.5.2 is often useful.
Theorem 1.5.3. Let p, a, b, a(x),f(x) satisfy the conditions of Theorem. 1.5.1
and let fix) be real-valued. Then we can find a polynomial p(x) which satisfies
A.5.5) and is such that p(x) remains between the upper and lower bounds of fix).
We refer also to the following property of Riemann-Stieltjes integrals which
plays a role in Chapter X.
Theorem 1.5.4. Let the real-valued function f(x) be bounded in [a, b], a and b
finite, a(x) non-decreasing, and let the Riemann-Stieltjes integral /* f(x) da(x)
exist. For every e > 0 there exist polynomials p(x) and P(x) such that
A.5.8) inf f(x) -e = p(x) ? f{x) = P(x) ^ sup/(x) +e,
and
Ja
A.5.9) / |P(.t) - p(x)}d«(x) <e.
See (for a(x) = x) Polya-Szego 1, vol. 1, pp. 65, 22S, problem 137.
Similar statements hold for approximations by trigonometric polynomials.
If f(x) is an even function, — ir g x ^ +t, the approximating trigonometric
polynomials can be chosen as cosine polynomials.
1.6. Linear functional operations
A) Let U(/) be an operation which makes a number ll(/) correspond to
every function fix), continuous in the finite interval [a, b). This operation is
called additive if
A.6.1) U(ci/i + c2f2) = ClU(/i) + c2U(/2)

12 PRELIMINARIES [ I ]
whenever ci and c2 are constants and /i(x) and /2(x) are arbitrary continuous
functions in [a, 6]. It is called continuous if U(/n) —* U(/) whenever fn(x) —> fix)
uniformly in [a, 6]. Additive and continuous operations are called linear.
An operation U(/) is called limited if there exists a constant M such that
U(/) | ^ M max | / |. The greatest lower bound of these constants M is the
norm of U(/). The class of additive and limited operations U(/) coincides with
that of the linear operations.
According to a theorem of F. Riesz A), any linear operation can be written
in the form
A.6.2)
where a(x) is of bounded variation, defined in [a, b] and independent of f(x).
It is obvious that A.6.2) always represents a linear operation. In A.6.2) the
function a(x) can always be so normalized that a(x — 0) ^ a(x) ^ a(x + 0)
or a(x + 0) ^ a(x) g a(x - 0) for a < x < b. Then the norm of U(/) is
given by /„ | da(x) \, which is the total variation of a(x).
B) Let K(x) be a given function continuous in [a, 6]. Then
A.6.3) / f(x)K(x) dx
defines a linear operation. Dirichlet's integral
n a a\ 1 f+X t( \ sin{Bn+l)(x-xo)/2}
A-6.4) — / f{x) :—r r-r— dx,
2tt J-t sin {(x - xo)/2\
where n is a non-negative integer and x0 arbitrary, is a special case of A.6.3).
It represents the nth partial sum of the Fourier expansion of fix) at x = xo.
Another important example is Fejer's integral
A6 5) * f+X fix)
(L6) 2(n + 1) ] m
f fix) ft
2x(n + 1) ]-. m \ sin{(x-xo)/2!
which represents the nth Cesaro mean of the Fourier expansion of fix). A
further illustration of linear operations is furnished by Lagrange's interpolation
polynomial
A.6.6) L(x) = /,(/; x) = /(xo)Zo(z) + ffaMx) + ¦ • • + f(xn)ln(x),
where k(x), h(x), ¦ • • , ln(x) are the fundamental polynomials associated with the
interpolation points x0, Xi , • • • , xn (see Chapter XIV). For a fixed value
x = ?, the expression L(f; ?) represents a linear operation on/(x). Finally, the
general mechanical quadrature formula,
A.6.7) Q(J) =
is also an example; here Xo, Xi , • • • , Xn are the so-called Cotes numbers (see
Chapter XV).

[ 1.6 ] LINEAR FUNCTIONAL OPERATIONS 13
C) We consider the sequence of linear operations
A.6.8) Un(/) = fbf(x) dan(x), n = 0, 1, 2, • • • ,
and the operation
A.6.9) U(/) = [bf(x)da(x),
where an(x) are normalized as in A.6.2). Then we have the following theorem:
Theorem 1.6. A necessary and sufficient condition that limn-.xVLn(f) = U(/),
where f(x) is an arbitrary continuous function, is that the following two relations
be satisfied simultaneously:
lim Un(xk) = U(xA), k = 0,1,2, ¦¦¦,
n—»w
A.6.10)
/ \dan(x)\ < A, n = 0, 1,2,
Moreover, if the second condition A.6.10) is not satisfied, a continuous func-
function/(x) exists such that the sequence {lln(/)} is unbounded.
This important theorem is due to E. Helly A, pp. 268-271). See also
Banach 1, p. 123. The first condition A.6.10) expresses the validity of the
limiting relation for an arbitrary polynomial. The second condition A.6.10)
states that the total variations of the functions an(x) are bounded.
D) Let b — a = 2w, and suppose that/(x), an(x), and a(x) are functions with
period 2-k. Then the first condition A.6.10) must be replaced by the following:
lim Un(cos kx) = U(cos kx),
A.6.11) ""°° k = 0, 1, 2, •••
lim Un(sin kx) = U(sin kx),
n —»oo
One of the most important applications of the preceding considerations is
to the theory of "singular integrals" of Lcbesgue:
A.6.12) Un(/) = f f(x)Kn(x)dx,
where \Kn(x)} is a given sequence of continuous functions. In this case we
are mainly interested in finding a necessary and sufficient condition that
l\n(f) —>f(xo), where x0 is a fixed point in [a, b] ?id/(x) an arbitrary continuous
function. According to Helly's theorem, this must hold if /(x) is an arbitrary
polynomial (or trigonometric polynomial in the periodic case) and the so-called
Lebcsgue constants (which arc the norms of Un(/)) are bounded:
A.6.13) / \Kn(x)\dx <A.

14 PRELIMINARIES [ I ]
See Lebesgue 1, 2; in particular articles 45, 46, pp. 86-88. See also Haar 1.
For Dirichlet's integral A.6.4) this condition A.6.13) is not satisfied; hence
there exist continuous functions whose Fourier expansions are divergent at a
preassigned point. (Du Bois-Reymond 1; Lebesgue 2, chapter IA', pp. 84-89.)
This condition is, however, satisfied in the case of Fejer's integral A.6.5) which
implies the Cesaro summability of the Fourier expansion of a continuous func-
function (Fejer 2, in particular p. 60). The same holds for the Cesaro means of
second order of the Legendre series (Fejer 4).
Regarding applications of Helly's theorem to the theory of interpolation and
mechanical quadrature, see Chapters XIV and XV.
1.7. The Gamma function
The Euler integral of the second kind
A.7.1) T(z) =
defines the Gamma function T(z) for 9i(z) > 0. By analytic continuation we
obtain a meromorphic function without zeros and with simple poles at z = 0,
— 1, —2, • • • . The functional equations
A.7.2) T(z + 1) = zT(z), r(z)r(l - z) = 7r/sin7r2
hold. Another important formula is
T(z)r(z + 1/n) • • • T{z + (n - l)/n)
A.7.3)
= nh~nz{2ir){n~l)nT{nz), n a positive integer.
In what follows we use mainly the cases n = 2 and n = 3.
The Euler integral of the first kind
A.7.4) B(p, q) = f xp-\l - xY~l dx, p > 0, q > 0,
Jo
can be expressed in terms of the Gamma function thus:
The integral A.7.4) exists also for complex p and q with positive real parts for
which A.7.5) remains valid. By means of A.7.5) the definition of B{p, q) can be
extended to arbitrary complex p and q. (See WThittaker-Watson 1, Chapter 12,
pp. 237, 239, 240, 254.)
The special case n = 2 of A.7.3) is as follows:
A.7.6) rB)rB + i/2)=21-2V
We mention also the formula
i i r(o+)
A-7.7) __= e'r
LTTl J - <=
1.71. Bessel functions
A) The Bessel function of the first kind of order a can be defined by

[ 1.71 ] BESSEL FACTIONS 15
a
Obviously, 3~Va0z) is an even integral function. Here a is arbitrary real.
If a is a negative integer, {T(v + a + I)) must be replaced by 0 whenever
v + a + 1 ^ 0. We then obtain the relation JaB) = (-l)V_o(z). If a is
not an integer, Ja(z) and J-a(z) are linearly independent. We notice the
special cases
A.71.2) J_jB) = (-) cos 3, />(z) = f-Ysinz.
The function A.71.1) satisfies Bessel's differential equation
A.71.3) y" + z'V +A - aV2)y = 0, y = Ja(z).
For non-negative integral values of a we introduce the Bessel functions of the
second kind
Ya(z) = - ( t + log - ) Ja(z) - -
TV ,-0_ V\
Here 7 is Euler's constant. The first sum is to be suppressed for a = 0, and the
curly brackets in the second sum are to be replaced by 1 for v = 0, a = 0, and by
1/1 + 1/2 + • • • + I/a for v = 0, a > 0. This function furnishes a second
solution of A.71.3) independent of A.71.1). (See Whittaker-Watson 1, Chap-
Chapter 17, pp. 370, 372.)
The formulas
A.71.5) J^z) + Ja+1(z) = 2az-1Ja(z), i {z~aJa(z)\ = - z-V.+i(z),
az
follow directly from A.71.1) on comparing the corresponding coefficients on
both sides. The integral representation
<L7L6) J-(z)' ffjrmii) CA ~er>e'"dl
holds for a > — |. This can be verified by introducing the development of
eltt and integrating by means of A.7.5).
B) The following important asymptotic formula is used in various appli-
applications :
= ( - )
\WZ/
A.71.7) Ja(z) = ( - ) cos (z - aw/2 - tt/4) + 0(z~j), z -¦+«>.
\wz/
This is only a special case (p = 1) of the asymptotic expansion (see Whittaker-
Watson 1, p. 368):

16 PRELIMINARIES [ I
Jab) = (-) cos (z - aw/2 - tt/4) { E a,*" + 0(z~2p) \
A.71.8) W V=°
+ (-) sin (z - aw/2 - tt/4) {S hz-2-1 +
here p is an arbitrary positive number, av and bv certain constants depending
only on v, and z —* + °o. Also, ao = 1.
This expansion holds also if z is complex, | arg z | ^ w — 5, 5 > 0, if we agree
that z1 = exp (| log 2) with | ^(log z) \ ^ w — d. We notice the following
important consequence of this formula, valid for —w/2 + 8 fs arg s 5s 3w/2 — 5,
5 > 0:
axil2 j / —ir/2 \ /q n—4 z r -. ¦ /-)/i l^ 1
(L7L9) *
+ Ba-z)-* exp [- z + (a + §)W] {1 + 0(| z T) }.
An asymptotic formula similar to A.71.7) holds for Bessel functions Ya(z)
of the second kind, with the only change that cosine is to be replaced by sine.
(See Whittaker-Watson 1, p. 371.)
C) It may be useful to notice th4: order of magnitude of Ja(z) and Ya(z) for
z —* 4- 0 and z —* + °o. From the preceding formulas we see that when z —* + 0,
Ja(z) ~ za, a real, a ^ — 1, —2, —3, ¦ • ¦ ,
A.71.10) Ya(z)~z~a, a = 1, 2, 3, ••-,
F0(z) ~ log A/z),
while when 3 —> + °°,
A.71.11) J.(z) = O(zTh), Ya(z) = (^(z).
1.8. Differential equations
We shall make frequent use of certain elementary transformations of homo-
homogeneous linear differential equations of the second order.
A) Let K(x), M(x), N(x) be functions defined in the interval a < x < b
in which K(x) and Mix) have continuous derivatives and K(x) ^ 0; let N(x)
be continuous. If in
A.8.1) K(x)y" + M(x)y' + N(x)y = 0
we introduce y = s(x)u(x), u(x) being the new unknown function, s(x) can be
determined so that u(x) satisfies an equation of the form
A.8.2) u" + \{x)u = 0.
Direct calculation gives
A.8.3) 2Ks' + Ms = 0; s(x) = exp I- /

[ 1.8 ] DIFFERENTIAL EQUATIONS 17
where the integration is extended from an arbitrary point x0 to x. Then
B) If we introduce into A.8.1) the new independent variable 0 defined by
x = aF), we obtain
A.8.5) K(x)a'@) ^ + {M(x)WF)f - K(x)a"(B)\ § + N(x)W(d)fy = 0.
If we apply the process in A) to A.8.5), the first derivative can be removed.
We write y = s*u; here, in view of A.8.3),
f, o «\ * / f Ma'2 - Ka" Jt\ , A}
A.8.6) s* = exp < - / —— dd> = (a'ys,
I J 2Ka )
where s has the same meaning as in A.8.3). Hence, y = (a'^su, and u satisfies
A.8.7) d-? + \*u = 0,
with
A.8.8) X* = — ( ° 2Ka' a ) " \ ° 2Ka' " ) + K ^'
As an application of the above, we note the following transformations of
Bessel's differential equation A.71.3), k ^ 0,
A.8.10) ^ + Q 4- ^^ u = 0;
Another elementary formula, important for further exposition is the repre-
representation of a solution y = y.(x) of the non-homogeneous equation
A.8.11) K(x)y" 4- M(x)y' + tf (x)y = /(x)
in terms of a fundamental system \yi(x), yz(x)\ of the corresponding homo-
homogeneous equation A.8.1). We have
where x0 is a fixed value and Ci, c2 proper constants. Now
A.8.13) yi(x)yz(x) - y[(x)yi(x) = const, exp < - / ^rfx>.
{ J K )
When we take M = 0, this expression becomes a constant.

18 PRELIMINARIES [ I ]
Applying this last remark to A.8.9), we obtain the important formulas
t' i \ t / \ r' t \ t / \ 2 sin air . , ,
Ja{x)J-a{x) — J-a{x)Ja{x) — , a non-integral,
¦wx
A.8.14)
j'a(x)Ya(x) - Y'a(z)Ja(x) = - — , a = 0, 1, 2, • • • .
TTX
As regards the evaluation of the constants on the right side, see Watson 3,
p. 43, B), p. 76, A).
1.81. Airy's function
An interesting transformation of Bessel's differential equation A.71.3) can
be obtained in the special cases a = ±1/3. If we use A.8.7) and A.8.8), there
is no difficulty in showing that both integral functions
*(*) = \ (x/3)V_J{2(x/3)i) = I ± -~~Lrv
3 3 ,=o v\ T(v + 2/3)
7(V) — _ (rl
o o o ?=o
satisfy the equation
A.81.2) ~ + \xy - 0.
For negative x we have k(x) > 0, Z(x) < 0. Using A.71.9),6 we obtain for
x < 0, x —> - oo
.o 1. .O J K\JLJ —^ ^K^J ^ O 7T I X cXp |^U X /O^ |.
Thus, but for a constant factor, the function
A.81.4) A(x) = k(x) + l(x)
is the only possible particular solution of A.81.2) which remains bounded if
x —* — oo. Indeed,
.Ol.O^ /I ^1^ ^^ Z o 7T 1 X I cXp \ —Z(J X
(See Watson 3, pp. 188-190, 202.) This function A (x) is called Airy's function;
it can be considered as the standard solution of A.81.2) and plays an important
part in numerous questions in mathematical physics. The function l(x)/k(x)
is increasing (see Fig. 1) so that an arbitrary real solution of A.81.2) has at
most one negative zero and infinitely many positive zeros. In particular, A(x)
6 If i is negative, we have
k(x) = I (| x |/3)l/2e-tVV_I/3{e-tV'22(| x \/S)>'>\,
o

1.82]
THEOREMS OF STURM'S TYPE
19
has no negative zero and infinitely many positive zeros. Since A (x) > 0 for
x < 0, we see from A.81.2) that A"(x) > 0 for x < 0; therefore A'(x) -* 0
as x —* — oo.
- 1
Fig. 1
1.82. Theorems of Sturm's type
The following "comparison theorems" of Sturm's type can be proved in the
usual way (see Szego 20, pp. 3-4):
Theorem 1.82.1. Let f(x) and F(x) be functions continuous in x0 < x < Xo
with f(x) ^ F(x). Let the functions y(x) and Y(x), both not identically zero,
satisfy the differential equations
A.82.1)
V" +f(x)y = 0, Y" + F{x)Y = 0,
respectively. Let x' and x", x' < x", be two consecutive zeros of y(x). Then the
function Y(x) has at least one variation of sign in the interval x' < x < x" provided
/Or) fk F{x) in [xf, x"\.
The statement holds also for x' = x0 [y(xo + 0) = 0] if the additional condition
A.82.2)
is satisfied (similarly for x" = Xo).
lim {y'{x)Y{x) - y{x)Y'{x)\ = 0
x-»jo+0
From Theorem 1.82.1 we readily derive (loc. cit., p. 4) the following theorem:

20 PRELIMINARIES | I j
Theorem 1.82.2. Let <j>(x) be continuous and decreasing in x0 < x < Xo,
and let y be a solution of
A.82.3) y" + <t>(x)y = 0
which is not identically zero. Then x' < x" < x'" being three consecutive zeros
of y(x), we have x" — x' < x'" - x"; that is, the sequence of the zeros of y{x)
is convex.
The last inequality holds also under the following more general coiidition:
A-82.4) <t>(x) > 4>(x") > <t>(y), for x < x" < y < x'".
In addition, it holds also if x' = x0 [y(x0 + 0) = 0] provided that
A.82.5) lim (x - xo)y'(x) =" 0.
z-»zo+0
In order to prove x" - x' = h < x'" - x", we compare A.82.3) with
Y" + <t>(x - h)Y = 0,
which has the solution Y(x) = y(x — h) in the interval x" ^ x ^ x'".
Another very elementary remark of a related nature is the following:
Theorem 1.82.3. Letf(x) be continuous and negative in x0 < x < Xo. Then
an arbitrary solution y of y" + f(x)y — 0, for which y —> 0 if x —> Xo , cannot
vanish in x0 ^ x < Xo .
Suppose the contrary. Now between two consecutive zeros sgn y" = sgn y
is constant, say positive; then y is positive and convex, which is a contradiction.
A further remarkable result of Sturm's type is the following (Watson 3, p. 518,
Makai 2):
Theorem 1.82.4. Let f(x), F(x), y(x), Y(x), x0, Xo, x', x" have the same
meaning and satisfy the same conditions as in Theorem 1.82.1. We denote by ?
the first zero of Y(x) to the right of x', x' < ? < x".
Assuming that y(x) > 0; Y(x) > 0 in x' < x < ?, and
A.82.6) lim |^^1,
we have y(x) > Y(x) in x' < x < ?.
We conclude as usual that the function y'(x)Y(x) — y(x)Y'(x) is increasing
in [xf, ?]; it is zero at x = x'(Y(x') = 0), thus positive in x' < x < ?. Hence
y(x)/Y(x) is increasing. In view of A.82.6) the assertion follows.
The statement holds also for x' = xy provided the condition A.82.2) is satisfied.
We prove now the following important consequence of the last theorem
(Hartman-Wintner 1; Makai 2):

[ 1.01 ] PRINCIPLE OF ARGUMENT; ROUCHE'S THEOREM 21
Theorem 1.82.5. Let 4>{x), x', x", x'" have the same meaning as in Theorem
A.82.2): x" - x' < x'" - x". Let y(x) describe a negative "wave" in [xf, x"} and
a positive one in [x", x'"\. The first "wave" is then entirely contained in the
second one.
The meaning of the last assertion is the inequality:
0 < -yBx" - x) < y(x), x" < x < 2x" - x'.
The proof proceeds as usual, taking into account that, Y(x) = —y{2x" — x),
,. y(x) .. v'(x)
hm j~jr ; = hm ,J,) '—r = 1.
x^x» -yBx" - x) x^x» y'Bx" - x)
1.9. An elementary conformal mapping
Let the complex variables x and z be connected by the relations
A.9.1) x = \{z + z~l), z = x + (x2 - 1)*.
The exterior of the unit circle, | z \ > 1, as well as its interior, is mapped onto
the whole z-plane except the closed interval [—1, 4-1] (the so-called cut plane),
with z = qo and 2 = 0, respectively, corresponding to x = °o. If we take that
branch of x 4- (x2 — 1) which becomes infinite at x = °o; we obtain \z\ > 1;
if we take the other branch, which vanishes at x ~ «j) we obtain | z \ < 1.
The unit circle z = e%e is carried over into the closed segment — 1^x^4-1
described twice since x = cos 0.
The circle | z | = r, or | z \ — r, 0 < r ^ 1, corresponds to the ellipse with
foci at —1, 4-1 and with semi-axes
|(r + r-i)f I | r _ r-i | .
Upon replacing z by eK or by e~K, we obtain the representation
A.9.2) x = cosh f.
It maps the tialf-strip
A.9.3) 5R(f) > 0, -7T < m) ^ t
onto the same x-plane cut along [—1, 4-1] as before; now, however, the point
x = oo has to be removed.
1.91. The principle of argument; Rouche's theorem; sequences of
analytic functions
Theorem 1.91.1 (Principle of argument). Let f(x) be analytic both inside and
on a Jordan curve C, and let f(x) 5* 0 on C. Then the variation of 3{log/(x)} =
arg/(x), as x describes C in the positive sense is 2-n-im, where m is the number of
the zeros off(x) in the interior of C, counted with the proper multiplicity.

22 PRELIMINARIES | I ]
Theorem 1.91.2 (Rouche"'s theorem). Let f{x) and g(x) be analytic inside
and on C, and let | g(x) \ < \f(x) | on C. Then f(x) + g(x) and f(x) have no
zeros on C and, the same number of zeros in the interior of C.
Theorem 1.91.3 (Theorem of Hunvitz). Let {fn(x)\ be a sequence of analytic
functions regular in a region G, and let this sequence be uniformly convergent in
every closed subset of G. Suppose the analytic function \imn-.o0fn(x) = f(x) does
not vanish identically. Then if x = a is a zero of f(x) of order k, a neighborhood
\ x — a\<8ofx = a and a number N exist such that if n > N,fn(x) has exactly
k zeros in \ x — a \ < 8.
The last theorem follows immediately from Theorem 1.91.2. Concerning
the preceding theorems see Polya-Szego 1, vol. 1, pp. 120-124.
In Theorem 1.91.3, let G be symmetric relative to the real axis, and let/n(x)
be real if x is real. If x = a is a simple real zero of f(x), then for n > N each
fn(x) has exactly one real zero in | x — a | < 5. For if fn(x0) = 0, then fn(xo)
is also 0.

CHAPTER II
DEFINITION OF ORTHOGONAL POLYNOMIALS;
PRINCIPAL EXAMPLES
2.1. Orthogonality
A) In what follows a(x) is a fixed non-decreasing function which is not con-
constant in the interval a ^ x ^ b. (See the remark at the beginning of §1.4 A).)
Definition. An orthonormal set of functions <t>o(x), <t>i(x), • ¦ • ,<t>i(x), I finite
or infinite, is defined by the relations
B.1.1) (<*>„,<*>J = / <t>n(x)<t>m(x) da(x) = 8nm, n,m = 0,l,2,---,l.
Here <t>n{x) is real-valued and belongs to the class L\{a, b).
Functions of this kind are necessarily linearly independent. If a(x) has
only a finite number N of points of increase (that is, points in the neighborhood
of which a(x) is not constant), I is necessarily finite and I < N.
Theorem 2.1.1. Let the real-valued functions
B.1.2) fo{x), f\{x)t'fz{x), ¦ ¦ ¦ ,fi(x), I finite or infinite,
be of the class L\{a, b) and linearly independent. Then an orthonormal set
B.1.3) <t>»{x), <t>i(x),<t>2 (x), ••• ,<t>i(x)
exists such that, for n — 0, 1, 2, • • • , I,
B.1.4) <t>n(x) = \nofo(x) + \Mx) + • • • + \nnfn(x), Xnn > 0.
The set B.1.3) is uniquely determined.
The procedure of deriving B.1.3) from B.1.2) is called orthogonalization.
(Cf. Stone 1, pp. 12-13.)
B) For the orthonormal functions B.1.4) the,following explicit representation
holds:
B.1.5)
where, for n ^ 1,
n = 0, 1, 2,
B.1.6)
Dn(x) =
(/o,/o) (/o,/i)
(/<,,/„)
(/n-1 , /o) (/n-1 , /l)
(/n-1 , /n)
fn{x)
23

24
DEFINITION OF ORTHOGONAL POLYNOMIALS
II
and, for n ^ 0,
B.1.7)
Dn = [(/»,/„)kM=0,l,2,....n > 0.
We write Z)_i = 1 and D0(x) = fo(x). The determinant B.1.7) corresponds
to the positive definite quadratic form
+ Uifl + • • • + Unfn
luofo(x) + Ux/iCz) + ¦ ¦ • + unfn(x)}2da(x),
B.1.8)
so that Dn > 0 for each n.
Furthermore, the following integral representations can be established:
fo(xo) fi(xo) ¦ ¦• ,
m = i r r... r
nl Ja Ja Ja
B.1.9)
/iCr0)
Mx) ¦
da(xQ) da(i
xn_i), n ^ 1,
Dn =
1
(n + 1)!
B.1.10)
fl(Xo) • • • fn(xo)
(Cf. Kowalewski 1, p. 326; Polya-Szego 1, pp. 48-49, 208, problem 68.)
C) Definition. Let {<t>n(x)} be a given orthonormal set, finite or infinite.
To an arbitrary real-valued function f(x) let there correspond the formal Fourier
expansion
B.1.11) /(*) ~/otf>o(z)
The coefficients /„ , called the Fourier coefficients of f(x) with respect to the given
system, are defined by
B.1.12)
/„ = (/, <*>„) = / /(*)*„(*) da(x),
Ja
n = 0, 1, 2, • • • .
Every finite section of the series B.1.11) has the following important minimum
property:

[ 2.2 1 ORTHOGONAL POLYNOMIALS 25
Theorem 2.1.2. Let <t>Jx), /Or), /„ have the same meaning as in the previous
definition. Let I ^ 0 be a fixed integer, and a0, ai , ¦ • • , at arbitrary real con-
constants. If we write
B.1.13) g(x) =
and the coefficients a, are variable, the integral
B.1.H) I' {f(x)-g(x)\2da(x)
J
becomes a minimum if and only if a, = /„ , v = 0, 1, 2, • • • , I.
The minimum itself is
B.1.15) P [f(x)}2da(x) - Zfl = Il/H2 - tfl,
so that
B.1.16) fo + fl + ...+fl
and (Bessel's inequality)
B.1.17) fl+fl+fl+ •¦• ^ ||/||2= I \f(x)\2da(x).
J
If the left-hand side of B.1.17) is an infinite series, it is convergent. The dis-
discussion of the equality sign in B.1.17) leads to the concept of closure (§1.5).
A classical example of Fourier expansions of this kind is the ordinary Fourier
series in terms of the trigonometric functions 1, cos nx, sin nx, n = 1, 2, 3, • • • ;
-TV ^ X ^ +7T.
D) Another important characterization of the orthonormal set B.1.4) can
be based on the preceding minimum property of the partial sums. Indeed, for
variable real values of Xo, Xi, • • • , Xn-i the expression
B.1.18) || Xo/o(x) + Xi/i(x) + • • • + K-ifn-i(x) + /„(*) ||
becomes a minimum if and only if
B.1.19) Xo/o(x) + Xl/l(z) + • • • + K-lfn-l(x) + /„(*) = X^Un^).
The extension of these considerations to complex function spaces is not
difficult. The scalar product of the functions f(x) and g(x) is then defined as
in A.4.3).
2.2. Orthogonal polynomials
A) Definition. Let a(x) be a fixed non-decreasing function with infinitely
many points of increase in the finite or infinite interval [a, b], and let the "moments"
B.2.1) cn =' I xnda(x), n = 0,1,2, ••• ,

26 DEFINITION OF ORTHOGONAL POLYNOMIALS [ II ]
exist. If we orthogonalize the set of non-negative powers of x:
B.2.2) 1, x, x2, ¦ ¦ ¦ , xn, ¦ ¦ ¦ ,
in the sense explained in §2.1 (the linear independence is shown below), we
obtain a set of polynomials
B.2.3) po(x), px(x), p2(x), -.., pn(x), ¦ ¦ ¦
uniquely determined by the following conditions:
(a) pn(x) is a polynomial of precise degree n in which the coefficient of xn is
positive;
(b) the system {pn(x)\ is orthonormal, that is,
B.2.4) / pn(x)pm(x) da(x) = 8nm, n, m, = 0, 1, 2, • • • .
The existence of the moments B.2.1) is equivalent to the fact that the func-
functions xn are of the class La(a, b).
A similar definition holds in the special case of a distribution of the type
w(x) dx. Here we assume that w(x) is non-negative and measurable in
Lebesgue's sense and that Ja w(x) dx > 0. Moreover, the moments must
exist again.
We call pn(x) the orthogonal polynomials6 associated with the distributions
da(x) and w(x) dx, respectively; in the latter case we also speak of the or-
orthogonal polynomials associated with the weight function w(x). The following
chapters are devoted to the study of these polynomials. Evidently if the
distribution is of the type w(x) dx, the system
B.2.5) {[w(x)]hpn{x)}, n = 0, 1, 2, •-.,
is orthonormal in the usual sense.
The linear independence of the functions B.2.2) can readily be shown. In
fact if p{x) is an arbitrary real polynomial, the relation
||p ||2 = f {p(x)}2da(x) = 0
is possible only if p(x) vanishes at all points of increase of a(x). Since there
are infinitely many such points, p(x) must vanish identically.
If a(x) has only a finite number, say N, of points of increase, the functions
1, x, x2, ¦ ¦ ¦ , xN~l are still linearly independent. Through orthogonalization
we obtain a finite system of polynomials- {pn(x)\, n = 0, 1, 2, ¦ ¦ • , N — 1,
satisfying similar conditions as required in the previous Definition. See §2.8
and §2.82.
B) Using the general formulas B.1.5) to B.1.8), we obtain, for n ^ 1,
6 Sometimes these are called Tchebichef polynomials. We shall reserve this terminology
for the important special cases A.12.3).

2.2
ORTHOGONAL POLYNOMIALS
27
B.2.6)
Co
C3
Cn
Cn+l
n—1 Cn
1 x x •¦' xn
where for n ^ 0
B.2.7) Dn = [c+M](,,M_0.i,2,...,n > 0.
In addition to B.2.6) we have po(x) = D~q> = Co*. The determinant B.2.7) is
associated with the positive definite quadratic form
B.2.8)
rb
Ja
+ Mia; + z^x2 + • • • + unxnJda(x),
which is called a form of Hankel or of recurrent type. (See Szego 1.)
The determinant in B.2.6) can be transformed by multiplying the next to the
last column by x, subtracting it from the last column, and repeating this opera-
operation for each of the preceding columns. In this way we obtain, n ^ 1,
B.2.9) Pn(x) = {Dn-lDn)~h
C\X —
c\x — a
dx — Cz
Cn-lX — Cn
CnX — Cn+l
Cn—1 X Cn Cn X Cn-f-1
n-2^ — C2»-l
Furthermore, according to B.1.9) and B.1.10), we have the following integral
representations:
<"b fb fb
••• / (X - Xo) (X ~ Xi) • • • (X - Zn_l)
Ja
B.2.10) " » '
{Dn-\Dn)~h fb
1% I
n (x,-
!,!,•• -,n —1
• • da(xn-i),
and
B.2.11)
i rb rb rb
(n + 1I Ja Ja Ja y,^0,l
n + 1
•da{x0) da(xi) • • • da(xn).
For B.2.10) and B.2.11) see, for example, Heine 3, vol. 1, p. 288. Formulas
B.2.6), B.2.9), B.2.10) are not suitable in general for derivation of properties
of the polynomials in question. To this end we shall generally prefer the
orthogonality property itself, or other representations derived by means of the
orthogonality property.

28 DEFINITION OF ORTHOGONAL POLYNOMIALS [II]
C) The Fourier expansion of an arbitrary function f(x) in terms of the
polynomials {pn(x)\ has the form
B.2.12) f(x) ~/opo(z) + fiPi(x) + • ¦ • + fnpn(x) + ...
with
B.2.13) /„ = f f(x)pn(x) da(x), n = 0, 1, 2, ....
The partial sums have the minimum property formulated in Theorem 2.1.2.
We notice as an important special case the following direct characterization of
the orthogonal polynomials (see §2.1 D)). Considering the set of all poly-
polynomials p(x) of degree n with the coefficient of x11 unity, we find that the integral
B.2.14) J* {p(x)}2da(x)
becomes a minimum if and only if p(x) = const. pn(x). Here the constant
factor is determined by normalizing p(x). If kn denotes the highest coefficient
of pn(x), the minimum is obviously k~2. From B.2.8) we find for this minimum
the value Z)n/Z)n-i, so that
B.2.15) kn = (Z
which also follows directly from B.2.6).
2.3. Further remarks
A) The restriction (a) in the definition in §2.2 A) concerning the highest
coefficient, and the restriction (b) concerning the integral of the square, is
only one of various possible ways of normalizing the polynomials in question.
Sometimes other kinds of normalization are appropriate, such as fixing the
value of pn(x) at x = a or at x = b,7 or fixing the highest coefficient of pn(x),
and so on. Since pn(x) has the precise degree n, every irn can be represented
as a linear combination of po(x), pi(x), • • • , pn(x) (see §1.12). Therefore
Pn(x), n ^ 1, is orthogonal to any 7rn_i. In particular,
B.3.1) / pn(x)xv da(x) = 0, v = 0, 1, 2, • • • , n - 1.
This condition determines pn(x) save for a constant factor. Frequently, this
wider formulation of the orthogonality property is used. Observe also that
if p(x) is a 7rn and
B.3.2) f pn{x)p{x) da(x) = c,
then the coefficient of xn in p(x) is ckn .
7 We have pB(a) ^ 0, pn(b) ^ 0 (see §3.3).

[ 2.5 ] A FORMULA OF CHRISTOFFEL 29
B) Let [a, b] be an interval symmetric with respect to the origin, that is,
a = —b, and let us consider a distribution of the type w(x) dx with an even
weight function, that is, w(-x) = w(x). Then pn(x) is an even or an odd
polynomial according as n is even or odd:
B.3.3) pn(-x) = {-l)npn(x).
It can contain only those powers of x which are congruent to n (mod 2). In-
Indeed, we have for v = 0, 1, 2, • • • , n — 1
/ pn(-l)x'w(x)dx = (-1)" / pn{x)x"W{x) dX = 0.
J-a J-a
Consequently, pn( — x) possesses the same orthogonality property as pn(x) (in
the wider sense). Therefore, comparing the coefficients of xn, we obtain
Pn(-X) = Const. pn(x) = (-l)"pn(x).
The linear transformation x = kx' + I, k ^ 0, carries over the interval
[a, b] into an interval [a', b'] (or [br, a']), and the weight function w(x) into
w(kx' + I). Then the polynomials
B.3.4) (sgn k)n\k\* pn(kx' + 0
are orthonormal on [a', b'] (or [b', a']) with the weight function w(kx' -f- I).
2.4. The classical orthogonal polynomials
1. Let a = -1, 6 = +1, w(x) = A - x)"(l + xf, a > -1, /3 > -1.
Then, except for a constant factor, the orthogonal polynomial pn(x) is the
Jacobi polynomial P'f'^ix) (see §4.1).
2. Let a = 0, b = +°°, w(x) = e~zxa, a > — 1. In this case pn(x) is,
except for a constant factor, the Laguerre polynomial L(na)(x) (see §5.1).
3. Let a=—oo)fe = +oo> w(x) = e"* . In this case pn(x) is, save for a
constant factor, the Hermite polynomial Hn(x) (see §5.5).
Some special cases of 1, except for constant factors, are:
The ultraspherical polynomials,, for a = /3.
The Tchebichef polynomials of the first kind, Tn(x) = cos nd, x = cos d,
for a = /3 = -? (see A.12.3)).
The Tchebichef polynomials of the second kind, Un(x) ~ sin.(n + 1H/(sin 6),
x = cos d, for a = j8 = +^ (see A.12.3)).
The polynomials ?/2n (cos @/2)) = sin (n + ^H/sin @/2) of cos d = x, for
a = -0 = * (see §1.12).
The Legendre polynomials Pn(x), for a = /3 = 0.
A detailed investigation of these polynomials will be given in later chapters.
2.5. A formula of Christoffel
A) Theorem 2.5. Let \pn(x)\ be the orthonormal polynomials associated
with the distribution da(x) on the interval [a, b]. Also let

30 DEFINITION OF ORTHOGONAL POLYNOMIALS [ II ]
B.5.1) p(x) = c(x - x,)(x - x2) ¦ ¦ ¦ (x - x,), c 9* 0,
be a 7T( which is non-negative in this interval. Then the orthogonal polynomials
\qn(x)}, associated with the distribution p(x) da(x), can be represented in terms
of the polynomials pn(x) as follows:
B.5.2) P(x)qn(x) =
Pn+l(x)
Pn(.Xi)
In case of a zero Xk, of multiplicity m, m > 1, we replace the corresponding
rows of B.5.2) by the derivatives of order 0, 1, 2, • • • , m — 1 of the polynomials
Pn(x), Pn+l(x), • • • , pn+l(x) at X = Xk .
This important result is due toChristoffel (see 1, actually only in the special
case a(x) = x). The polynomials qn(x) are in general not normalized.
The proof is almost obvious. The right-hand member of B.5.2) is a 7rn+i
which is evidently divisible by p(x). Hence it has the form p(x)qn(x), where
qn(x) is a 7rn . Moreover, it is a linear combination of the polynomials pn(x),
pn+i(x), ¦ ¦ ¦ , pn+i(x), so that if q(x) is an arbitrary irn-i, then
B.5.3) / p(x)qn(x)q(x) da(x) = f qn(x)q(x)p(x) da(x) = 0.
J a Ja
Finally, the right side of B.5.2) is not identically zero. To show this, it suffices
to prove that the coefficient of pn+i(x), that is, the determinant \pn+v{x^i)],
v, fi = 0, 1, 2, • • • , I — 1, does not vanish. Suppose it to vanish; then certain
real constants Xo, Xi, ^2, ¦ ¦ ¦ , Xi-i exist, not all zero, such that
B.5.4) XoPn(z) + Xipn+i(:r) + • • • + Xi
vanishes for x = X\, x2, • • • , xt. Hence B.5.4) is of the form p(x)G(x),
where G(x) is a, irn^i . Since B.5.4) is orthogonal to an arbitrary irn_i , we have
the relation
P(x)G(x).G(x)da(x) = 0;
whence G{x) = 0, a contradiction.
B) The representation B.5.2) enables us, for instance, to reduce ultra-
spherical polynomials with a = /3 = an integer, or with a-\-\ = fi-\-\ =
an integer, to Legendre and Tchebichef polynomials, respectively [cf. §4.21 C)].
Another illustration may be obtained in connection with the polynomials con-
considered in §2.6.
By using some special properties of da(x) or of p(x), formula B.5.2) can be
simplified. For example, let da{x) = w{x) dx, w{x) and p(x) be even func-
functions, and a = —b. Then, instead of B.5.2), we have the representation (I even)

2.6
A CLASS OF POLYNOMIALS
31
Pn+i(x)
p(x)qn(x) =
where {±xi, ±x2, • • • , ±xy2} is the total set of zeros of p(x). For instance,
the orthogonal polynomials qn(x) associated with the weight function 1 — x2
in [—1, +1] can be determined from
A - x2)qn(x) =
Pn{x)
Pn(l)
Pn+2{X)
Pn+2A)
= Pn(x) - Pn+2(X).
(Cf. D.7.27), X = i)
2.6. A class of polynomials considered by S. Bernstein and G. Szego
Let p(x) be a polynomial of precise degree I and positive in [ — 1, +1]. Then
the orthonormal polynomials pn{x), which are associated with the weight func-
functions
'a - x2rHP(x)}-\
B.6.1)
w{x) =
c
1-1
can be calculated explicitly provided I < In in the first case, I < 2(n + 1) in
the second, and I < In + 1 in the third. The polynomials of the first case
play an important role in the proof of Szego's equiconvergence theorem (9; cf.
Theorem 13.1.2). All three cases were later investigated by S. Bernstein C)
in connection with his asymptotic formula B; cf. Theorem 12.1.4).
Theorem 2.6. Let p{x) be a m of precise degree I and positive in [ — 1, +1].
Let p(cos 0) = 1 h{el9) B be the normalized representation of p (cos d) in the sense
of Theorem 1.2.2. Writing h(eie) = c{6) + is{6), c{6) and s{6) real, we have the
following formulas:
< 2n;
pn(cosd) =
B.6.2) = B/7r)i{c@) cos nd + s(d) sin nd\,
w(x) = A -s2r'{pC«0
r\
pn(cos0) =
B.6.3)
sin
sin"
W\X) = 11 —
I <2(n + 1);

32 DEFINITION OF ORTHOGONAL POLYNOMIALS [ II
pn(cos 0) = tT
- * L<n\ sin ^n + ^6 e(a\ cos
B.6.4) " * \C-W " sin @/2) " "W l
w(x) = (j^) {p(x)}-\ K2n+1.
These formulas must be modified for I — 2n, I = 2(n + 1), and I = 2n + 1,
respectively, by multiplying the right-hand member of B.6.2) by A + hi/ho) ,
and those of B.6.3) and B.6.4) by A — hi/ho) , where ho = h@) and hi is the
coefficient of zl in h(z).
First we observe that the right-hand members of B.6.2), B.6.3), B.6.4) are
cosine polynomials with the highest terms
B.6.5) B/tt) /io cos nd, B/tt) /i0 -—:— , it ho
sin 6
sin0 ' sin @/2) '
respectively. In the first of these expressions, if I = 2n > 0, h0 must be re-
replaced by ho + hi ; in the second and last, if I — 2(n + 1) and I = 2n + 1,
respectively, we have ho — hi in place of ho.
We give the proof of B.6.2). First we show that
r+i
\ pn{x)x"{\ — x ) {p(x)} * dx = 0, v = 0, 1, • • • , n — 1,
J-i
or, what amounts to the same thing,
Now,
B/tt)
2
since
J.
l
_B
the
Pn (COS
f f ine -
{Jo
/tt)* / /
4 V-
functidn
e)
id
l(C
(z
cos vd\
- M
re.(n+r)«
h
n+r + z
P (cos
+ eiin~
(eie)
n~"){zh
"^ dd
dd
,«)
}-
lis
= 0,
2 X
B/7r)J^/l f
4 it ir-
irregular for 1 z 1
0,1,
zn+"
1 Z
^ 1.
2, ••• ,n - 1.
+ z"~" ^^ _ n
h ( \ I '
Furthermore,
= j pn(cos0)B/7r)i/iocosn0{p(cos0)}~1d0
Jo
\ B/)UB/)J dt \\ f ^ dz) =
\ f
J\z\-i
= \ B/7r)U0B/7r) dt \\ f ^ dz) = ]
4 U J\z\-i zh{z) j 4
The proofs of B.6.3) and B.6.4) are similar. In place of cos vd we use
sin (v + lH/sin 0 and sin {v + |H/sin @/2), respectively. The modifications

[2.8] STIELTJES TYPE; ANALOGUE OF LEGENDRE POLYNOMIALS 33
necessary for I = 2n, I = 2{n + 1), and I = 2n + 1, in B.6.2), B.6.3), and
B.6.4), respectively, are also obvious. Finally, we notice that B.6.2) arises
from B.6.4), B.6.4) from B.6.3), and B.6.2) from B.6.3) by replacing P(x)
by A - x)p(x), A + x)p{x), and A - x2)p(x), respectively.
2.7. Stieltjes-Wigert polynomials
Wigert B, p. 7; also Stieltjes 11, pp. 507-508) found a very elegant explicit
representation for the orthonormal polynomials pn(x) associated with the
weight function
B.7.1) w(x) = ir~hk exp (-A;2 log2 x) = ^kx'"' logx, 0<x<+oo;k>0.
Using the notation (cf. Gauss 1, p. 16)
—^^
where
B.7.3) q = exp {-BA;2)-1},
we have
B.7.4) pn(x) = (- l)V/i+M(l -?)A -9*) ¦•¦ A -?B)}"*
For n = 0 the product in the braces must be replaced by 1.
The proof can be based on the identity of Gauss:
B.7.5) E H q^+l)nu* = A + qu){\ + q\) • • • A + qnu).
See Szego 12, where' other similar polynomials (related to the theory of theta
functions) are also considered. Also see Hahn 5.
2.8. Distributions of Stieltjes type; an analogue of Legendre polynomials
Tchebichef D) investigated a remarkable finite set of orthogonal polynomials
associated with the distribution da(x) of Stieltjes type, where a(x) is a step
function with jumps of one unit at the points x = 0, 1, 2, • • • , N — 1 (N is a
fixed positive integer). This is a distribution of the type mentioned at the
end of §2.2 A). The associated polynomials are, except for constant factors
(see B.8.3)),
B.8.1) Ux) = n! AnQ(X ~ *) , n = 0, 1, 2, • • • , N - 1.
Indeed, Tchebichef shows D, pp. 547, 552; see also A. Markoff 1, pp. 21 -22) that
/ + OO
tn{x)tm{x) da(x) = Z Ux)tm{x) =0, if n ^ r»,
oo x~*0, 1. 2, •• -,JV-1

34 DEFINITION OF ORTHOGONAL POLYNOMIALS | II ]
and
r°V(x)}ada(x) = Z {tn(x)}2
7-oo z=0. 1,2, •••,Af-l
B.8.3) = N(N2 - 12)(N2 -22) • • • (JV2 - n2)
2n + 1 '
n, m = 0, 1, 2, ••• ,N - 1.
These formulas hold for all non-negative values of n and m, but they are trivial
for n ^ N or m ^ iV, since in(z) = 0 for a; = 0, 1, 2. • • • , N - 1, if n ^ iV.
In B.8.1) we used the symbols
= /(x + 1) - /(x),
B.8.4) An/(x) = A{An-V(x)}
= f{x + n) - Q/(x + n - 1) + ¦ ¦ ¦ + (-1O(*).
By the mean-value theorem
B.8.5) An/(x) = fn\x + 6n), 0 < 0 < 1,
(see, for example, Polya-Szego 1, vol. 2, pp. 55, 241, problem 98), we obtain
for a fixed value of n the remarkable formula
B.8.6) lira N~ntn(Nx) = PnBx - 1),
JV-»oo
where Pn(x) is the Legendre polynomial of degree n (see §4.1 C)). The repre-
representation B.8.1) is the "difference" analogue of D.3.1), a = /3 = 0. The
proofs of B.8.2) and B.8.3) are analogous to those in §4.3.
Tchebichef also considers A, 2) the more general case in which the points
0, 1, 2, • • • , N — 1 are replaced by an arbitrary set of N distinct points. In
this connection he obtains an interpolation formula having a certain significance
in mathematical statistics. (See Jordan 1.)
2.81. Poisson-Charlier polynomials
These polynomials have become important in some recent investigations con-
connected with the calculus of probability and statistics (see Doetsch 2 and the
literature quoted in E. Schmidt 1, also Meixner 1, 2). They belong to the
distribution da(x) where a(x) is a step function with the jump
B.81.1) j(x) = e~aax{x^~l at the point x, x = 0, 1, 2, • • • ; a > 0.
Obviously, the total variation of a(x) is
«(+«) -a(-oo) = f)j(x) = 1.
z=0
The corresponding orthonormal polynomials are:

[2.82] KRAWTCHOUK'S POLYNOMIALS 35
t s pn(x) = a'\n^ ± (- l)-fn) A a~{X)
B.81.2) r-o W W
= an/Vr*(-l)"{j(z)TA'jU - n).
A simple proof of B.81.2) can be given by means of the method of generating-
functions (see §4.4; cf. Doetsch 2, p. 260, and Meixner 1, 2). Let, for a suf-
sufficiently small | u> | >
G(x, w) = ± a-nl\n\yiPn(x)wn =
P ±± ^ (Yi
n=0 n=0 i—O n! \V/ \V
w
; e = e A + a 1
„=o w
Then
23 j(x)G(x, u)G{x', v)
z-0,1.2,---
B.81.4) ~
z=0,1.2.---
—a—u—v a(l+a~1u)(l+a~1v) a~^uv
so that
B.81.5) zl j(x)aTnl2(nty pn(x)a~m/2(m!)~*pm(a;) = a
z=0,1.2,---
n, m = 0, 1, 2, • • •.
The polynomials B.81.2) are connected with Laguerre polynomials (§5.1)
by the relation
B.81.6) pn(x) = o~n/a(n!)*Lir~n)(a).
Concerning the expansion problem associated with Charlier-Poisson poly-
polynomials, we refer to E. Schmidt 1.
2.82. Krawtchouk's polynomials
Considerations in the calculus of probability lead also to the following dis-
distribution da(x).
Let a(x) be a step function with the jump, at the point x, of
B.82.1) j(x) = (^j pxqN~x, x = 0, 1, 2, ¦ ¦ ¦ , N.
Here N is a positive integer, p > 0, q > 0, and p + q = 1.
See Krawtchouk 1. The total variation of a(x) is 1. The associated set
of orthogonal polynomials is again finite as in §2.8.

36 DEFINITION OF ORTHOGONAL POLYNOMIALS [ U
A) The method of generating functions yields the formula
B.82.2) pn(x) = <(N)\ (pq)~nl2 E (-:
[\n/j r—0
n = 0, 1, 2, • • • , JV.
Indeed, let
K(x, w) = E
B.82.3)
In case a; is an integer, 0 ^ x ^ JV, the last summation can be extended over
all values of v, n = 0, 1, 2, • • • ; v ^ n, since the general term vanishes if
N — x < n — v or \{ x < v. (From JV — x 5; n — v, x ^ v, we have JV 5; n.)
Thus
TTf -\ V ^ ' ' V ( I^- /^ - A n-v n-v
iv(a;, w) = 2^i [ I q w JLi {— V { JP w
i-o V/ n-» \n — v/ ^
B.82.4)
) A - pw) ,
from which
E i(x)X(x, w)iC(x, w)
1=0.1.2. •¦•.A'
B.82.5) = J2 (N) pxqN~x{\ + qu)'(l - pu)N~x{\ + qv)'(\ - pv)N~x
so that in fact
B.82.6) «-o.i.2.-".w
??,, w = 0, 1, 2, • • • , JV.
If n > JV, obviously pB(x) = 0 for x = 0, 1, 2, • • • , JV.
B) Two other classes of polynomials can be derived from the polynomials
B.82.2) by two different limiting processes.
(a) Let z be real and let x denote the greatest integer less than or equal
to pN + z{2pqN)i where p, q, z are fixed, and JV —¦ <x>. Then for a fixed n
B.82.7) lira pB(s) = Bn?i\yiHn(z)
N-"»
if Hn(z) denotes the nth Hermite polynomial (§5.5). This follows readily from
B.82.3) and B.82.4), since for x an integer, 0 < x < N,

[2.91 FURTIIKIt SPECIAL CASKS 37
lim Z i(N))\pq)nliPn(x)WN)iw}n
W — oo n=0 lAn/J
= lim {1 + B/N)iqiv\x{l - B/N)h pw\N~x
= lim exp \B/N)hqwx ~ (VN)lpw(N - x) - N "V/V-c - N~lp2w2(N - x)\
= exp [2z{pq)hw - pqw2} = ? Il^l \{vq)
n=0 Tl\
)\v\n,
n=0
s
s
(Use Titchmarsh 1, p. 95; see E.5,7).) It is instructive to observe that the
same limiting process applied to the given distribution rfa(x) leads to the distri-
distribution e~z dz of the Hermite polynomials; more precisely,
B.82.8) j{x) ^ {2TtVqN)~he~z\ or j(x) dx ^ tt^** dz.
(b) Let pN = a, where a is a fixed positive number, N —> &, p —-> 0, q —> 1,
Then for a fixed n and a fixed integer x 2: 0, we find that lim.v_» pn(x) exists
and is identical with the Poisson-Charlier polynomial B.81.2). In fact (B.81.3))
lim A + qw)x(l - pxv)"-r = lim A + qwY(i - y>i*;)~'(l - p^O" '"
B.82.9) N-x p~*°
2.9. Further special cases
Concerning other distributions da Or) of Stieltjes type, see A. Markoff 1, pp.
7-18; Stieltjes 11, pp. 546 555; and Gottlieb 1.
Markoff considers the ease for which a(x) is a step function with the jump,
at the point qx, of j(x) — qx, x = 0, 1, 2, • • • , N — 1, and q > 0, q ?* 1. Thi
distribution is very much similar to that in §2.8, Analogues of B.8,1) and
B.8.6) hold.
Stieltjes and Gottlieb investigate the case for which a Or) is a step function
with the jump qT at the point x, x = 0, 1,2, • • • , 0 < 7 < 1.
In addition to these "discrete" distributions as well as to those studied in
§2.8 and §2.82, see Karlin-McGregor 2, Eagleson 1, and Gasper 6, 7.
A remarkable distribution can be defined by the weight function
B.9.1) w(x) = {x(a - z)(/3 - x)}~% 0 < x S a, a < /3.
Heine C, vol. 1, pp. 294-296) derives a second order linear differential equation
for the associated orthogonal polynomials which are related to the Jacobian
elliptic functions. Recently Achieser A) investigated the orthogonal poly-
polynomials associated with the weight function
({A-x2)(a-x)F-x)}"J|c-x|, -l?x?a, b^x^+l,
B.9.2) w(x) = {
[0, a < x < b,
where —I<a<6<+1 and c depends in a proper way on n and b. These
polynomials are also related to the elliptic functions.
In some cases the condition of the positiveness of the weight function can be
removed to a certain extent. (Cf. Szego 19,)
Concerning the polynomials of Polluczek A-4), see Appendix.

CHAPTER III
GENERAL PROPERTIES OF ORTHOGONAL POLYNOMIALS
In this chapter we shall deal with properties of orthogonal polynomials
which hold for distributions restricted only by certain conditions of integrability.
Usually, we shall consider distributions of the Stieltjes type daix), but at times
we shall be concerned with distributions of the special type wix) dx. However,
aix) and w(x) will always be taken subject to the conditions formulated in
§2-2 A).
3.1. Extremum properties; closure
A) Let fix) be a given function of the class L2a(a, b), and let xn belong to
Laia, 6) for ?t = 0, 1, 2, .... Then it is evident that the integrals
/& rb
\fix) |2 daix), I fix)xn daix), n = 0, 1, 2, • • • ,
exist in the Stieltjes-Lebesgue sense. Next, denoting by [pnix)\ the ortho-
normal set of polynomials associated with the distribution da(x) in [a, b], we
state the following theorem:
Theorem 3.1.1. The weighted quadratic deviation
C.1.2) jf |/(x) - pix)\2daix),
where pix) ranges over the set of all irn , becomes a minimum if and only if pix)
is the nth partial sum of the Fourier expansion
Six) ~foPt)ix) + fptix) + f2piix) + • • • + fnPnix) + • • • ,
C.1.3) /•«-
Sn = / f(x)pn(x) daix), n = 0,1,2, ...
See Theorem 2,1.2 and §2.2 C). The minimum itself is
C.1.4) !\fix)\2daix) - El/, I2-
Ja c=0
This implies Bessel's inequality, that is,
C.1.5) \j
B) On replacing n by n — 1 and taking fix) = xn, we obtain the following
direct characterization of pn(x):
38

[3.1 ] EXTREMUM PROPERTIES; CLOSURE 39
Theorem 3.1.2. The integral
C.1.6) f \p(x)\2da(x),
J
where p(x) ranges over the set of all irn with the highest term xn, becomes a minimum
if and only if p(x) = const. pn(x).
See §2.2 C). If kn is the coefficient of x" in pn(x), the minimum of C.1.6)
is attained for p(x) = KZlpn(x).
C) Theorem 3.1.3. Let xo be an arbitrary complex constant, p{x) an arbitrary
¦Kn with complex coefficients, normalized by the condition
C.1.7) / \p(x) \2da(x) = 1.
The maximum of \ p(x0) \ is given by the polynomials
C.1.8) p(x) = e{Kn(x0, Xo)\-iKn(xo, x), \ e \ = 1,
where
Kn(x0, x) = po(xo)po(x) + pi.(a:o)pi(x) + • • • + pn(x0)pn(x).
The maximum itself is Kn{xa, Xo).
If we write p{x) = Aopo(x) + \iPi(x) + • • • + \npn(x), condition C.1.7)
becomes | Xo |2 + | Ai |2 + • • • + | An |2 = 1, and by Cauchy's inequality it
follows that
n
2
C.1.10) | p(*o) P ^ Z) I A, I' Z) I P,(*o) P = ^nUo,
c=0 c=0
The latter bound is attained for A, = Xp»(xo), where A is to be determined
according to the condition
Thus the statement is established.
The "kernel polynomials'' Kn(x0 , x) = Kn(x} x0) = Kn(x, x0) can be used
for the representation of the nth partial sum sn(x) of the Fourier expansion
C,1.3) in the form of an integral. In fact we have
C.1.11) =<Tpv(x) ["f(t)p,(t) da(t) = fbf(t)Kn(t,x)da(t).
c=0 Ja Ja
As a consequence of C.1.11) we obtain

40 GENERAL PROPERTIES OF ORTHOGONAL POLYNOMIALS [ III 1
C.1.12) / Kn(t, x)p{t) da(t) = p(x),
Ja
where p(x) is an arbitrary irn . We may easily show that this is a characteristic
property of Kn(t, x) as a ?rn in I. As a further consequence we notice the
following theorem:
Theorem 3.1.4. Let a and xo be finite, xo ^ a. Then the polynomials
{Kn(xo, x)} are orthogonal with respect to the distribution (x — xo) da(x).
This follows immediately from C.1.12) by writing x = x0, p(t) = (t — xo)r(i),
where r(t) is an arbitrary 7rn_i. A similar result holds if b is finite.
D) According to the previous results the expression C.1.4) decreases as n
increases, and consequently it tends to a non-negative limit as n —> °o. We
have Parseval's formula
... = f
Jo-
C.1.13) |/0|2+ |/1|2+ |/2|2+ ¦•¦ + |/n|2+ ... = f |/(x) |2 da(x)
J
when and only when this limit is zero.
The validity of C.1.13) is evidently equivalent to the fact that the integral
C.1.2) can be made arbitrarily small by a proper choice of the polynomial p(x).
This is, however, the same as the closure in L2a(a, b) of the system {pn(x)\ or
of the system {xn\ (see the definition in §1.5 A)), Thus, according to Theorem
1.5.2 we have the following:
Theorem 3.1.5. The set of the orthogonal polynomials {pn(x)\,n = 0, 1,2, • ¦ • ,
associated with the distribution da(x) on a finite interval [a, b], is closed in L2a(a, b).
More generally it is closed in Lpa(a, b), p ^ 1.
For a function f(x) of the class L2a(a, b) Parseval's formula C.1.13) holds.
A function f(x) of the class La(a, b), for which /„ = 0, n = 0, 1, 2, • • • , is
necessarily a zero-function.
The finiteness of tho interval considered is an essential restriction. Some
cases of infinite intervals will be studied later. (See §5.7.)
The assumption /„ = 0 in the last part of Theorem 3.1.5 is equivalent to
the fact that
C.1.14) / f(x)xn da(x) = 0, n = 0, 1, 2, ••..
The discussion of this condition is closely connected with the uniqueness of
Stieltjes' problem of moments. An example showing that Theorem 3.1.5 does
not hold generally in case of an infinite interval is the following:
C.1.15) da(x) = exp (-x" cos nit) dx, f(x) = sin (a/1 sin hit), 0 < m < 1/2.
Here C.1.14) is satisfied (cf. Polya-Szego 1, vol. 1, pp. 114, 285, 286, problem
153), and yet/(x) is not a zero-function. In the same case, if p(x) is an arbitrary
polynomial,

[3.11] GENERALIZATIONS 41
A = f lf(x)\2da(x) = [ f(x) {f(x) - p{x))da{x)
Jo Jo
which shows that the integrals
/»«
|/(x) - p(x)\da(x) and |/(x) - P(x) |2 da(x)
Jo
cannot be made arbitrarily small.
3.11. Generalizations
Numerous analogous problems arise if the weighted quadratic deviation
C.1.2) is replaced by other types of deviations. The most interesting cases are
C.11.1) f \f(x) - P(x)\pda(x),
Jo-
p being a fixed positive number, and the limiting case p —> °o8 (called also the
"Tchebichef deviation"):
C.11.2) max {|/(x) - p(x)\w(x)}.
In the last case we assume that f(x) and w(x) are continuous. Similarly, the
integral C.1.6) might be replaced by the expression
C.11.3) f |.P(x)fda(x),
Jo.
or by
C.11.4) ¦ max {| p(x) \ w(x)}.
The polynomials of fixed degree which minimize C.11.1) and C.11.2) represent
a generalization of the nth partial sum of the expansion of f(x) in terms of the
orthogonal polynomials associated with da(x) or w(x) dx. The polynomials
of fixed degree, and with highest coefficient unity, which minimize C.11.3)
and C.11.4) represent a generalization of the orthogonal polynomials them-
themselves.
Since the number of investigations which can be classified under this general
point of view is very considerable, only the most important aspects can be
indicated here.
A) For p = 2 the polynomials minimizing C.11.1) are the partial sums of the
8 Replacing da(x) by {w(x)}p dx, we have (a and b finite, f(x) and w(x) continuous)
lim / | /(*) - P(x) j» {w(x) \* dx\ = max {| /(*) - P(x) | w(x)}.

42 GENERAL PROPERTIES OF ORTHOGONAL POLYNOMIALS [ III ]
expansion of a given function/(x) in a series of orthogonal polynomials (Theorem
3.1.1). In case a = — 1, b = +1, w(x) = A — z2)~*, we obtain the expansion
of f(x) = / (cos 0) in a cosine series; in case a = — 1, b = +1, w>(z) = 1, we
obtain the Legendre series. (Cf. Chapters IX and XIII.)
B) In the general case C.11.1) the existence and uniqueness of the minimizing
polynomials have been investigated. See Jackson 1, 2, 3; Shohat 1, pp. 509-513,
4, pp. 160-161. Both authors consider only distributions of the type w(x) dx.
For general distributions see Tamarkin 1, p. 118.
C) Let a, b be finite, da(x) = dx, and p = 1. For problems C.11.1) and
C.11.3) see S. Bernstein 2, in particular pp. 135-137, where references are also
given to the earlier literature. (Cf. also Geronimus 5.) Recently Achyeser B)
discussed the problem of minimizing
C.11.5)
/ | p(x) \dx + I | p(x) | dx,
7p Jr
where [p, q] and [r, s] are given disjoint finite intervals and p(x) ranges over
all ¦Kn with the highest term xn. The minimizing polynomials can be repre-
represented in terms of elliptic functions.
D) In the case where a and b are finite, f(x) continuous, and w(x) = 1, the
minimum problem corresponding to C.11.2) leads to the closest approximation
of continuous functions by polynomials. The connection between the closeness
of this approximation (as n —> oo) and the continuity properties of f{x) has been
investigated in great detail. (See Jackson 4.)
E) If a, b are finite,/(x) continuous, and da{x) = dx, the minimizing polyno-
polynomials of C.11.1) (for a fixed n) tend to the minimizing polynomial of C.11.2) as
p —> oo. (We have existence and uniqueness in both cases.) See P61ya 2;
also Shohat 1, pp. 513-514, 4, p. 171. Both authors consider only distributions
of the form w(x) dx. For general distributions, see Tamarkin 1, p. 125.
F) If a = —1, b = +1, w(x) = 1, then problem C.11.4) has the solution
p(x) = 2[~~nTn(x) (see the notation in A.12.3)). This is a classical result due to
Tchebichef and is the starting point of various investigations of the highest
interest. (Cf. S. Bernstein 1.)
3.2. Recurrence formula; Christoffel-Darboux formula
A) Theorem 3.2.1. The following relation holds for any three consecutive
orthogonal polynomials:
C.2.1) Pn(x) = (AnX + Bn)Pn-l(x) ~ CnPn-*{x), U = 2, 3, 4, • • • .
Here An , Bn , and Cn are constants, An > 0 and Cn > 0. // the highest coeffi-
coefficient of pn(x) is denoted by kn , we have
h A kn kn—2
C.2.2) An - r— , Cn - , 2
/fcn_l /ln-1 "T—1
For the proof, we first determine An so that pn(x) — Anxpn-i{x) is a 7rn_i .

3.2 ] RECURRENCE FORMULA 43
This can be represented as a linear combination \opo(x) + XiPi(x) + • • • +
Xn_ipn_i(z), and because of the orthogonality it is readily seen that Xv = 0
if v < n - 2. Therefore C.2.1) follows. The first part of C.2.2) is a conse-
consequence of C.2.1); the second part follows from
/b rb
Pn{x)pn^{x) da{x) = 0 = An I XPn-l(x)pn-2(x) da(x) - Cn,
I
since the integral of the right-hand member is equal to
/ pn^(x)(kn-2Xn~l + •••)<*«(*) = f^ / {pn-l(x)\2da(x).
Ja kn-l Ja
'The recurrence formula C.2.1) is valid also for n = 1 if we write p~i(x) = 0,
with the understanding that Ci is arbitrary. The first formula in C.2.2) then
holds for n = 1.
Concerning a converse of Theorem 3.2.1, see Favard 1.
B) Theorem 3.2.2. We have
Po(x)po(y) + Pxix)pi(y) + • • • + Pn(x)pn(y)
=
kn+i x — y
For the special case da{x) = dx, see Christoffel 1; see also Darboux 1. This
important identity can be easily derived from the recurrence formula. For
we have
pn+i{x)pn{y) - pn(x)pn+i(y)
= {(An+1x + Bn+l)pn(x) - Cn+ipn^(x)}pn(y)
- pn(x){(An+iy + Bn+l)pn(y) - Cn
= An+i(x — y)pn(x)pn(y) + Cn+i{pn(x)pn^1(y) -
By C.2.2), this becomes
kn_ Pn+\(x)pn(y) — pn(x)pn+i(y)
kn+i x - y
— Pn\x)pn\y)
n\y) + 5j
kn x — y
which holds also for n = 0, with the understanding that k-i is arbitrary. On
replacing n by 0, 1,2, • • • , n and adding, we obtain C.2.3).
We notice the special case x = y:
iPo(x)}2+ [pi(x)\2+ ••¦ + {pn(x)\2
C-2-4) fcn , , , , , , >, , I m
= \Pn+l(x)pn(x) — pn{x)pn+l(x)\.

44 GENERAL PROPERTIES OF ORTHOGONAL POLYNOMIALS [ III ]
C) The left-hand member of C.2.3) is identical with the "kernel" Kn(x, y) =
Kn(y, x) introduced in C.1.9). Using C.1.12), we may derive C.2.3) in a
different way by showing that the right-hand member of C.2.3) (replacing y
by t) satisfies C.1.12). For we have
K+l Ja
p(t)da(t)
kn+l Ja X - t
= ^ f \pn+i{x)pn{t) - pn(x)pn+1(t)} >^^J^l da(t)
Kn+l Ja
j~- p(x) /
"'n+1 Ja
X —
X —
da(t)
M Mt).
Kn+1 Ja X ~
Here the first and third integrals of the right-hand member vanish (also for
n = 0). The second term is p(x) since
K_ Pn+l{x) — Pn+l(t) _ fc ^ _ ,
Another proof of C.2.3) may be obtained by combining Theorem 3.1.4 with
Theorem 2.5.
3.3. Elementary properties of the zeros
Theorem 3.3.1. The zeros of the orthogonal polynomials pn(x), associated
with the distribution da(x) on the interval [a, b), are real and distinct and are
located in the interior of the interval [a, b].
In special instances, particularly in the classical cases (see §2.4), we shall
obtain later more exact information concerning the position of the zeros. (See
Chapter VI.)
As a consequence of Theorem 3.3.1 we have a(a) < a(xi — 0) and a(xn + 0) <
a(b), where xi and xn are the least and the greatest zeros of pn(x), respectively.
A) The usual proof of the preceding theorem is based on the orthogonality
property. From
'•b
pn(x)da(x) = 0, n ^ 1,
f
Ja
we are assured of the existence of at least one point in the interior of [a, b) at
which pn(x) changes sign. (The function a{x) lias an infinite number of points
of increase.) If X\, x2, • • • , xi denote the abscissas of all such points, the
product pn{x){x — Xi){x — x2) • • • (x — xt) has a constant sign (that is, is non-
negative or non-positive throughout [a, b]); we have / ^ n. On the other hand,
if / < n,
I
C.3.1) I pn(x)(x - xi){x - x2) ¦ ¦ ¦ (x ~ xi) da(x) = 0.

[ 3.3 ] ELEMENTARY PROPERTIES OF THE ZEROS 45
Since the integrand is not a zero-function, this is impossible. Therefore we
have / = n.
B) A slight variation of this argument may be made as follows. Let x0 be
an arbitrary zero of pn(x). The coefficients of pn(x) being real, we see that
pn(x)/(x — f0) is a 7rn_i . On the other hand
= [\x-
Ja
C.3.2) / pn(x) -^^r da{x) = I (x-xo)\^^2da(x) = 0,
Ja X — Xo Ja \X — Xo
so that
C.3.3)
2
x — x0
da(x).
In other words, x0 is the centroid of a mass distribution on the interval [a, b].
The integral in the left-hand member of C.3.3) being positive, xQ is real. From
C.3.2) we see that a < xQ < b.
If xo were a multiple zero, we should have
r
which is a contradiction.
C) The statement concerning the location of the zeros (not their simplicity)
follows also from the minimum property formulated in Theorem 3.1.2. Were
a zero x0 to lie outside [a, b], the distance | x — x0 | could be diminished simulta-
simultaneously for all x in [a, b] by a proper displacement of x0. Hence the corre-
corresponding integral C.1.6) could not be a minimum.
For an extension of this argument to polynomials possessing an analogous or
a more general minimum property in the real or complex region, see Feje"r 7,
Szego 5.
From the orthogonality property of the kernel Kn(x0, x) (Theorem 3.1.4),
we can similarly derive some theorems concerning the location of its zeros in x
if Xo is regarded as a parameter. (See Szego 5, pp. 241-244.)
D) The reality and simplicity of the zeros (without the more exact statement
concerning their location in [a, b]) follow from the recurrence formula by means
of Sturm's theorem (Ferron 4, vol. 2, pp. 7-9). For, the polynomials
C.3.5) po(x), pdx), p2(x), ¦¦¦ , pn(x)
form a Sturmian sequence in [a, b] since (a) if pv{xo) = 0, v ^ 1, it follows from
C.2.1) that p,-i(a:o)p,+i(a:o) < 0; (b) Po(x) is a constant ^ 0, and pn(x) is of
precise degree n ; (c) at a point xo where pn(xo) = 0, we have pn(xo)pn-i(xo) > 0.
The latter fact follows from C.2.4) if n is replaced by n — 1 and x by x0 (see
below). Now the number of variations of sign in C.3.5) is n if x < 0 and | x |
is sufficiently large; it is 0 if x > 0 and sufficiently large. (Cf. §6.2 A) and the
footnote 32.)
E) From C.2.4) we obtain the important inequality
C.3.6) p'n+1(x)pn(x) - p'n{x)pn+i{x) > 0, x real.

46 GENERAL PROPERTIES OF ORTHOGONAL POLYNOMIALS [ III ]
As a first consequence we point dut that pn(x) and pn+i{x) cannot have common
zeros. Furthermore, we obtain the following separation theorem:
Theorem 3.3.2. Let xx < x2 < ¦ •¦ < xn be the zeros of pn{x), x0 = a, xn+l = b.
Then each interval [xv, x,+i], v = 0, 1, 2, • • • , n, contains exactly one zero of
Vn+i{x).
In fact if ? and ??,$<??, are two consecutive zeros of pn{x), we have
Pn(?)Pn(v) < 0. On the other hand, C.3.6) yields -p»(?)P«H-itt) > 0,
-p'n(v)Pn+\(v) > 0, so that pn+i(ij)pn+i(v) < 0. This indicates an odd number,
that is, at least one zero of pn+i(x) in f < x < r\. Now let f = xn be the greatest
zero of pn(x); then pl(f) > 0, and C.3.6) yields pn+i(f) < 0. Since pn+i(b)
is positive, we obtain at least one zero of pn+i(x) on the right of f = xn , and
similarly at least one on the left of the least zero xx of pn(x). Consequently,
we can have only one zero of pn+x{x) between xv and xv+i, v = 0, 1, 2, • • • , n.
By interchanging the role of pn(x) and pn+i(x), we can prove as before the
existence of at least one zero of pn(x) between two consecutive zeros of pn+i(x).
This shows again that we cannot have more than one zero of pn+i(x) between
two consecutive zeros of pn(x).
F) Theorem 3.3.3. Between two zeros of pn{x) there is at least one zero of
pm{x), m > n.
See Stieltjes 11, pp. 414-418. For the following proof see Popoviciu 1.
Let ?i ,&,•••, ?m be the zeros of pm(x) in increasing order. According to
Theorem 3.4.1 we have
C.3.7) E KPn(Qp(Q = / pn(x)p(x) da(x) = 0,
»<-l Ja
where {XM} are the Christoffel numbers associated with {^} (see §3.4) and
p(x) is an arbitrary ?rn_i . Now an argument similar to that used in A) shows
that the sequence {pn(?i), ?>«(&)> • • • . ?>«(?<«)) displays at least n, and therefore
exactly n, variations of sign. Here sgn pn(?i) = (—1)", pn(?m) > 0. Thus
there are n distinct intervals
?m, < x < ?m,+i , v = 1,2, ¦ ¦ ¦ ,n; 1 S mi < to < • • • < /"«+i ^ ™,
containing exactly one zero of pn{x), respectively. This establishes the state-
statement.
Other simple consequences of C.3.6) are:
Theorem 3.3.4. Let c be an arbitrary real constant. Then the polynomial
C.3.8) Pn+i(x) - cpn{x)
has n + 1 distinct real zeros. If c > 0 (c < 0), these zeros lie in the interior of
[a, b), with the exception of the greatest (least) zero which lies in [a, b) only for
c ^ pn+l(b)/pn{b), [c ^ p»+i(o)/p»(o)].

[3.4] GAUSS-JACOBI MPJCHANICAL QUADRATURE 47
Indeed, the function pn+i(x)/pn(x) increases from — °°to + °o in the intervals
xv < x < x,+i, v = 0, 1, 2, • • • , n, where x0 = — <x>, xn+\ = + °° •
Theorem 3.3.5. The following decomposition into partial fractions holds:
^ = T l' 7 ¦> 0
o x ?'
pn+i{x) v=o x -
^v} denote the zeros of pn+i(x).
For we have
C 3 10) / = P"^ = P»+i(€>)P»(€r) ~ pU^)Pn+ife) 0
3.4. The Gauss-Jacobi mechanical quadrature
A) Theorem 3.4.1. If xi < xi < ¦ ¦ • < xn denote the zeros of pn(x), there
exist real numbers Xi , X2, • - • , Xn such that
C.4.1) / p{x) da(x) = \ip(xi) + \2P(X2) + • • • + Kp(xn),
whenever p(x) is an arbitrary 7T2n-i . The distribution da(x) and the integer n
uniquely determine these numbers Xv.
The set \xv = xvn) of zeros, as well as the set of numbers {Xv = Xvn}, depends,
of course, on n. Sometimes the numbers Xv are called Christoffel numbers. See
Gauss 2, Jacobi 1, Christoffel 1, Tchebichef 1, Mehler 1; Heine 3, vol. 2,
pp. 1-31.
It suffices to prove C.4.1) for the special cases p{x) = xk, k = 0, 1, 2, • • • ,
2n — 1. These cases represent 2n conditions which uniquely determine, as
we shall prove, the Christoffel numbers Xv and the points xv. (If the distinct
points xv are given arbitrarily, the numbers Xv can be determined so that C.4.1)
holds for every 7rn_i.)
To prove C.4.1) we construct the Lagrange interpolation polynomial L(x)
of degree n — 1 which coincides with p(x) at the points xv, that is,
C.4.2) L{x) = t, p(*0 / / fo*}_ r , = t, p(xX(x),
v=l Pn\Xv)\X Xv) v=\
where the lr(x) are the fundamental polynomials associated with the abscissas
Xi, xi, • • • , xn of the Lagrange interpolation (see §14.1). Now p(x) — L(x)
is divisible by pn(x), so that p{x) — L(x) = pn(x)r(x), where r(x) is a 7rn_i .
Therefore
rb rb rb
/ p(x)da(x) = / L(x)da(x) + / pn(x)r(x) da(x)
Ja Ja Ja
= f' L{x)da{x) = EpW fb
Ja v—1 Ja
Ux)da(x).

48 GENERAL PROPERTIES OF ORTHOGONAL POLYNOMIALS [ III ]
This establishes C.4.1) with
/b [b V (x)
Ux) da(x) = / -rr-rr-3—\ da(x), v = 1, 2, • • • , n.
Ja Pn\Xv)\X Xp)
Conversely, let C.4.1) hold for an arbitrary irin^i, called p(x). Then we
choose p(x) = l(x)r(x), where l(x) = (x - Xi)(x — x2) ¦ ¦ ¦ (x - xn) and r{x)
is an arbitrary 7rn_i . We find from C.4.1) that
/ l(x)r(x) da(x) = 0,
so that l(x) = const. pn(x).
The interpretation of the left-hand member of C.4.1) as a mechanical quad-
quadrature is obvious. For an arbitrary function f(x) defined in [a, b] it may
be written (cf. §15.1)
C.4.4) Qn{f) = Xj/(;Ti) + X2/(?2) + • • • + \nf(Xn)-
Then Theorem 3.4.1 can be formulated as follows: Qn(f) = jaf(x)da{x)
provided 'f(x) is an arbitrary 7T2n_i. Further, from C.4.3) the Christoffel
numbers Xv are the values of Qn(f) for f(x) = l,(x). Also we can discuss for
a fixed function f(x) the convergence of the sequence {Qn(f)} as n —-> <». (Com-
(Compare Theorem 15.2.3 and also Problem 9 below.) Concerning mechanical quad-
quadrature formulas holding for an arbitrary Tr2n-A, see Shohat 7, p. 465.
B) Theorem 3.4.2. The Christoffel numbers X,, are positive, and,
C.4.5) Xj + X2 + • • • + XB = / da(x) = a(b) - a(a).
Ja
The following representations hold:
C.4.6) X, =
C.4.7) X, =
C.4.8) X71 = \po(xy)}2 + {Pi(a)}2 + ••• + {pn{xr)\2
= Kn{Xv, Xv).
Here the previous notations are used.
Concerning C.4.8) see Shohat 3, p, 456. The special case a = —l,b = +1,
da(x) = dx is particularly important. Here the abscissas xv are the zeros of
the nth Legendre polynomial, and the sum of the Christoffel numbers is 2, the
length of the interval of integration. This is the case originally considered by
A'n+l — 1 l\'n

[3.4] GAUSS-JACOB I MECHANICAL QUADRATURE 49
Gauss and Jacobi. Another important special case, namely a = — 1, b = 4-1,
da(x) = A - x2)~h dx, is due to Mehler A).
The positiveness of Xv is clear from any of the representations C.4.6), C.4.7),
C.4.8). In case C.4.7) we take into account C.3.6). According to C.4.5)
the sum of the X, is the total mass of the distribution da(x) spread over the
given interval.
The discussion of the representation C.4.7) can be carried further in the case
of the classical orthogonal polynomials. (Cf. §15.3 A).)
For the proof of Theorem 3.4.2 we write p(x) = {L(x)\2 in C.4.1); this
furnishes C.4.6). Furthermore, writing y = x, in C.2.3), multiplying by
da(x), and then integrating, we obtain, according to C.4.3),
Kn+1 Ja
X—Xy
kn+i
This establishes C.4.7). Combining C.4.7) with C.2.4) for x = xv, we get
C.4.8).
C) As an application of C.4.1) we obtain, for arbitrary real constants
Uq , U\ , • • • , Un-l ,
= I (uo + U\X + • • • + uv-\Xn~1Jda(x)
Ja
F(u)
C.4.9)
G(u) = I x(uo + uix 4- • • • 4- Un-ix"*1)'2 da(x)
= zl KxXuo + u\xv 4- • • • 4- un_\xn, 1J.
Therefore, the characteristic values of the pencil
n
C.4.10) G{u) - E
are precisely ^ = Xi , xi, • • • , xn . With the notation B.2.1), the quadratic
form of the left-hand member becomes
n-l n-1
C.4.11) Z) Z) (cv+mh-1 - ^cv+m)mvmm.
Its determinant is a ?rn in ? which vanishes for ? = xi, xi, • • • , xn, and is
therefore a constant multiple of pn(?). We thus arrive at a new proof of equation
B.2.9).
See also Problem 10.

50 GENERAL PROPERTIES OF ORTHOGONAL POLYNOMIALS [ III ]
3.41. Separation theorem of Tchebichef-A. Markoff-Stieltjes
In 1874 Tchebichef stated a very remarkable property of the Christoffel
numbers (see 6, 8), proofs of which were given independently by A. Markoff
and Stieltjes. Let n ^ 2. In view of the positiveness of Xv and of C.4.5), there
exist numbers j/i < 3/2 < • • • < yn-i, a < yi, yn-\ < b,9 such that
C.41.1) X, = «({/,) - a(j/r_i), v = 1,2, ¦¦¦ ,n;y0 = a, yn = b.
Theorem 3.41.1. The zeros Xi, xi, ¦ ¦ ¦ , xn , arranged in increasing order,
alternate with the numbers J/i, 3/2, • • • , yn-\ ', that is,
C.41.2) xv < y, < Xy+i ;
more precisely
a(x, + 0) - a(a) < ct(yv) - a(a) = Xi + X2 + • • • + X,
C 41 3)
. < a(xv+i — 0) — a(a), v = 1, 2, • • • , n — 1.
In view of C.41.1) the quadrature formula C.4.1) becomes
f
Ja
C.41.4) fp(x) da(x) = E p(x,)\a(yv) - a{y,-x)).
Ja x=l
Since y,-\ < xv < yv, the right-hand member has the character of a "Riemann-
Stieltjes sum."
As a further consequence of the inequalities C.41.3) we notice that
a{xv + 0) < a(xv+i — 0). Thus we have proved the following:
Theoeem 3.41.2. In the open interval {xv, sv+i), between two consecutive zeros
of pn(x), the function a(x) cannot be constant.
Or in other words: In an open interval in which a(x) is constant, pn(x) has
at most one zero.
3.411. First proof of the separation theorem10
Let v be an integer such that 1 ^ v ^ n — 1. Choose for p(x) in C.4.1)
a special iv-in-i subject to the following 2n — 1 conditions:
l if k = 1, 2, • • • , v,
p(*) I
C.411.1) [0 if Jfc = v + 1, v + 2, ... ,n;
P'(xk) = 0 if k = 1, 2, • • • , v - 1, v + 1, • • • , n.
Then this polynomial is uniquely determined.
9 To find such numbers yv it might be necessary to modify a(y) at some of its points of
discontinuity, which has, of course, no influence on C.4.1). It should be also observed
that yv is in general not uniquely determined.
10 Cf. A. Markoff 1, 2, Stieltjes 1, A. Markoff 3.

3.412
SECOND PROOF OF SEPARATION THEOREM
51
By Rolle's theorem p'{x) has at least one zero in each of the open intervals
C.411.2) {xy, xz), fa, x3), • • • , (x,-!, x,)) (avu, xr+i), • • • , (xB_i, xn).
These zeros together with xk , 1 ^ k ^ n, k 9s- v, furnish (n — 2) + (n — 1) =
2n — 3 zeros for p'{x). Since p'{x) is a 7r2n_3 , it follows that these are the only
zeros of p'(x), and also that they are all simple zeros. Hence p{x) is monotonic
between any two consecutive zeros of p'{x); in particular, it is monotonic
between the zero in (x,,_i, xr) and xv+i, and therefore also in [x,, x,+i]. Fur-
Furthermore, p{x) is decreasing in [x,, xr+i\ since p(xv) = 1, p(x,+\) = 0. From
Fig. 2
these considerations the graph of p(x) is easily seen to have the shape given in
the figure. Therefore we have
C.411.3)
p(x) =i 1 in a ^ x ^ xv,
p(x) 2: 0 in xv %. x %. b.
For this special case the general formula C.4.1) gives
+ X2 +
+ K =
p{x) da{x)
/ p(x) da(x)
r
da(x),
which establishes part of the inequalities C.41.3).
To prove the remaining part we consider the distribution d[ — a( — x)] in
[ — b, —a]. The associated orthonormal set is {(— l)"pB( — x)} with the zeros
— xn < ~Xn-i < • ¦ • < — x\ . In place of the numbers y\, yi, • • • , yn-i we
now have — yn-x, — yn~z, ¦ • ¦ , —yi . Then, according to the preceding result,
— a{;Xn-v+i - 0) < ~a(yn-v), or a{y,) < a{x,+i - 0).
3.412. Second proof of the separation theorem11
Let the non-decreasing step-function V(x) be defined by the following
conditions:
11 See Stieltjes 12, especially pp. 588-592.

52
GENERAL PROPERTIES OF ORTHOGONAL POLYNOMIALS
III
C.412.1)
0, a ^ x < Xi ,
Xi, Xi ^ x < x2 ,
V{x) = -jXi + X2 , x2 ^ x <x3,
X2
Xn , xn ^ x ^ b.
Then C.4.1) can be written in the form
C.412.2)
j
\(x)d\a{x) - V(x)} = j\(x) d{a(x) - a{a) - V(x)\ = 0.
An integration by parts (see A.4.4)) yields
C.412.3) J {a(x) - a(a) - V(x))p'{x) dx = 0,
since V(a) = 0 and (see C.4.5)) a(b) - a(a) - V(b) = 0.
The function V(x) is constant in the open intervals (a, Xi), (zi, x-i), ¦¦¦ ,
(zn_i, xn), (xn , b); hence a(x) — a(a) — V(x) = j3(x) is non-decreasing there.
We have /3(z) ^ 0 (but not /3(z) = 0) in the first interval, and /3(z) ^ 0 (but
not /3(z) = 0) in the last interval. In the other intervals {xv, xv+i) the function
/3(x) is either of constant sign (constantly non-negative or non-positive), or
there exists a point y,xv <y < xv+i , such that j3(y — 0) < 0 and I3(y + 0) > 0.
Thus the total interval [a, b] can be subdivided in at most 2n intervals in which
/3(z) is non-negative and non-positive alternately, without being identically zero.
The end-points of these intervals are some of the zeros xv and some of the points
y previously defined. Now from C.412.3) we conclude, by means of an argu-
argument similar to that in §3.3 A), that the number of these intervals is at least 2n,
and then exactly In. Less precisely, /3(z) has exactly 2n variations of sign in
[a, b] which are located at the zeros xv as well as at the points y mentioned,
whose number is n — 1. At the points y there is a transition from negative to
positive values. Hence at the points xv there is a transition from positive to
negative values.
Consequently,
C.412.4)
${xv - 0) > 0 > 0(z, + 0),
v = 1, 2,
n,
which is equivalent to the statement of Theorem 3.41.1. (The first inequality
is trivial for v = 1; the same holds for the second one for v = n.)
We can also prove the above statement by a slight modification of the argu-
argument used. Let yv denote the point in {xv, xv+i) with the variation of sign of
the type y; then we have
C.412.5) 0(z, + 0) <
(Cf. the footnote above.)
= 0 <
- 0).

[3.42] ANOTHER SEPARATION THEOREM 53
3.413. Third proof of the separation theorem12
Let the functions 4>k(x, t) be denned as follows
Ux - t)k if x?t,
C.413.1) 4>k{x, t) =
[ 0 if x > t,
where A; is a non-negative integer, x and t arbitrary and real. Let13
/&
4>k(x, t) da(x) -
Then according to C.4.1)
C.413.3) Fk(a) = Fk(b) =0, k = 0, 1, 2, • • ¦ , 2n - 1.
Also Fk(t) is continuous if A; ^ 1; furthermore, we readily see that F0(t) = /3(i),
where j3(t) has the same meaning as in §3.412. Now
C.413.4) F'k(t) = -kFk^(t), a < t < b, k ^ 1.
[For k = 1, t = xv, this means F[{xv ± 0) = —F0(xv ± 0).] If we take into
account C.413.3) and C.413.4), Rolle's theorem furnishes for the number of
zeros of Fk(t) (including t = a and t = b) the lower bound 2n + 1 — k,
1 ^ k ^ 2n — 1; this holds also for k = 0 in the sense that Fo(t) has at least
2n — 1 variations of sign. From this point on the statement follows by an
argument similar to that in the second proof.
3.42. Another separation theorem
If .Tin < x2n < • ¦ • < xnn denote the zeros of pn(x), we know (Theorem 3.3.2)
that the system [xvn) alternates with the system [xv,n+i\, that is,
C.42.1) xv^.,n < xv,n+1 < xvn , v = 1, 2, • • • , n + 1; xo,n = a, xn+i,n = b.
Let now {X,nJ denote the system of Christoffel numbers associated with
pn(x), and let {yVn) be the numbers yv denned in C.41.1). Stieltjes showed
A2) that in addition to Theorem 3.41.1 the following separation theorem holds:
Theorem 3.42. We may assert that
C.42.2) y,-.i.n < yv,n+i < yvn ;
or
Xln + Mn + • • • + A,_l,n < Ai.,,+1 + \2,n+l + • • • + A,,n + 1
C.42.3)
< XiB + X2n + • ¦ • + X,n , v = 1, 2, • ¦ • , n.
For v = 1 the inequalities involving yon and Xon must be disregarded.
12 This proof is due to Polya and Uspensky (written communication).
13 In case k = 0, t = a the integral in the right-hand member should be replaced by 0.

54 GENERAL PROPERTIES OF ORTHOGONAL POLYNOMIALS [ III ]
The analogy between C.42.1) and C.42.2) is obvious.
For the proof we use the same function V(x) as in §3.412, denoting it now
by Vn(x), and introducing the corresponding function Vn+i(x) associated with
the system {X,,n+ij. Then by C.412.3) we have
C.42.4) f \Vn(x) - Vn+1(x)}P'{x) da(x) = 0,
where p{x) is an arbitrary 7r2n_i . Hence Vn(x) - Vn+i(x) has at least 2n — 1
variations of sign. Such variations can occur only at the points xvn and xv,n+\ .
In the first case Vn+i(x) is constant in the neighborhood of this point, and
since Vn{x) increases, the variation of sign is necessarily a transition from
negative to positive values. The opposite is true for x,,n+i . The total number
of points {xpn} and {Xy,n+i} is 2n + 1. Now Vn{x) and Vn+i{x) are identical
in the intervals a ^ x < x' and x" < x ^ b, where x' is the minimum and x"
the maximum of all the zeros xvn and xv,n+i . Hence no variation of sign is
possible at x' or x". This means that a variation of sign actually occurs at
each of the other zeros and is of the type described above.
As a first consequence of this, we again obtain C.42.1), that is, Theorem
3.3.2, and as a second consequence, the inequalities
C.42.5) Vn(xvn - 0) - Fn+](z,n - 0) < 0 < Vn{xvn + 0) - Vn+1(xvn + 0).
These are the same as the inequalities C.42.3).
3.5. Continued fractions
Historically, the orthogonal polynomials {pn(x)} originated in the theory
of continued fractions. This relationship is of great importance and is one of
the possible starting points of the treatment of orthogonal polynomials. See
Tchebichef 1-8, Heine 3, vol. 1, pp. 260-297, Stieltjes 11.
A) For an infinite continued fraction we use the notation
C.5.1) f>o + ,^— + |^- h • • • + pr- + • • • .
| Oi | f>2 \bn
Here, as usual, the convergent Rn/Sn, n = 0, 1, 2, • • • , is defined as the finite
fraction obtained from C.5.1) by stopping at the term bn. (See, for example,
Perron 3.) We have
Ro = bo, Ri = bobi + Oi, • • • ,
C.5.2)
So = 1, Si = bi, • • - ,
and the recurrence formulas
C.5.3) Rn = bnRn-i + anRn-2 , Sn = bnSn^ + anSn^ , n = 2, 3, 4, • • • ,
which hold also for n = 1 if we define R-i = 1, &_i = 0. Also, we easily obtain
(see Perron, loc. cit., p. 16)

[ 3.5 ] CONTINUED FRACTIONS 55
RnSn-i — Rn-iSn = ( — 1)" cii a2 • • • an ,
or
C.5.4) - = ^ ' J n = 1, 2,3, • • • .
B) Let {pn(z)} be the orthonormal set of polynomials associated with the
distribution da(x) on [a, b]. The recurrence formula C.2.1) then suggests the
consideration of the continued fraction
C.5.5) XI - . . Cz I - C»[ , . Cn' .
Here ^4n , 5n , Cn have the same meaning as in C.2.1). Therefore,
C.5.6) ba = 0, bn = Anx + Bn, n ^ 1; Ol = 1, an = -Cn, n ^ 2.
Next we prove the following theorem:
Theorem 3.5.1. TTie convergents Rn/Sn of C.5.5) are determined by the
formulas
Rn = Rn(x) = CoKcoC2 - c\)h \ ?^ 2l^lda(t),
C.5.7) U x-t
Sn = Sn(x) = c\Vn{x), 71 = 0, 1, 2, • ¦ • .
Here cn has the same meaning as in B.2.1).
Accordingly, the orthogonal polynomials are identical with the denominators
of the convergents of the continued fraction C.5.5).
The second part of the statement follows immediately by comparing C.2.1)
with C.5.3) for n ^ 2 and observing that the statement is true for n = 0 and
n = 1. As regards the first part, we notice that it holds also for n = 0 and
for n = 1. (Since pi(x) = hx + const., the corresponding integral becomes
. Then we use B.2.15) and B.2.7).) Finally, if n ^2, we have
f
Ja
n - bnRn-\ —
(x) - Vn{t) - (AnX + Bn){Vn-l{x) - Vn-l{t)\
X - t
x - t
-(Ant + Bn)Pn-l(t) + (AnX + Bn)Pn-l(t)
X - t
= An f Vn-iit) da{t) = 0,
Ja
which establishes the statement.

56 GENERAL PROPERTIES OF ORTHOGONAL POLYNOMIALS [ III ]
Therefore, the numerators of the convergents are expressible in terms of
Vn{x). Obviously, Rn is a polynomial of degree n — 1 in x,
C) On expanding the rational function Rn(x)/Sn(x) in descending powers
of x, we obtain for n ^ 1
C.5.8) f=^ = Jon* + d,nX'2 + dknX~Z + • • • .
According to C.5.4) this expansion agrees with that of Rn-i(x)/Sn-i(x) up to,
and including, the term x~Un~2) Whence there exists a power series
C.5.9) dox'1 + dix~2 + d2x~z + ¦ ¦ •
such that, for n ^1,
C.5.10) f^4 = ^ + d>x~2 + • • ¦ + d2n^x-2n + E dvnx-v-\
On\X) ,-2n
This is generally true for the convergents of any continued fraction of the
type C.5.5).
Theorem 3.5.2. The equality
C.5.11) dv = co~2(coc2 - c\fcv, v = 0, 1, 2, • • • ,
is valid.
In fact, if d, is defined by these equations and we use C.5.7), we find
= coKcoC2 - ?)h{faVn{Xl~_Vtn{t) da® - Vnix^cox-1 +¦¦¦+ c^
Here the expression in the braces can be written in the form
mo
— t Ja X — t
= / Vn{x) - Mt) x-*nfnda(t) - f Pn(t) l~% l da®
Ja X - t Ja X t
J X - t
x - t
'"-1 + x2n~2t + • • ¦ + xt2n~2 + i2"-1) da(t).
Since the first integral of the right-hand member is a Kn-i, the expansion of
the first term starts with x~n~x. Since the contributions of the powers
1, t, t2, ¦ • • , in-1 in the second integral vanish, the expansion of the second
term starts with xnxn~x = x~n~1. On dividing by Sn(x), we obtain an expan-
expansion of the form C.5.10). This requirement uniquely determines the numbers
do, di, • • • , d2n-i, and therefore, the whole sequence {d?}.

[ 3.5 ] CONTINUED FRACTIONS 57
D) Theorem 3.5.3. The following decomposition into partial fractions holds
for the convergents of C.5.5):
A ^ 19\ ">n\X) _ -2/ „ 2\i V X,n
\O.O.li,) — Co \CqC2 — CJ 2-d ,
(~>n\X) j/=l X Xvn
where X, = X,n and x, = xvn have the same meaning as in §3.4.
For, we find from Theorem 3.5.1
D /"_ \ —i/ 2\i rb /,\
Kn\.Xm) _ Co \C0C2 — Ci) I Pn\t) i /\
S'n(Xyn) C\p'n(xvn) Ja t — Xvn
Now we can apply C.4.3).
From C.5.12) we see that the zeros of Rn(x) are real and that they alternate
with those of Sn(x).
E) Finally, we consider the special case in which [a, b] is a finite interval.
Then the expansion C.5.9) represents the function
/o c io\ j?^\ -2/ 2\j / da(t)
{6.0.I0) t{x) = co (coc2 — Ci) / ——-
t
2si f da(f.
Ja X -
provided | x | is sufficiently large. In various special cases, a function F(x)
representable in the form C.5.13) may be developed directly into a continued
fraction of the type C.5.5), the denominators of this fraction being the or-
orthogonal polynomials associated with the distribution da(t). Such an approach
to these polynomials is essentially different from that used in Chapter II.
Theorem 3.5.4. Let [a, b] be a finite interval. Then
C.5.14) ^
if x is an arbitrary point in the complex plane cut along the segment [a, b]. The
convergence is uniform on every closed set having no points in common with [a, b].
This theorem is due to A. Markoff E, p. 89).
If x be real, x > b, we may combine Theorem 3.5.3 with Problem 9 to get
C.5.15) F(x) - f^ = C
This tends to 0 as n —> <x> (Problem 52) provided x is sufficiently large. On
the other hand, the left-hand member of C.5.15) is uniformly bounded in the
exterior of an arbitrary closed curve containing [a, b] in its interior, since X,n > 0
and C.4.5) holds. Now the statement follows by use of Vitali's theorem
(Titchmarsh 1, p. 1E8). Another proof can be based on Theorem 15.2.3.
Concerning further properties of the convergents we refer to Sherman 1
and the bibliography given there. Regarding the relation of the continued
fraction C.5.5) and of the orthogonal polynomials to the problem of moments,
see Hamburger 1, 2, M. Riesz 2, and the bibliography quoted in these papers.

CHAPTER IV
JACOBI POLYNOMIALS
In this chapter we shall be concerned with the main properties of Jacobi
polynomials, which include as special cases the ultraspherical polynomials,
particularly, the Legendre polynomials. Among the topics which are not con-
considered here, but which are reserved for later study, are the properties of the
zeros, asymptotic expressions, expansion problems, and properties connected
with interpolation and mechanical quadrature.,
Addition theorems for Legendre and ultraspherical polynomials have also
been omitted, as have the relations of these polynomials to spherical and surface
harmonics of various dimensions. Limitations of space and the existence of
exhaustive treatises on these subjects are the chief reasons for such an omission.
The interested reader may well consult Whittaker-Watson A, pp. 326-328,
335) and Hobson A).
4.1. Definition; notation; special cases
A) The definition of the Jacobi polynomials Pi"'e)(x) has been given in
§2.4, 1; they are orthogonal on [—1, +1] with the weight function w(x)
= A — x)a(l + %Y- Assurance of the integrability of w(x) is achieved by
requiring a > — 1 and /3 > —1; the normalization of Pl"'e)(x) is effected by14
D.1.1) P^U)
)
)"
The orthogonal polynomials with the weight function (b — x) "(x — a)" on the
finite interval [a, b] can be expressed in the form
D.1.2) const. P(na^ 12 ?—-^ - 1
{ b — a
(see the last remark in §2.3). The case a = 0, b = 1 is often used (Jacobi 3;
Jordan 1, vol. 3, pp. 231-234; Courant-Hilbert 1, pp. 76-77).15
Stieltjes F, p. 75) writes a and /3 for (/3 + l)/2 and (a + l)/2, respectively,
in terms of our notation. The same notation is used by Feje"r A3, p. 42).
Jordan's function Zn(u) in our notation is
14 According to §3.3 the zeros of P^3)(x) are in -1 < x < +1, so that P^Q) ^ 0.
16 The statement on p. 76 in Courant-Hilbert must be corrected so as to read
p(x) = x«-»(l — x)m, q > 0, p — q > —1.
58

[4.1] DEFINITION; NOTATION; SPECIAL CASES 59
Courant-Hilbert's function Gn(p, q, u) is the same as Zn(u) with p = a, q = y.
The important identity
D.1.3) Plna^(.x) = (-iyp^a)(-x)
is readily derived by means of the last remark in §2.3. Combining D.1.3) with
D.1.1), we have
B) For a = /3 we have the ultraspherical polynomials. They are even or
odd polynomials according as n is even or odd (§2.3 B)).
Theorem 4.1. The following formulas hold:
+ a +
r(* + a + )( + )
_r 1Vr(a> + a + i)r(y + D pM,,)
-(} r(, + « + i)rB, + i) P'
2"+1 (:c) " r( + +iyrB + 2) ^ B:c "
+ 2)
As a consequence of these important relations, Jacobi polynomials with
a or /3 = ±f may be expressed by ultraspherical polynomials. In order to
establish the first relation, it suffices to prove that
P
- x2)adx = 0,
where p{x) is an arbitrary tt2,-i . This is trivial if p{x) is odd. Let p(x) be even
and equal to r(x ), where r(x) is an arbitrary -k,-\ . Then we have
f+1 Pla'-i)Bx2 - I)r(z2)(l'- x2)adx = 2 I P^'-*)Ba;2 - l)r(a;2)(l - x2)"^
- 2-"-} P
= 0.
A similar argument may be used to prove the second relation. The constant
factors are determined according to D.1.1), D.1.3).
The case a = —l,b= +1, w{x) = | x \2k, k > —\, can also be reduced to
Jacobi polynomials. The corresponding orthogonal polynomials are (see
Szego 2, p. 349):

60 JACOBI POLYNOMIALS [ IV ]
f const. Pi°-k~i\2x2 - 1) if n = 2v,
D"L6) ?n{X) = [ const. xPr+i\2x2 - 1) if n = 2v + 1,
where the constant factors are different from zero; they depend on v and k.
The proof is similar to the previous one.
C) The simplest cases of ultraspherical polynomials are16
<_,._,> _ l-3---Bn-l) ( l'3---Bn-l)
^n W~ 2-4 ...2n nW 2-4---2n " '
D 1 7^ P(M)(r) - 2 !'3 ••• Bn + D r/ /_n
D.1.7) Pn ^-22.4... Bn + 2) U*x)
- 9 1-3 •'• Bw + !) sin (n + 1H
2-4 • • • Bn + 2) sin0 '
where x = cos 6, and !Tn(a;) and t/n(a;) denote the Tchebichef polynomials of
the first and second kind [A.12.3)]. This follows from
/+
Tn{x)Tm{x){\ - x)^dx = \ cos nd cos mti dd = 0,
r-r
Un(x)Um(x)(l - x2)hdx = / sin (n + 1N sin (to + 1Hd0 = 0, n 5^ to,
because of D.1.1).
In this connection, two "mixed" cases of importance may be mentioned:
pU.-i)(T) = 1-3 ••• Bn - 1) sin {Bn + 1H/2}
n K J 2-4 ••• 2n sin @/2)
D.1.8) x
p(-4.j)/ x = 1-3 ••• Bn - 1) cos {Bn + 1H/2}
n K) 2-4 ..-2n cos @/2)
The proof is similar to that of D.1.7) (or is obtained by setting a = \ and a = — \
in D.1.5)). Formulas D.1.7) and D.1.8) also follow from §2.6 by putting
p(x) = 1 there.
Another important ultraspherical case is a = C = 0, that is, the Legendre
polynomials Pi°'0>(a;) = Pn(x). Less elementary cases are a = /3 = — f, and
a = /3 = — f, for which Koschmieder A) gave representations in terms of elliptic
functions.
4.2. Differential equation
A) Theorem 4.2.1. The Jacobi polynomials y = P(na '?)(x) satisfy the following
linear homogeneous differential equation of the second order:
D.2.1) A - x2)y" + [0 - a - (a + /3 + 2)x]y' + n(n + a + P + l)y = 0,
16 In the first equation the coefficient of Tn{x) is 1 for n = 0.
17 For n = 0, the numerical factor on the right side is 1.

4.2 ] DIFFERENTIAL EQUATION 61
or
D.2.2) di
+ n(n + a + 0 + 1)A - x)a(l + zA = 0.
To prove this, we note that since y is a irn , the expression
d{(l - x)a+1(l + x)"+1y'}/dz
has the form A — a;)a(l + x)?z, where z is also a 7rn. In order to show that
z = const, y, we prove the orthogonality relation
[ > I V-1- •"/ V-1- I ¦«'/ tf >H\-^J """ — ",
where p(z) is an arbitrary 7rn_i . An integration by parts reduces the left-hand
member to
since a + 1 and /3 + 1 are positive. A second integration by parts gives
i -j
In the last integrand the coefficient of y is of the form A - x)a(l + x)^r(x),
where r(x) is a 7rn_i . Hence this integral vanishes and the statement is estab-
established. The constant factor —n(n + a + j3 + 1) may be determined by com-
comparing the highest terms.
An alternative form of D.2.1) is
A - x'2)Y" + [a-/3+(a + /3- 2)x]Y'
D.2.3) + (n+ l)(n + a + /3)F = 0,
Y = A - x)a(l + x)"y = A - x)a(l + ^Vi"'^^).
B) Replacing n(n + a + 13 + l)in D.2.1) by 7, we may ask: For what
values of 7 has this equation a polynomial solution which is not identically
zero?
Theorem 4.2.2. Let a > — 1,/3 > — 1. The differential equation
D.2.4) A - x2)y" +[/3-a- (a + /3 + 2)x]yt + y y = 0,
where 7 is a parameter, has a polynomial solution not identically zero if and only if
7 has the form n(n +a + /3 + l),n = 0, 1, 2, •••. This solution is const.
•Pl"'p\x), and no solution which is linearly independent of Pi"'?)(x) can be a poly-
polynomial.
To prove this, substitute y = ^Ho ciy(x — 1)' in D.2.4). We find

62 JACOBI POLYNOMIALS [ IV ]
-Cr+ 1) Y,v(v -l)a,(x - I)"
- [2(« + 1) + (a + 0 + 2)(s - 1)] I] **,(* - I)"-1 + 7 E aXx - 1)' = 0,
which yields the recurrence formula
[y ~ v{? + a + 0 + 1)K - 2(» + 1)(. + a + l)ar+1 = 0,
D.2.5)
, = 0,1, 2, ....
Assuming that y is a polynomial, let us suppose that an is the last nonzero
coefficient. Then we see from D.2.5) that for v — n the coefficient of an must
vanish, that is, y = n{n + a + /3 + 1). Conversely, if this condition is satis-
satisfied, then On+i = On+2 = • • ¦ =0 since the coefficient of ay+i never vanishes.
Now let 7 = n(n + a + /3 + 1), and let z be a second solution of D.2.1) or
D.2.2). If we let x —> dbl in the relation
D.2.6) A - x)a+1(l + xf+\y'z - yz') = const.,
we see that y and z cannot both be polynomials unless the constant in the right
member is zero, that is, unless y and z are linearly dependent. This argument
shows that z cannot even be regular at ? = -1 or at i = +1, unless y and z
are linearly dependent.
4.21. Hypergeometric functions
A) Substitution of x = 1 - 2x' in D.2.1) yields
x'(l - x') pL + [a + 1 - (a + 0 + 2)x'] %
D.21.1) dx" dx'
+ n(n + a + j8 + \)y = 0,
which is the hypergeometric equation of Gauss. On account of the second part
of Theorem 4.2.2, for n = 1, we obtain the important representation:
1 v^ (n
D-212) - a S (.
•(« + » + 1) ••• (o + n)
18 The general coefficient
X — 1 \ 18
f * J(n + a
+ A + 1) - • • (n + a + A + ,)(« + i» + 1) • • • (<* + n)
is to be replaced by (a + l)(a + 2) ¦ ¦ ¦ (a + n) for v = 0, and by
(n + a + |8 + l)(n + a + j8. + 2) • • • Bn + a + /3) for «, =n.

[ 4.22 1 GENERALIZATION 63
Here, and in what follows, F(a, b; c; x) is the usual notation for the hyper-
hypergeometric series
F(a, b; c; x)
D.21.3) =l,y o(o + 1) • • • (o + y - 1) 6F + 1) • • • F + r - 1) ,
ti 1-2---K c(c + l) ••• (c + k- 1) *'
convergent for j a; j < 1 and satisfying
D.21.4) x(l-x)ll + [c- (a+b + l)x] ^ - aby = 0.
¦ax2 dx
(See Whittaker-Watson 1, p. 283.) For latter reference we observe that
D.21.3) is without meaning if c is a non-positive integer. However, it is readily
seen that if to is a positive integer,
lim (c + to — \)F(a, b; c; x)
c-*— (m— 1)
D.21.5) _ (_ nm-i a(a + 1) • • • (a + m - 1NF + 1) • • • F + m - 1)
to!(to — 1)!
•xmF(a + to, 6 + to; to + l;x),
and the function xmF(a + to, 6 + to; to + 1; z) satisfies the equation D.21.4)
with c = — {m — 1).
B) In the formula D.21.2) the hypergeometric series stops with the term
in xn. The constant factor in the first part of D.21.2) is determined by D.1.1).
Using D.21.2), note that the coefficient l[a'?) of the highest term xnmP(naJ)(x) is
D.21.6) Zi-" = lim x-nPla^(x) = 2~
C) Another application of D.21.2) is the useful formula
D.21.7) ^ {Pi-»(s)} = \{n + a + /3
which follows immediately when we expand both sides of D.21.7) according
to D.21.2).
As an application of D.21.7) we observe that the successive derivatives
T'n(x), T'L{x), T'"(x), ••• of the Tchebichef polynomial Tn{x) are, but for
constant factors, Pi*lV(rc), Pn-a(x), Plnl$(x), ¦¦¦. The first is, except for a
constant factor, Un-i{x) (see D.1.7)). We note also that the derivatives P'n{x),
P'n{x), • • • of the Legendre polynomial Pn(x) are constant multiples of P(n1-i(x))
P™(x), ¦¦¦ , respectively.
4.22. Generalization
A) The second formula D.21.2) furnishes the extension of the polynomial
P^'^ix) to arbitrary complex values of the parameters a and /3. It is a poly-

64
JACOBI POLYNOMIALS
IV
nomial in x, a, and /3. In the following, we again denote this irn by P^'^^x).
Many of the properties of Pia'0>(x) may be extended to this general case. The
polynomial P\?'fi){x) satisfies the differential equation D.2.1), and the formulas
D.1.1), D.1.3), D.1.4) hold. Spme other results, however, (for instance, the
theorem on the location of the zeros, cf. §6.72) must be essentially modified.
Using D.1.3), the representation of P[aJ>(x) as a wn in x + 1 can easily be
derived.
B) By comparison of the corresponding powers of x — 1, we obtain the
identity
D.22.1)
2n + a
n
•* n
X —
~2~
x + 3
x —
Furthermore,
D.22.2) (j) P*rl'»(x) = (n
and
= -2n - a - 0 - 1.
I an integer, 1 ^ I ^ n,
L
D.22.3) \k ~
n + a + /3 + /c = 0, fcan integer, 1 ^ A; ^ w.
In connection with D.22.2) see D.21.5).
C) Let n ^ 1. A reduction of the degree of P(na'?)(x) occurs if and only if
n + a + /3 + /c = 0fora certain integer k, 1 ^ k ^ n. In this case "x = x,
is a zero of order n — k + 1," this being the precise order unless a = — I, I an
integer, k S I ^ n. If
D.22.4) n + a + /3 + A; = 0, a = -I, 1 ? k ? I ? n,
the polynomial Pi"'®(x) vanishes identically.
By setting n + a + /3 + /c = e, a -\- I = i], it can be shown that P\?'^{x)
= e?-(:r) + »?6'(a;), except for terms of higher order, if e —> 0, tj —> 0. Here r(x)
and s(x) are certain ?rn independent of e and -q. In view of D.22.2) and D.22.3)
it follows that, apart from constant nonzero factors,
YPi'rr'^ix), s(x) = PiZ['~n+l-k\x),
or
D.22.5)
r(x) = A -
r(x) = A - x)-Pftf (x),
(x) = PLa/2
a, /3, n integers, a ^ —n, /3 2; —n, a + /3 ^ — n — 1, n 2; 1. Both polynomials
r(x), s(x) are solutions of D.2.1); they are linearly independent since r(l) = 0,

[ 4.23 ] SECOND SOLUTION 65
s(l) 5^ 0. Also, they have the precise degrees n and k — 1 = — n — a — /3 — 1,
respectively. In this instance the general solution of D.2.1) is a polynomial.
D) Once more let n ^ 1. We then see that P(na^\l) 5^ 0 unless a = -I,
1 ^ I ^ n. If a = — I, a; = 1 is a zero of order I, and this is the precise order
unless n + a + /3 + fc = 0, 1 ^fc^Z, in which event we recognize the ex-
exceptional case D.22.4).
According to D.1.3) we have P(na^(-1) * 0, unless /3 = -I, 1 S I ? n.
In this case x = — 1 is a zero of order I, and here this is the precise order unless
n + a + /3 + fc = 0, 1 ±5&±sZ, which is again essentially the case D.22.4).
E) Let n ^ 0. From C) a second case may be derived in which the general
solution of D.2.1) is a polynomial. If we replace nby— n — a — 0 — 1, the
differential equation D.2.1) remains unchanged, which leads to the linearly
independent polynomial solutions:
D.22.6) Tl X X ~n~^~l X ' "
a, /3, n integers, a < — n, /3 < — n, n ^ 0.
4.23. Second solution
A) According to the theory of hypergeometric functions, a second solution
of D.2.1) is given by
D.23.1) A - xTaF(-n - a,n + p + 1; 1 - a;-
unless a is an integer. (See Whittaker-Watson 1, p. 286; cf. in particular
yx and ?/2 -19 The functions D.21.2) and D.23.1) are then linearly independent;
Now let a be an integer. If a = — I, 0 52 I ^ n, the function D.23.1) is,
but for a constant factor, identical with Pia'?)(x) (see D.22.2)). The same is
true if a —* «o = a positive integer, provided we multiply D.23.1) by a — a0
before passing to the limit a —> a0 (see D.21.5)).
Finally, for integral values of a, a < —n, P(naJ\x) and D.23.1) are linearly
independent, since
n /
and D.23.1) vanishes for a; = 1. The latter function is a polynomial if and
only if n + /3 + lisa non-positive integer, that is, /3 is an integer less than — n.
This is the case referred to in §4.22 E).
B) Numerous other representations are obtained for the solutions of D.2.1)
by using the classical transformation formulas of hypergeometric functions.
The only singularities of this differential equation are at a; = +1, —1, and °o.
Interchanging a and /3, and replacing a; by —x, we obtain the expansions about
x = -1.
19 This can be readily shown by introducing in D.21.1), y = x'~"z. Analogous methods
can be used in the cases D.23.2), D.23.3).

66 JACOBI POLYNOMIALS [ IV ]
The expansions about x = x are especially important. From Whittaker-
Watson 1, p. 286,20 we obtain the solutions
D.23.2) A - x)nF(-n, -n - a; -2n - a - 0; ^-\
D.23.3) A - a:)-"—'-1 F(n + a + 0 + l,n + 0 + l;2n + a + 0 + 2; -?—) .
\ 1 — x/
The first function is, except for a constant factor, Pia^\x) [cf. D.22.1)]. The
second function is obtained from the first by replacing n by — n — a — 0 — 1.
Apart from constant factors, the expressions D.23.2) and D.23.3) arise from
D.21.2) and D.23.1), respectively, by replacing a by — 2n — a — 0 — 1,
A - x)/2 by 2/A - x), and then multiplying by A - x)n. Consequently,
D.23.2) and D.23.3) are linearly independent unless — In — a — 0 — 1 is an
integer not less than — n.
Theorem 4.23.1. Let a, 0 be arbitrary, n ^ 0 an integer. The general solution
of D.2.1) can be represented in the forms
if a ?± —n, —n + 1, —n -f 2,
APiaJ\x) + 5A + x)->F (_n - 0, n + a + 1; 1 - 0; l±?
D.23.4)
ifp * -nt-n+ l,-
5A -
a + 0 ^ -n - 1,-n - 2,
respectively. Here A and B arc arbitrary constants.
C) The preceding results enable us to prove the following:
Theorem 4.23.2. // a and 0 arc arbitrary and n ^ 0 is an integer, then
D.22.5) and D.22.6) arc the only cases in which the general solution of D.2.1)
is a polynomial. They can be characterized by one of the following sets of conditions:
(a) a, 0 negative integers, a ^ — n, 0 ^ — n, a + 0 S — n — 1, n ^ 1,
D.23.5)
(b) a, 0 negative integers, a < — n, 0 < —n,n 2; 0.
We see from D.2.6) that in the case in question a and 0 must be negative
integers. Now, let a < —n, 0 ^ — n; then D.23.1) is a non-polynomial solu-
20 In particular see the functions denoted by yn and 2/22 . (There is a misprint in the
corresponding formulas: the exponent of — x should be —A in 3/21 and — B in 2/22.)

[ 4.3 ] RODRIGUES' FORMULA; ORTHONORMAL SET 67
tion. The case a 5; — n, /3 < — n can be excluded by making use of D.1.3).
Finally, let a ^ — n, /3 ^ — n, a + /3 ^ — n. Then D.23.3) is a non-polynomial
solution.
D) Let a be an integer. x In the exceptional cases, excluded in Theorem 4.23.1,
we can show that the second solution contains logarithmic terms in its repre-
representation about x = +1 (similarly for x = — 1, x = «). (See D.61.6).)
An extension of the preceding discussion to arbitrary values of n is also
possible. However, in what follows, we shall confine ourselves to non-negative
integral values of n.
The consideration of the second solution, properly normalized, will be resumed
in §4.61, where some other representations will also be given.
4.24. Transformation of the differential equation
Applying §1.8 to D.2.1),.we obtain the following important transformations
of the differential equation of Jacobi polynomials:
d2u A JL-jx_ 1 1-/32
1~ 1 A / 1 __\O ~T~ 4
dx2 |4 A - xJ 4A+ xJ
D.24.1) n(n + a+
u =¦ u(x) = A -
D.24.2)
u = u{8) = ^sin |j [cos |J i^ai/"(cos 6).
The special cases a - ±5, /3 = ±5 arc to be particularly noted.
4.3. Rodrigues' formula; the orthonormal set
A) Given a and /3 arbitrary, we have
D.3.1) A - x)"(l + rfP^ix) = (-~^~ (J^f {A - x)n+a(l + x)n+fi}.
First, take both a and /3 greater than — 1. A simple application of Leibniz' rule
then shows that the right-hand member is of the form A — x)"(\ + x)Bp(x), p(x)
being a Tn . To show that p(x) = const. Pla'^(x), it suffices to prove that
(?fn+a(
((i ¦x)n+a(i+x^r^dx=°'
where r(x) is an arbitrary 7rn_i . But integration by parts n times yields the
result

68 JACOBI POLYNOMIALS [ IV
(-1)" J^ A - x)n+a(l + x)n+ssrw{x)dx,
which vanishes since r(n)(x) = 0. The constant factor can then be determined
by setting x = 1 and using D.1.1) (see D.3.2)).
Since Pia'®(x) is a polynomial in a and /3 by D.21.2), and since the same is
true for the right-hand member of D.3.1) when divided by A — x)a(l + xf,
it follows that D.3.1) is valid for arbitrary a and /3.
On calculating the nth derivative in D.3.1) by Leibniz' rule, we obtain the
important representation
(i,9, _(* + «\(x + *Y y w(n - 1) • • • (n - y H- D (n + p\(x - lY
l4^ "V n A~>/ ^(a+l)(a + 2)--.(^F7)V * A^TV
n
B) The argument used in A) readily leads to the formula
_ 2 +^+ r(n -f a -f l)T(n -f 0 -f 1) _ , (a,^
~~ 2n -f a + /3 -f 1 T(n + l)T(n + a + 0 -f 1) "
(Here we have the inequalities a > —1,/3 > — 1, and for n = 0 the product
Bn H- o H- j8 H- l)r(n -f a + j8 -f 1) must be replaced by r(o + j8 H- 2).)
In fact, because of D.3.1) and D.21.6), we have
r A - z)"(l H- x?\P{na^{x)}'dx = ll-v j^ A - x)a{\ + x?P(naS){x)xn dx
iiOl/!) T1 A - x
= 2-t^> / A - x)n+a(l + x)n+ti dx.
Now we employ D.21.6) and A.7.5),
Using the notation D.3.3), we obtain as the orthonormal set associated with
the weight function A - x)a(l + xf in [-1, -fl]
pn(x) = {/iio>/S)r*Pia>/S)(a;)
\2n + a + j8 + 1 T(n + l)F(n + a + j8 H- 1)V D(a,^)/ x
n
n = 0, 1,2,

[ 4.4 ] GENERATING FUNCTION 69
4.4. Generating function
A) Formula D.3.2) can be written as follows:
D.4.1) P?» <*) . -1.
where we assume that x 5* ±1. The integration is extended in the positive
sense along a closed curve around the origin, such that the points — 2(x ± I)
lie neither on it nor in its interior. (We define the first and second factors of
the integrand to be 1 for z = 0.) Hence for sufficiently small values of | w \ ,
1 r
T = i, /
27rt 7
/. , x +1 \Vi . * - 1 V
I1 "i—o— z)\1+ —*— z
V 2 A 2 /
D.4.2) E Pi"n(x)vT = i, / V , 2 Aw 2 / . dz.
2rt 7 / + l \ / l \
The denominator is
D.4.3) - \{x2 - \)wz2 - z{xw - \)-w = \{\ - x2)w{z - zo)(z - Zo),
where
D.4.4) Zo = Zo(w) = — , R = R(w) = A — 2xw -+- w2) .
1 — x2 w
For Zo = Z0(w) there is an. analogous expression with — R instead of R.
Here zo and R are regular analytic functions of w provided | w | is sufficiently
small; we take R@) = 1. At w = 0 the function zu has a zero, and the function
Zo has a pole. For sufficiently small | w | , zo lies in the interior, and Zo in the
exterior, of the integration curve of D.4.2), so that by Cauchy's theorem
oo r i -j—i / _i_i\a/ i\0
Z PiaS)wn = - A - x2)w ( 1 + ^—- zo ) ( 1 + X-~ zo) («o - Zo)".
Now, we readily get
1 + i±i zo = 2A - w + R)~\ 1 + x- ~ zo = 2A + w + R)~\
so that
00
r^n (,x)iy =2 /c A — it; -f- it) A -f- it; -f- it)
= 2a+ A — 2xw -f- to2) {1 — w -f- A — 2zu> -f- ty') p"
• {1 -f- w -f- A — 2zw -f- ly2M)""^.
This is the generating function (series) of the Jacobi polynomials (Jacobi 3,
pp. 193-194) which may be established directly for x = ±1. The expressions
{ }~a and ( p^ must be taken positive for w = 0.

70 JACOB I POLYNOMIALS ( IV ]
B) A slight variation of this argument may be made by writing D.3.1) in
e form
the form
-1. fY(Y(Y .
2« J \2 t — x / \1 — x) \1 + x) t — x
Here x ?± ±1, and the integration is extended in the positive sense around a
closed contour enclosing the point t = x, but not the points t = ±1. Also the
functions (A — t)/{\ — x))a and (A + t)/(l + re))" are assumed to reduce to 1
for t = x. We next write
D.4.7) l)t~^=w~1, t = w~1{l-(l-2xw + w2f] =x + \{x2 - l)u> H .
Here that branch of A. — 2xw -j- w2)* must be taken which is equal to +1 for
w = 0. Then if m> describes a small closed curve around the origin, t describes
a curve of the type mentioned above. Furthermore,
D.4.8) \^ = 2{1 - w + A - 2xw + to2)*}-1,
X JO
JO
w
- 2xw + w2f}~\ -^- = A - 2zw + w2rhw~l dw,
t X
X. "]— X 1/ X
so that
IAA s - • ¦ .... :i-w + (l-2xw +
D.4.9)
- m;2)j}^A - 2xw + w2)~hw~l dw,
which is the desired result.
C) A third method of deriving the generating function is based on the fol-
following remark. If the function F(x, w) of the right-hand member of D.4.5) is
developed in a power series in w, it is seen that the coefficient of wn is a poly-
polynomial of degree n in x. To identify this polynomial with P(na"ff)(x) we show
that
r+i
D.4.10) / A - x)a(l + xfF(x, u)F(x, v) dx,
considered as a function of u and v, is a function only of the product uv, which
is equivalent to the orthogonality property. For x = 1, the identity D.4.5)
can be proved directly, and this procedure furnishes the normalization of the
coefficients.
In Legendre's case: a = /3 = 0, the integral can be calculated explicitly
(Legendre 1, p. 250). In the general case, Tchebichef E) transformed this
integral into the form
D.4.11) 2a+m /"V(l - 0"(l - uvtya(l - uvt2)~ldt,
Jo
from which the statement follows.
Concerning a fourth method based on Lagrange series, see P61ya-Szego 1,
vol. 1, pp. 127, 303, problem 219.

[4.5] RECURRENCE FORMULA 71
4.5. Recurrence formula
A) In the present case the general formula C.2.1) becomes
2n(n +a+ 0)Bn?+ a ^
t
D.5.1)
- 2(n + a - l)(n + 0 - l)Bn +« + 0)Pi°-2)(x), n = 2, 3, 4,
Here, the coefficient of xPli^x) may first be verified by means of D.21.6);
then, by alternately setting x = -j-1 and x = — 1, the coefficients of P^i^x)
and Pi-2}(x) may be calculated. Actually, the formula is but a special case
of the relations between contiguous Riemann P-functions (see Whittaker-
Watson 1, pp. 294-296).
B) Using the notation D.3.3), we obtain the following expression for the
"kernel" (cf. C.2.3)):
^« ' O«n 4 t/2tC\T*f*nt
l)r(n H- j8
x - y
In particular, for y = 1:
Ki-^Oc, 1) = Ala'^(x)
-f ^r^-f /3 -f 1)
r(n H- a H- j8 H- 2)
D.5.3) 2n + a + j8 ¦+- 2 r(a -f l)r(n -f j8 -f 1)
a
1 - x
_ o/Sl l\(n + Ct +
-a-/S-l l\(n + Ct + 0 + 2)
The last representation in D.5.3) is a consequence of Theorem 3.1.4, since
A - x)a(l + x)\l - x) = A - x)a+1(l + xf. We also note that
:J>)(x) - (n+ l)Pi+'f(x)
v 2n + a + f3 + 2 1 - x
D.5.4) , _
P(a,C+1)/ ^ ¦"
n (X
2n-fa-f0-f2 1+x
The second formula follows from the first one if we interchange a and 0 and use
D.1.3). Finally, by using D.21.7) and the last formulas, we obtain

72 JACOBI POLYNOMIALS [ IV J
A - x2) 4- [Piajn(x)\ = i(n + a + p +
D.5.5) ' ' dx
where
A = 2{jl + a)(n + ^ + <* + 0 + 0
B +
D.5.6)
c = _ 2n(n + l)(n + a + 0 + 1)
Bn + a + 0 + l)Bn + a + 0 + 2)'
Here Pi+i'(x) [or P(na-fi(x)] can be expressed by means of D.5.1) in terms of
xP^>(x), P{na*\x), PiaJ.?(x) (or P^f (x)). This yields
' Bn + a + 0)A - x2) — }Pia^(x)}
= -n{Bn + a + 0)x + 0 - a}PJ,ai/})(x) + 2(n + a)(n -
D'57) Bn + a + 0 + 2)A - x2) — {P{"S){x)\
ax
= (n + a + 0 + l){Bn + a + 0 + 2)x + a - /3}Pia^(x)
- 2(n + l)(n + a + 0 + l)Pi+f (x).
In addition, we notice the following consequence of the last formula D.5.3):
D-5-8) o-a-fi-i r(n + a + 0 + 2)r(n + a + 2)
= 2
T(a + l)r(a + 2)r(n +
4.6. Integral representations in general
The representation D.3.1) and its integral form D.4.6) are closely related to
a classical method used for the integration of the hypergeometric equation and
others of similar type. We again start from the formula D.4.6):
A - x)a(l + xfPi^ix) = —H- / A - 0"+a(l + 0n+^(< - x)~n-ldt,
D.6.1)
where x ^ ± 1. The integration is extended in the positive sense over a closed
curve enclosing x, but not the points t = ±1. Using an idea of Euler A), we
try to integrate D.2.1) and D.2.3) by means of
D.6.2) Y = A - x)a(l + xfy = / A - 0"+a(l + 0"+P(< - *)
with a proper choice of the contour of integration. Here x ^ ±1, and the
path of integration must avoid the points — 1, +1, and x. However, we allow
— 1 and +1 as end-points of a path provided the integrals D.6.2) and D.6.3)
are convergent.

[ 4.6 ] INTEGRAL REPRESENTATIONS IN GENERAL 73
Substituting D.6.2) in D.2.3), we obtain
A - z2)F" + [a-/3+(a + /3- 2)x]Y' + (n + l)(n + a + 0O
= f A - 0"+a(l + t)n+\t - x)—3{(n + l)(n +
D.6.3) + (n + l)[a - 0 + (a + 0 - 2)x](f - x)
+ (n + l)(n + a + 0H - xJ} dt
= -(n + 1) j jt {A - i)"+a+1(l + 0n+^+10 - x)—2} dt.
Therefore, we see that D.6.2) satisfies D.2.3), provided one of the following
two conditions is fulfilled:
(a) The path of integration is a closed contour along which the expression
A - t)n+a+\l + t)n+p+1(t - x)~n~\ or what amounts to the same thing,
A — 0"(l + 0" returns to its original value.
(b) The integration is extended along an arc, finite or infinite, such that the
first expression mentioned vanishes at the end-points.
Specialization of the contour according to these restrictions yields numerous
important integral representations for Jacobi polynomials as well as for other
solutions of D.2.1). (Cf. §4.61, §4.82.) For a special contour integral allowed
in the sense of (a) and (b), we must first show that y is not identically zero; then y
can be identified with a constant multiple of Pi"'p)(x), or with some other
particular solution of D.2.1); finally, the constant factor must be determined.
The resulting integral representations hold, save for some exceptional values of
a and 0.
Further integral representations are obtained by replacing nby— n — a — 0 — 1
in D.6.2); this does not affect D.2.3). Thus,
D.6.4) Y = A - x)a(l + xfy = j A - O^^d + O""""^ - x)n+a+l>dt,
where the contour is chosen as in (a) or (b) above. Instead of the first expres-
expression in (a) we now have
t)"n-a(t - x
+a+l)-\
Rodrigues' formula D.3.1) is a special case of D.6.2), the path of integration
being a closed curve which encloses x but not =fc 1. Condition (a) is then satisfied.
4.61. Application; functions of the second kind
A) Theorem 4.61.1. Let x be arbitrary in the complex plane cut along the
segment [— 1, + 1]. Let a > — 1, 0 > — 1, n ^ 0. Excluding the case n = 0,
a + 0+1^0, a solution y = Q(na'^(x) of the differential equation D.2.1), which
is linearly independent of P(na'^(x), can be obtained in the form
D.61.1)
r+i
A - t)n+
l + t)n+\x - t)-n~ldt.
In the exceptional case: n = 0, a + 0 + 1 = 0, we have Qoa'^(x) = const.;
then a non-constant solution is given by

74 JACOB I POLYNOMIALS [ IV ]
Qw(x) = log (.t + 1) + 7T-1 sin xa (x - \Ta{x + I)""
D.61.2) /•+! d _ ty(x + j)*
• / ——^ _ t log A + 0 dt.
The function Qia'P)(x) is called Jacobi's function of the second kind. We use
this notation also if n = 0, a + /3 + 1 = 0, for the function Q(a)(x). In the
special case a = /3 = 0 we write Qi°'0)(x) = Q»(x) (Legendre's function of the
second kind). (Cf. Jacobi 3, pp. 195-197.)
Both D.61.1) and D.61.2) are multi-valued (except if a and /3 are integers).
Both integrals are single-valued and regular in the complex plane cut along the
segment [— 1, + 1]. Obviously, Q(na'$)(x) ~ x-n-<*-0-i as x _> ^ which shows
that Qla'0)(x) is linearly independent of P(naJ)(x) (except if n = 0, a + /3 + 1 =0,
see below). The corresponding property of Q(a)(x) is clear. We have, as is
easily seen,
D.61.3) Q(a)(x) = 2ir~l sin Tra {— Q^lW(:
The function Qi"'w(x) satisfies the differential equation D.2.1); this follows
from D.6.2) since the segment [— 1, + 1] which is the path of integration,
satisfies the condition (b) of §4.6. Differentiating D.2.1) with respect to /3, and
substituting /3 = — a — 1, we obtain the solution Q(a)(x) since Qo"'^(x) = const.
if/3 = - a - 1.
B) Theorem 4.61.2. The following representations hold:
D.61.4) Qia^(x) = i(x - ir%r + 1)"" / ' A - 0"(l + 0" Pn ^- dt
J-\ x — t
_ on+a+^ T(n + a + l)T(n + /3 + 1) , _ ._„_„_!, ._(}
i x/
; 2n + a
(a)(x) - log (x + 1)
1 - x>/ ^ V A"l 2
D.61.6) + V 1 - x>/ ^ V " A"l 2 + +
c = r'(l) - ^^ - log 2.
T(-a)
By use of Rodrigues' formula, D.61.1) may be integrated by parts n times.
This establishes D.61.4). From D.61.1) we readily obtain
(x-l)a(x+l/Qia^(x)
= (-2)-"-1 E (n + ") A - Z)-"-'-1 / +1 A - 0n+'+"(l + t)n+fidt
„-<> \ w / 7-i
+ ^ + a+ l)r(n + /3+l) / 2 V+>+1
TBn + v + a + 0 + 2) \1 - x/

[4.61 ] APPLICATION; FUNCTIONS OF SECOND KIND 75
This, in connection with the notation D.21.3), yields D.61.5).
Turning to the exceptional case, we first notice that for n = 0, a + /3 + 1 ?¦* 0,
D.61.5) becomes
D.61.7)
For a + /3 + 1 =0 this will, in fact, be a constant since (A.7.2), second formula)
- T(a + l)r(|8 + l)(x - l)~a~\x + 1)""
= - \ ~- (* - 1)—\x +
2 Sin ira
Now from D.61.7), taking into account D.61.3), we obtain D.61.6).
C) The case n = 0 may be treated in another way by means of the relation
D.2.6). For z = 1 this becomes
D.61.8) A - x)a+1(l + z)"+y = const.
This yields an integral representation for y.
Theorem 4.61.3. Let a > — 1, /3 > — 1. We then have the following integral
representations:
D.61.9) ° T(a+^+l) U
if a + p +¦ 1 > 0,
|j[*[«-ir^1(« + ir/|-1
D.61.10)
¦a + ^ + 1 <0;
a
Q(a)U) = f'\(t- iya-\t + i)-^-1 -r^-Adt + log(x +1)
D.61.11) > L < + 1J
ifa + p+ 1 = 0.
In the first case the integrand is ~ Ca~~^~~2, as i —> <», so that the integral is
convergent. The constant factor in D.61.9) can be obtained by comparing the
principal terms in D.61.7) and D.61.9).
In the second case the principal term of the integrand is (a — I3)t~a~e~3, so
that the integral is convergent. In the third case the principal term of the
integrand is 2(a + l)(t + 1)~2, so that here too we have convergence.
D) Another very general integral representation of a second solution of D.2.1)
is obtained by choosing the path of integration in D.6.2) as in Jordan-Poch-
hammer's integral for the Gamma function (see the figure in Whittaker-Watson
1, p. 257). This path can be defined by the scheme

76 JACOBI POLYNOMIALS [ IV ]
D.61.12) (- 1 -), (+ 1 +), (- 1 +), (+ 1 -).
Condition (a) is then satisfied, x ^ =fc 1. The principal term is af" ~a-'3~1 pro-
provided the integral / A - t)n+a(l + t)n+$dt, extended over the contour in question,
does not vanish. This is the case (see loc. cit., p. 257) unless one of the following
conditions is satisfied:
D.61.13) a ' ' ' ' ' >>>•••>
2n + a + j8 + 2 = 0, - 1, - 2, ....
In the special cases, where a + /3 or a — /3 is an integer, the contour can be
simplified.
4.62. Further properties of the functions of the second kind
In the following considerations we again assume that a > — 1 and /3 > — 1.
A) The possible singular points of Qia'®(x) are + 1, — 1, <*>. In order to
discuss this function near x = + 1 (and also for later purposes), we write D.61.4)
in the form
/+1 p(a,0)/ \ _ p(a,/S)/,\
A - 0°(l + 0" — ———=~—- dt
i x t
D.62.1) + | (x — l)~a(x + l)~"Pla'w(a;) / -—-— dt
J-x x - t
= — \{x — \)~a{x + I)"" / A — t)a(l + t)e— ~^- ~ dt
J-i x — t
The last integral is a irn-\ in x (a constant multiple of the numerator Rn(x) of the
nth convergent of the continued fraction defined in §3.5; see the first part of
C.5.7)). Therefore, if x approaches + 1, the behavior of Q(na'^(x) is to a certain
extent determined by that of Qo"' \x).
The discussion of Qi"'?)(x) near x = + 1 is not difficult. Expanding the
factor (t + I)"" in the integrand of D.61.9) into a power series in t — 1, we
obtain for a + /3 + 1 > 0, a not an integer,
Qla>"\x) = const. + (x - iyaM
here M(u) is a power series of u, convergent for j u \ < 1, and M@) 9^ 0. A
similar representation holds ifa + /3-fl<0 [D.61.10)]. (In the exceptional
case a + |8 + 1 = 0, this is not true for Ql0"'$\x); however, it is true for Q(a\x),
cf. D.61.11).) Now, let a be an integer; then we use D.61.9) again. In view of
+...+(^+^(^y+
the integration furnishes a logarithmic term. Thus,

4.62
FURTHER PROPERTIES
77
const. + (x - l)"aM,
D.62.2) Qf-» (x) = ¦
if a > - 1, |8 > - 1; a ^ 0, 1, 2, • • • ; a + /3 + 1 ^ 0,
(-1)'
2
if a =-- 0, 1, 2, • • •;/8 > — 1.
Here Mi (it) and M2(u) are power series convergent for | u \ < 1 with Mi@) ^ 0,
M2@) ^ 0 (see below). A representation similar to the first one holds for Qia)(x).
We have, for instance,
D.62.3)
dt
1 , x + 1
2 6 x - r
[The statement M2@) ^ 0 requires further comment for a = 0. Taking
x > 1, and then integrating by parts, we have
[' (t- l)-'(
J<*>
= (x + l)-^1 log (x - 1) + (/3 + 1) f* (/ + I)-*-2 log (t - 1) dt,
J
M2@) = lim {Q(oaJ)(x) + ¦§ log (x - 1)}
so that
= -OS + 1J* / (t + I)"* log (t -\)dt* 0.]
B) Now we prove the following theorem:
Theorem 4.62.1. Let x be real, x > 1, and take (x — l)a, (x + l/ real and
positive. We then have, for x —> 1 +0,
'(x - l)~a, a > 0,
D.62.4) Qia^(x) ~ log(x - 1), a= 0,
1, a < 0.
More precisely,
' ! r(a)r(n + 0 + 1) , . iV_a
D.62.5) QiaJ)(x)
(-1)", 1
^ x - r
The behavior near x = — 1 of Qia'^(x) is similar.
The case a > 0 follows from D.61.1):
a
= 0.
f+
n+p
t)n+pdt.
In the case a = Owe use D.62.1), D.62.2), and P^lW(l) = 1. In the case a < 0
the first term of the right-hand member of D.62.1) vanishes as x —> 1 + 0.

78 JACOBI POLYNOMIALS f IV ]
Thus the statement is equivalent to Qo"'fi){x) ~ 1. This is immediately clear
from D.61.9) if a + 0 + 1 > 0. If a + 0 + 1 < 0, we use D.61.10) and show
that
D.62.6) r [(t - \ya-\t + i)-"-1 - r*-*-2] dt + —i—, < o.
Ji a + /3 + 1
On writing
we see that D.62.6), as a function of a and /3, increases with a if a + /3 is constant.
But when /3 approaches — 1, we find for the left-hand member of D.62.6) the
result
f* 1
/ [(t - I)"" - f"] dt + ~ = 0.
We have, as an instance, [D.62.1), D.62.3)]
D.62.7) QBM)(z) = Qn(x) = R(x) + \Pn(x) log ^-±-j <
where R{x) is a ?rn_i. The logarithmic factor is chosen so that it tends to 0
as x —> oo.
C) Theorem 4.62.2!. Let a be an integer, a > 0. We consider Q(na^(x) (real
and positive for x > 1) in the complex plane cut along the line [— oo, + ]]. Then
D.62.8) Qia^(x + iO) - Qia^(x - ?0) = (- \)a~\iP{na^(x),
- 1 < x < + 1.
This follows from D.62.1) and D.62.2).
On the other hand, the function
D.62.9) Qi"'»(x) = \[Qna'»{x + ?0) + Q(na^(x - ?0)}
is analytic on — I<x< + 1 and satisfies the differential equation D.2.1).
As x —> 1 — 0, it displays a behavior similar to that of Q(na'^(x). In particular,
wefindforQiM)(z) = QB(x)
D.62.10) Qn(x) = R(x) + \Pn(x) log \~^x>
D.62.11) QB(- x) = (- l)n+1Qn(x), - 1 < x < + 1,
D.62.12) lim Qn(x) = + ».
x-»l—0
Here /?(x) has the same meaning as in D.62.7).
In general, if a and /3 are both integers, the function Qi"'®(x) is regular and
single-valued in the whole plane cut along [— 1, + 1].
D) The functions of the second kind satisfy the same recurrence formula as
P{na^(x) [D.5.1)], that is,

4.62 ] FURTHER PRORERTIES 79
2n(n + « + |8)Bn + a + |8 (^
D 62 13) = Bn+a + /3- 1){ Bn + a + 0)Bn + a + 0 - 2)x + a2 - /32)
•Q?f(x) - 2(n + a - l)(n + 0 - l)Bn + a + 0)Q?? (x),
n = 2, 3, 4, ....
This follows from D.62.1) on account of C.5.3) and Theorem 3.5.1. There is,
however, an essential difference if n = 1. We then have, according to D.62.1),
Qi"^\x) = h [(« + 0 + 2)x + a - 0] Q(oa^(x)
- 2—(a + 0 + 2) ^±^^ (x - D-(x + 1)-.
Therefore, both systems of functions [D.3.3)]
D.62.15) M) l: V" {X)>
qn(x) = lh{na-f>\-iQi'"»(x), n = 0,1,2,..-,
satisfy the same recurrence formula of the type C.2.1) for n ^ 1, provided we
define
D.62.16) p_x(x) = 0, q^(x) = (x - iya(x + 1)^.
Thus a procedure similar to that used in §3.2 B) furnishes, for n ^ 1,
qn{y) - Pn(x)qn+1(y)
_
kn+i x — y
D.62.17)
= Pn(x)qn(y) + *^
k
kn x — y
Here kn denotes the coefficient of xn in the "normalized" polynomial pn(x).
This formula also holds for n = 0 if we modify it as follows:
D.62.18) ?° P'(*>ftfa)nP>(*>g'(y> = Mx)qa(y) + const. ?^ .
Adding, we obtain the important result:
D.62.19) =i
The constant | in the right-hand member can be determined by substituting
n = 0, multiplying by ya+?+1, then permitting y —> <», and finally using D.61.5).
We shall return to this formula in §9.2, where it will be used in a classical
manner for the expansion of an analytic function in terms of Jacobi polynomials
or of Jacobi functions of the second kind.

80 JACOBI POLYNOMIALS [ IV ]
4.7. Ultfaspherical polynomials
A) If a = /3, Jacobi's polynomial Pna'p)(x) is called an ultraspherical poly-
polynomial.21 The following is the customary notation and normalization:
7
( ) (n + 2X) (x_i,x-|), >. ,
I) n (r)> a = ~ '•
r(n + x +
Here we assume first that a > - 1, or X > — ?. Some important special cases
are (cf. A.12.3))
D.7.2) P<*>(z) = Pn(x), P[l\x) = Un(x).
If a = - \, or X = 0, the polynomial PnK\x) vanishes identically for n ^ 1.
This case will be treated later. (Cf. D.7.8).)
We next observe a number of formulas and theorems which can be obtained
immediately from the theory of general Jacobi polynomials by setting a = /3 =
X - i X > - i;
D.7.3)
D.7.4) P{n\- x) = (- l)nPiX)(x);
D 7 5) A ~ X2)V" ~ BX + 1)Xy> + n(n + 2X)^ = °' V =
A - x2)F" + BX - 3)xY' + (n + l)(n + 2X - 1)F = 0,
F= A _x2)x-}
+ 2X " 1)(
( n )(- „, n + 2X; X + -
D.7.6)
The last formulas define P™(x) for all values of X. If necessary, for some
special values of X, say X = Xo, the formulas may be interpreted as limits for
X —¦> Xo .22 For X = — m, m = 0, 1, 2, • • • , we obviously have (cf. the first
formula D.7.6)) PnX)(x) = 0 if n > 2m. In this case
lim " •= <-j-
x-»—m K-\- m la A jx——m
D.7.7)
,B771IG1-2771-1)! „, o . !
CTf—n, n —2m; —m+l;
n!
exists. These polynomials are, except for constant factors, again the Jacobi
polynomials Pna<a\x), a = - m - $. For instance (cf. A.12.3))
D.7.8) lim \"lP{n\x) = B/n)Tn(x), n ^ 1.
21 They are sometimes called Gegenbauer's polynomials. See the papers of Gegenbauer
A-7). See also Heine 3, vol. 1, pp. uy7-301, 449-464. Occasionally, the notation Cnx(x)
is used instead of Pn (x).
22 This should also be done in all the subsequent formulas.

[4.7] ULTRASPHERICAL POLYNOMIALS 81
In case a = — I, (n + l)/2 ^ Z ^ n, the polynomial P^'a)(x) vanishes identi-
identically (§4.22 C)). However, the corresponding expression PiX)(z), as a limit, has
a meaning and does not vanish identically.
B) Further formulas involving P^ (x) are
D.7.9) lim x~nP^(x) = 2n(n + X
*-» \ n
D.7,0) g J^
D.7.11)
D.7.12, a - ^,f w - (-# r4g^±^ (|)- a -
(,7,3) PrW
D-7.14) ~
D.7.15)
X > - i X ?* Of
r n
D.7.16) "o r(X + h) r(n + 2X)
= 2X-}A - 2xw + ^2r}{l -xw+(l- 2xw
U717, '(X) = 2(n + X - D^-i(*> - (n + 2X - 2)Pixi2(x), n = 2,3,4, • • • ;
D.7.18)
1 — x
C) We obtain as a second solution of D.7.5), which is linearly independent
of PlX)(x), (cf. D.23.1), D.23.3), D.61.1), D.61.4), D.61.5))
V = A ~ X^F(~ n ~ X + *' n
X 9* - n + 1/2, - n + 3/2, - n + 5/2, • • • ;24
23 Use A.7.3), n = 2. In the limiting case X —> 0, n ^ 1, we multiply D.7.15) by X~2,
X -> 0 [D.7.8)].
24 For X = — n — 1/2, — n - 3/2, • • • this is a polynomial linearly independent of P^ (x)
(cf. §4.23 A)).

82 JACOB I POLYNOMIALS [ IV J
D 7 20)
V={1~ x>~n~2XF(n + 2X, n + X + I; 2n + 2X + 1; j-
X ^ - n/2, - n/2 - 1/2, - n/2 - 1, • • • ;25
y = A - z2)}-x f+1 A - e)n+x~\x - I)-"-1 dt
D.7.21)
= const. A - z2I-* / A - t2)^^— dt, X > - 1/2, X ^ 0;
y = A - x)-n-x-}(l + x)*-xF(n + X + §, n + 1 ;2n + 2X + 1; .-J
D.7.22)
X ^ - n/2, - n/2 - 1/2, - n/2 - 1, •...26
According to Theorem 4.23.2 the general solution of D.7.5) is a polynomial if
and only if X — | is an integer and X ^ — n/2.
D) Another generating function, essentially different from D.7.16) (cf.
Problem 16), and much simpler, is often used as a definition of the ultraspherical
polynomials, namely:
PiX)(s) + P?\x)w + P?\x)w2 + • • • + P{n\x)wn + ...
D.7.23)
For the proof, we consider the recurrence formula D.7.17), from which
00 00 00
E nP(n)(x)wn~1 = 2x E (n + X - DP^,^)/ - E (n + 2X - 2)P^l2{x)wn~l
n"l n"l n = l
and in which we define P-i(x) = 0. If the left-hand member of D.7.23) be
denoted by h(w), the hist equation may be written in the form
h'{w) = 2xwl-\v?h{w))' - w2-2\wnh(w))'
= 2x{\h(w) + wh'{w)\ - \2\wh(w) + w2h'(w)};
or h'(w)/h(w) = 2\(x - w){\ - 2xw + w2). Since h@) = P^(x) = 1,
D.7.23) follows by integration.
On differentiating D.7.23) with respect to X, we obtain
D.7.24) - A - 2xw + w2)~x log A - 2xw + w2)
which, for X = — m, m = 0, 1, 2, • • • , gives the generating function of the
polynomials D.7.7), provided only those terms are considered for which n > 2m.
These polynomials are, as mentioned in A), constant multiples of the Jacobi
polynomials P{n'"\x) with a = — m — \. The polynomials of degree n ^ 2m,
defined by this new generating function D.7.24), are essentially different from
26 Cf. §4.23 B).
26 Formula D.61.5) has been considered only under the restriction a > — 1, 0 > — 1.
We see immediately, however, that except for the indicated values of X, the function D.7.22)
is a solution which is ~/x"""~2X as x —+ »,bo that it cannot be a polynomial.

[4.7]
ULTRASPHERICAL POLYNOMIALS
83
the P(n\x) which are given in this case by the corresponding terms of D.7.23).
We have, for instance, [D.1.7)],
iO)
PiO)(x) = 0,
D.7.25) X = 0
~ l0g (l ~ 2xW
)w
2-4 ••• 2n
27
D.7.26) X = - 1
»-in 1-3 ••• Bn~ 1) " " ' '
P^ix) = 0, n ^ 3,
— A — 2z«; + w2) log A — 2xw + w2) = 2xw — Bx2 + l)^2
- 32.T 2-4 •••Bn-6) p(_|iH
P
1-3 •¦¦Bn-3) "
E) From D.5.5), D.5.6), D.5.7), D.7.14), we obtain the relations
x)±{Pl\x)}
= [2(n + X)]-1!^ + 2X - l)(n
= _ nxP™(*) + (« + 2X -
= (n + 2X)xPiX)(x) - (n + l)P(X(x)
= 2XA - x*)P^\x),
We can then derive the following identities:
- n(n
n ^ 0, Pi'i'Ca;) = 0.
D.7.28)
ax
(n
ax
ax
- x~
ax
Adding these formulas we find, by use of D.7.14),
— [Pnli
dx
D.7.29)
- PlCliix)} = 2(n
n ^ 1, Pl-i(x) = 0.
F) Finally, we give some special formulas involving hypergeometric functions.
Combining D.1.5) and D.21.2), we obtain
X; X + 1; 1 -
D.7.30)
27 The first of these identities can be derived directly by writing 1 — 2w cos 0 + w2
A - u>e'»)(l - we"")-

84 JACOB I POLYNOMIALS [ IV
Jf)xF(~ v, v + X + 1; X + 1; 1 - x2)
+ 1 /
D.7.30)
The constant factors can be calculated by substituting x = 1 and comparing
the highest powers [or by computing Pu{0) and P?+i@) from D.7.23)].
Another way of writing the second part of D.7.30) is the following:
(.4.7.31) Pn {x) = 2^ ( —1; r/.>,r/ i nTV o . n Bs)
1 {X)L {m\ l)r{n 2m + 1)
m-\- l)r{n — 2m )
This last expression is an explicit representation of the ultraspherical poly-
polynomials. (Cf. Problem 15.)
The formulas D.1.5) may be proved by a more general consideration. For
instance, in the first case, we may start from the corresponding differential
equations
A - x2)y" - 2(o + l)xy' + 2^B^ + 2a + 1J/ = 0,
D.7.32)
A _ zV - {« + ? + (« + \)x)z' + v{y + a + h)z = 0
and show the relation y{x) = zBx2 — 1) between their general solutions. This
argument furnishes at the same time relations between the non-polynomial
solutions of D.7.32) similar to D.1.5). To be specific, let us replace the quanti-
quantities n, a, /3, x in the expression D.23.3) first by v, — |, a, 1 — 2x2, respectively,
and then by v, -\-\, a, 1 — 2x , respectively; we then find (in the second case
after multiplying by x)
y = x-n-2a~xF{[{n + l)/2] + a + 1/2, [n/2] + a + 1; n + a + I; aT2)
D-7-33)
~n-2XF((n + l)/2 + X, n/2 + X; n + X + 1; x~2),
as a second solution of D.7.5), which is linearly independent of P(n\x), provided
X^ -[(n+ l)/2] ~k;k = 0,1,2, ....
Starting from the first formula in D.7.21), we find, save for a constant factor:
y = A - n-V-V-1 E (" + 2v\x-2' l+X A - fy^f'dt
- n _ 2>.j-x -n-i V /« + 2A r(n + X + h)T(v + I) -*,
D.7.34) '-V 2^/ T(n + X + .+ D
_ r(n + x + l)r(|), 2xj-x-n-i
- r( + x+i) [l~x) x
¦F((n + D/2,. 1 + n/2; n + X + l;x~2).
For further properties of ultraspherical polynomials the reader may consult
Whittaker-Watson 1, pp. 329, 330, 335, and the literature quoted there. See
also Wangerin 1, pp. 730-731. The function Cn(x) of these authors is identical
with P^ (x) in our notation. The Legendre associated functions P™(x), m an
integer (cf. Hobson 1, p. 90), can be represented in the form (cf. D.21.7))
D.7.35) Pmn(x) = const. A - x2)ml2P{^i\x).

.4.8] REPRESENTATIONS FOR LEGENDRE POLYNOMIALS 85
4.8. Integral representations for Legendre polynomials
In the important case a = /3 = 0, that is, for Legendre polynomials, the
method of §4.6 leads to various integral representations. According to D.6.1),
D.8.1) RW I
2iri J \2 t - x ) t — x
Here the path of integration encloses the point x. At present, the position of
this contour with reference to the points t = ± 1 is immaterial.
A) Integrals of Dirichlet-Mehler. (Cf. Dirichlet 1, Mehler 5.) Take x in
the interior of the interval [— 1, + 1], so that x = cos 0, 0 < 0 < ir. For the
contour in question we choose the circle
D.8.2) 11 - 1 | = | eiB - 1 | = 2 sin % t=l + 2 sin d--e
1 2
so that
2 _ - 1 + sinr-e1'*
D.8.3) l«l 2
i\
2t ~ cos0 . e _V
1 + sin ~e *
Next let ^ vary from — ?r to + *", then 1 + sin Ifl-e** describes the small circle
in the figure. The expression in the right-hand member of D.8.3) has the
a = sin 9/2
Via. 3
absolute value 1 and an argument twice that of the numerator. Thus, if we
write
D.8.4) I ^
2 t — cos0
the quantity <f> varies from — 0 to + 0 and again from + 0 to — 0. Solving
D.8.4), we obtain
D.8.5) t =-- e* + eim{2 cos <f> - 2 cos 0)}.
Here the positive value of ( )' corresponds to | \p | < (ir + 0)/2, that is, to
the "exterior" arc, while the negative value corresponds to | ^ | > (ir + 0)/2,
that is, to the "interior" arc. Furthermore, from D.8.4) it follows that
tdt = e{*dt + (t - cos 0)iV*d<f>;
whence

86 JACOBI POLYNOMIALS [ IV
dt _ ie^df _ ie^nd<t>
t - cos 0 ~ r^e** ~ B cos <f> - 2 cos 0)*"
Finally,
el*/2d<f>
- 2cos0)»'
or,
D.8.6) P. (cos 0) = 2 f ° °2^L+1>* ^,
7t > Bcos<^. - 2cos0)J
where the square root of 2 cos <i> — 2 cos 0 must be taken with positive sign.
This is the first formula of Dirichlet-Mehler. Substituting v — 0 for 0 we ob-
obtain, because of D.1.3), the second formula
D.8.7) Pn (cos 0) = 2
[!Li d+.
7r Je B cos 0 — 2 cos <?)'
B) Fzrsi integral of Laplace. (Cf. Whittaker-Watson 1, pp. 312-313.) Let
x be different from ± 1, and choose the circle
D.8.8) 11 - x | = | x - 1 I*
as the contour of integration. Writing t = x -{- (x2 — 1)V* (with an arbitrary
but fixed determination of (x2 — 1)*), we find that
D.8.9) \ l~^ ==x+ (x2 - 1)» cos 4, -*-
Consequently,
D.8.10)
JiTT J-t
= IT
~J ' 'x + (*2 - iy cos<t>\nd<i>,
an expression known as Laplace's first integral. It holds for arbitrary values
of x.
C) Second integral of Laplace. (Cf. Whittaker-Watson 1, p. 314; Jacobi 2,
p. 153.) This integral is given by
(x2 -
D.8.11)
/ "
= 7r-1 I {x + (x2 — 1) COS^J""^.
0
It may be derived from the first integral of Laplace in the following manner.
Let 0 < r < 1 and
D.8.12) 1 ~ r2> 1 ~ r2>
x + (x - 1)J cos 4> = A - r2) 11 + rz |2, z = e'*,
whence,

4.8
REPRESENTATIONS FOR LEGENDRE POLYNOMIALS
87
D.8.13)
Pn(x) =
2tt
rz
hl\dz\.
Now, substituting
w — r
D.8.14) z =
1 — no'
1 + rz =
1 -r2
dz
1 -r2
1 — rw' dw A — rtt>J'
28
we obtain
D.8.15) Pn(x) =
2tt
Substituting W = —el* , we obtain D.8.11). Using analytic continuation, this
formula may be immediately extended to arbitrary complex values of x.
D) As a special case of D.4.9) we note the representation
D.8.16) Pn (cos 0) = —. / w
-n-l
A - 2w cos d + w2) hdw.
Fir,. I
The integration is extended in the positive sense along a contour enclosing the
origin but neither of the points e±%\ By a proper choice of the contour, formu-
formulas D.8.6) and D.8.10) can be derived again from D.8.16) (see Polya-Szego 1,
vol. 1, pp. 115, 287-288, problem 157).
E) Integral representation of Stieltjes. (Stieltjes 8.) Suppose 0 < 0 < w.
This important representation can be obtained from D.8.16) by using the con-
contour in the figure. (The derivation from D.8.1) seems too complicated.) Since
the integrand is O(w~n~2) as w —¦> «>, the contribution of the large arcs tends to
zero as the radius becomes infinite. The same is true of the contributions of
the small arcs around e±tS (the integrand is O[\ w — e±l8 \~h\ there). Thus,
we have
Pn(cos 0) = 29? ~ I w~n~\l - 2w cos 0 + w2)~h dw,
Zwl J
extended twice over the straight line w = t~le\ where t increases from 0 to 1
and then decreases from 1 to 0. We have, then,
- 2w cos d
te~i9{\ - t)~\l -
28 Concerning this argument see Szego 21.

88 JACOBI POLYNOMIALS [ IV ]
where the signs + and — correspond to the first and second cases, respectively
[A - tyh = A - te~2ieyh = 1 for f = 0]. Therefore,
Pn(cosd) =49?—. / tn+le~i{n+l)ete~ie{l - t)~\l - te~2ie)~\-1~2eie) dt
2m Jo
M-8'17) 2 f r 1
= - -3^ ' t A — t) \1 - te ) 'dt}.
ir { Jo )
F) Further integral representations are obtained by replacing n in D.8.1) by
— n — 1 (cf. D.6.4)) and observing the conditions (a), (b) formulated in §4.6
(here the expression A — t )~"(t — a;)" must be considered). We then have
D.8.18) y =
1 / fit2 - A""'1 dt
2ri J \2 t - x 1 t - x
Integrals of this type may represent Legendre polynomials as well as Legendre
functions of the second kind, or a proper linear combination of these, according
to the special choice of the path of integration.
By using the contour in A), we again obtain Dirichlet-Mehler's formula
D.8.6), since the expression in the right-hand member does not change if we
replace n by — n — 1. The same procedure transforms Laplace's first integral
D.8.10) into Laplace's second integral D.8.11), and conversely. Thus, choos-
choosing the contour in D.8.18) as in B), we obtain the second integral. The ex-
expression in the right-hand member of D.8.11) cannot represent a solution other
than Pn(x), since it is finite and equal to + 1 at x = + 1.
4.81. Legendre functions of the second kind
A) Let x be in the complex plane cut along the segment [— 1, + 1], and x =
§(z + z~l), I z I < 1- We deform the path of integration in D.61.1) into the
circular arc through — 1, z, + 1 (cf. Whittaker-Watson 1, p. 320, example 1).
This deformation is permitted since sgn !$x = —sgn !$z. Then
/ = (z + De + (z - 1)
(z+ IV - (z - 1)'
D.81.1)
It — 1 dt 1 . , 2 ,\ u i-i
2T^X=drX-^-t=[X'+{X ~l) C°ShTl •
The new variable t is real and ranges from — co to + °° ; furthermore {x2 — 1)*
^ x as x —> oo. This furnishes the following integral representation for
Qi°'0)(x) =- Qn(x), which is very similar to Laplace's second integral:
D.81.2) Qn(x) = h / [x + {x2 - l^coshTp'1^.
—00
B) Let x be real, x > 1; we write x = cosh f, f > 0. Introducing in D.81.2)
cosh f + sinh f cosh r = e ,
D.81.3) dr = e» = e9/2
d0 sinh f sinh t B cosh 6 — 2 cosh f )*'
we obtain

[4.82] GENERALIZATIONS 89
D.81.4) Q»(coshf)=/ ~ r-Kide.
J[ B cosh 0 — 2 cosh f)>
In this formula (Watson 2, p. 154) we first assume that f > 0. By analytic
continuation it can he extended to the half-strip A.9.3). The path of integra-
integration is the horizontal"line 9t@) ^ 9?(f), $@) = $(?).' This formula corresponds
to Dirichlet-Mehler's formula.
Flo. 5
4.82. Generalizations
A) Generalization of the Dirichlet-Mehler integral. (See Fejer 12.) Suppose
0 < 0 < ir and 0 < X < 1. According to the generating function D.7.23) we
have
D.82.1) PiX)(cos d) = ~ f w~n~\l - 2w cos 6 + w2)~x dw.
Here we take the unit circle \w\ = 1 as the path of integration, avoiding the
singular points w = e±x , in the neighborhood of which the integrand is
0{\w- e±ie Tx}. (See Fig. 5.) For w = j\ 0 ^ <f> < 6, we have
A - 2w cos 6 + w2)~X = eriH(w~l - 2 cos 6 + wO* = e~iHB cos <f> - 2 cos 0)~x,
and for w = e1*, 6 < <f> ^ ir,
A - 2w cos 6 + w2)~X = e~a<*~T)B cos 0 - 2 cos <^.)"x,
so that
PiX)(cos0) = 29?(^-. re-l'(B+1)Vx*B cos <^» - 2 cos
[Z7TZ >
27TZ
whence
- 2 cos
PiX)(cos 0) == tt" / F{<f>) | 2 cos <^ - 2 cos 0 |~x d<?,
D.82.2)
cos {n + \)<t> if 0 ^ <^» < 0,
cos [(n + X)<^> — Xtt] if 0 < <f> ^ 7T.
The expression resulting from D.82.2) for X = 1/2, is the sum of the expres-
expressions D.8.6) and D.8.7).
B) Generalization of the integral of Stieltjes. (Stieltjes; Hermite-Stieltjes 1,
vol. 2, p. 122, no. 284.) If we choose in D.82.1) the same contour as in §4.8 E),
an argument similar to that used there leads to the representation

90 JACOBI POLYNOMIALS [ IV
D 82 3) F"X)(COS 6) = B/7r) Sin Xir${eim jf r+2X"^1 - O"XA ~ te™)-*
1@) = (n + 2XH +(i - X)tt; 0 < 0 < tt, 0 < X < 1.
The formulas in §4.8 A) and B) can be extended to Jacobi polynomials
P(naJ)(x) for integral values of a. The resulting representations, however, are
rather complicated. Finally, we notice the following generalization of D.81.2)
arising from D.61.1) in a manner similar to that used in §4.81 A) (notation the
same as there):
•[X-\-(x — 1) COShrj dr,
X = |(z -\- z~l), I 2 I < 1; a > — 1, /? > —1,
with a proper determination of the functions involved.
4,9|. Trigonometric representations
A) Finite cosine expansion of Legendre polynomials. From the generating
function D.7.23) we obtain, for X = §,
A - 2w cos 0 + w2)~h = A - iee")~*(l - we~ie)'h
D 9 1) °° °°
V ' ' ' _ V m imS V^ m -itrf
where
D.9.2) g0 = 1, ^^^T^l^' m = 1,2,3, •••,
so that
Pn(cos 0) = i: gmeimegn-me-i{n-m)e
n n
gmgn~me = /_, gmgn-m cos [n — 2mH
= 2^-. cos nd + 2grigrn_i cos (n - 2H + • • •
|(s(+, nodd,
I 2
[n even.
Consequently, Pn(cos 0) is a trigonometric cosine polynomial with non-negative
coefficients.
Relation D.9.3), as an identity in 0, can be written as follows in terms of
Tchebichef polynomials:
Pn(x) = 2gognTn(x) + 2gign^Tn^(x) + • • •
D.9.4) j2gr(n_i)/2Sr(n+i)/2Ti(a;), n odd,
[gin, n even.
B) Infinite sine expansion of Legendre polynomials. (Heine 3, vol. 1, pp.
19, 89.) We have

[4.9] TRIGONOMETRIC REPRESENTATIONS 91
4 9.4... 2n
Pn(cos0) = - —' j~-r\ {/«sin (" + l)° +/» sin (« + 3H + • • •
X o • 0 • • • Bn + 1)
D.9.5) + /,sin(n + 2i/+1H+ •••},
f =1. f =f = (" + D(n + 2) ••• (n.+ r) „ = 1 2 3 • • •
For n = 0 the factor B-4 • • • 2n)/C-5 • • • Bn + 1)) must be replaced by 1.
Abel's transformation (§1.11 D)) shows this expansion to converge for the values
0 < 0 < 7r, and even uniformly for e ^ 0 ^ 7r — e, 0 < e < t/2. The conver-
convergence may be deduced also from the elements of the theory of Fourier series. In-
Indeed, D.9.5) is the formal expansion of the function denned by Pn(cos 0) for 0 < 0
< 7r, andi by — Pn (cos 0) for — v < 0 < 0 (see below). It is a generalization
of the classical expansion (n = 0)
D.9.6) tt/4 = (sin0)/l + (sin 30)/3 + (sin 50)/5 + • • • .
First proof (Heine, loc. eit.; cf. also Feje"r 20, pp. 24-26). Using the notation
A.12.3), we have
D.9.7) / Pn(cos 0) sin (m + 1N do = / Pn(t)Um(t) dt.
This integral vanishes if m < n, or if m ^ n, and m — n odd. Now write
m = n -\- 2v, using D.9.3), we find
/ Pn(cos 0) sin (m + 1H dd =*= 22 gkgn-k / cos (n - 2A;H sin (m + 1H dd
Jo *-o J0
D.9.8) = t,
k
-k _ V1 Qkgn-k
Considering casa continuous variable momentarily, we easily verify (by calcu-
calculating the residues) that
U Q 0\ V 9kgn-k = (y+l)(y + 2) ••• (y + n) 2 • 4 • • • 2n
^ ; «^+A;+§ (»'+i)(i' + |)(»'+n + J)
§ (i)( |)( J) 3-5-.• Bn+l)J"
which establishes the statement.
Second proof (see Fej^r 19, pp. 202-203). Expansion of the last factor in
the integrand of D.8.17) gives
D.9.10) Pn(cos d) = - ^L^+l)e E gjimi I tn+m{\
IT { 0 JO
IT { m-0
- t)~h dt),
)
where gm has the same meaning as in D.9.2). The last integral can be calcu-
calculated by means of A.7.5), and the statement is thus established.
Third proof. We use mathematical induction with respect to n. According
to the recurrence formula of Pn(x) (D.7.17), X = \), it suffices to show that
n2
= 0, 1,2, ••-,« = 1,2, 3,

92 JACOBI POLYNOMIALS [ IV ]
For n = 0 we have D.9.6). The identity D.9.11) can be verified by direct
calculation.
Remark. Formula D.9.5) is closely related to certain expansions of the
functions of second kind. We find from D.61.4), a = /3 = 0,
D.9.12) Q(.°'0>(x) - Qm(x) = I f'1 ^ dt.
2, 7-i x — t
If we substitute x = §(w + uT1) (§1.9), the function Qn{?(w + w)} will be
regular for | w \ S 1, w ^ rt 1. Using D.7.23), X = 1, we obtain for \ w\ < 1,
+1)r j® r
r dtw r
\ 1 — itVO -\- WL J-\
so that, because of D.9.7), D.9.8), D.9.9),
D.9.13) Qn[h(w + vT1)) - 2 -J;' 4
Now let | w \ < l,w-> e", 0 < 8 < ir. Then \(w + iv) -> cos fl - zO, so that
D.9.14) Cn(cos 6 - iO) = 2 . 2'4Vo'2In E/^(n+2'+1)9, 0 < 6 < x.
o • 5 • • • Bn + 1) v-=o
(Here use was made of the convergence of the last series and of Abel's con-
continuity .theorem; see Titchmarsh 1, pp. 9-10.) Replacing i by — i, we obtain
D.9.15) Qn(cosd + iO) = 2 . l'*'?,, E f.e^*"*1"\ 0 < 6 < r.
o {Zn \ l)
r-=o
From here D.9.5) follows again because of D.62.8), a = P = 0. By use of
D.62.9), a = 0 = 0, we find (Heine 3, vol. 1, p. 130)
2 • 4 • ¦ • In i^»
D.9.16) Qn(cos0) = 2— ___E/,cos(n + 2J, + lH, 0 < 6 < ¦*.
6 • o • • • {Zn -\- 1) v-=o
Another variation of these considerations (see Hobson 1, pp. 57-58) is to
introduce x = %(w + itf1) and y = wn+lz into D.2.1). Then z, as a function of
w2, satisfies a hypergeometric differential equation from which D.9.13) (therefore
also D.9.5)) again follows.
By means of D.9.9) we see that the sequence {/,.} is completely monotonic
(cf. §6.5 D)).
C) Another trigonometric representation oj Legendre polynomials. (Stieltjes
7, 8.) Starting again from D.8;17), we write
A - O~* = eWi-emB sin flHl - A - t) e——\ .
^ A sin x) j
Consequently, for 2 sin 0 > 1
°° -iV(«-ir/2)
D917)
D.9.17)
where

4.9 ] TRIGONOMETRIC REPRESENTATIONS 93
D.9.18) h=l; h, = hm = g,. * *''.; (" 7 ?}^1N, v = 1,2,3, •
Expansion D.9.17) is convergent if tt/Q < 6 < 57r/6. Its importance will be
more fully realized in discussing the asymptotic behavior of Pn(cos 6) for large
values of n (§8.5). Concerning the connection of this representation with that
of Heine [D.9.5)], see Stieltjes 8, p. 244.
D) Ultraspherical polynomials. From D.7.21) we obtain as in A):
PiX)(cos d) = 2a0an cosnd + 2ctian-i cos (n - 2H -\
D.9.19) |2a(n_i)/2a(n+i)/2 cos 0, nodd,
[«n/2, n even,
where
D.9.20) A - u)~X = a0 + axu + a2u + • • • + anun + ¦ ¦ ¦ .
Therefore,
D.9.21) «" = (n + n ~ 0' n = 0, 1, 2, ....
Here the cases X = 0, — 1, — 2, • • • are to be excluded. In particular, if
X > 0, the coefficients an are positive, so that PiX)(cos 6) as a trigonometric
cosine polynomial again has non-negative coefficients.
Expansion D.9.5) can likewise be extended. We have (Szego 19, pp. 508-
509) for X > 0, X ^ 1, 2, 3, • • • , 0 < 6 < it,
D.9.22) ,(x) _ .. ,(x> _ ,(\)
Jo — 1, Jv — Jvn
A - X)B - X) • • • (v - X) (n + l)(n + 2) • • • (n + v)
\-2---v (n + X + l)(n + X + 2) • • • (n + X + v)'
v = 1, 2, 3, • • • .
The generalization of the third proof given in B) is particularly simple. The
special case n = 0 is
D.9.23) (sin OJ^1 = 27Tir(X + \) E {~~ X)_(,2'~ X) ,' \^v " X) sin B, + 1H.
It can be verified in various ways. (Cf. Whittaker-Watson 1, p. 263, problem
40; multiply this formula by cos x, and substitute x = r/2 — 6, s = 2X — 2.)
Another extension of D.9.5) is the following:
P^(cos0) = -1 f)«rf^ + L±^) cos{(n + 2v + 2XH - X,},
D 9 24) r(X) ,=o T(n + v + X + 1)
o < x < i,o <e <tt.
It follows readily from D.82.3) by means of the argument used in the second
proof given in B).
The extension of D.9.17) (Szego 17, pp. 57-60) is

94 JACOBI POLYNOMIALS [ IV ]
P<x>(cos0) = B/,) sin Xr dl±2X) ? r(, + \)r(, -• X + 1)
Di925) rM '"» "!T(n + , + X + l)
cos{(n+y+\)fl-(y+\),r/2} n x
(glHTdF* L , 0 < X < Ijtt/6 < 0 < 5tt/6.
It follows from D.82.3) by the same argument as used in C).
4.10. Further properties of Jacobi polynomials
A) Rodrigues' formula. A generalization of Rodrigues' formula D.3.1) is the
following:
—T)m+aC\ iT)m+0p(m+a,m+P)
X) {l+X) ^nm
D.10.1) (-l)m / d\m
= 1 Zl ( _ 1
2™n(n-\)---{n-m+\)\dx)
Here n ^ m. The proof is similar to that given in 4.3 A).
B) Integral representations for ultraspherical polynomials, (a) Combining
D.1.5) and D.3.2) we find
Pfra){x) = P?-a)(l) ¦ x»F(-v, -v + \; a + 1; 1 - a;-2),
Ptfi(x) = PlttflA) • x*+lF(-v, -v - i; a + 1; 1 - ar2)
so that, in view of D.7.1) and D.7.3),
x»F(-n/2, -n/2
D.10.3) T(X) n!
oi-2X r^r?
(n\ r(fc + §)
We used A.7.3), n = 2, and A.7.5). This generalization of D.8.10) holds pro-
provided that X > 0. See Gegenbauer 1, Seidel-Szasz 1.
(b) Another remarkable integral representation valid for non-negative integral
values of X — ? is the following:
D..0.4) _ ^^ ^^ I
• cos (X — \)<p d<p.
For the sake of simplicity we assume here that x > 1, (a;2 — 1)* > 0.
For the proof we use D.4.6) with the circle D.8.8) as contour of integration.
Since
t2 - 1 = 2(x2 - l)V>{a; + (a;2 - 1)* cos<p|
we have, a = A — i,
— x
1 /
= — / {a; + (a;2 - 1)* cos <?}"+«{2(a;2 - I)* ei<e\a d<p
Zir J
which yields easily D.10.4).

[4.10] FURTHER PROPERTIES OF JACOBI POLYNOMIALS 95
C) Generating functions, (a) A slight modification of the argument in
4.4 C) (proving the formula D.4.5) for the generating function F(x, w)) is the
following. Let —1 < w < 1; we write D.4.7) as follows:
D.10.5) x = t + \w{\ - t2)
so that x and t describe the interval [ — 1,-f-l] at the same time. Hence, using
the formula D.4.7) and the two first formulas of D.4.8), we find
A - x)a(l + XyF(x, w) dx = A - t)a(l + ty dt
so that for any non-negative integer n
r+i r+i
I A - x)a(l + xyF(x, w)xn dx = I A - t)a(l + ty\t + hw(l - t2)]n dt,
J-i J-i
i.e., a polynomial of degree n in w. This shows that the coefficients of F(x, w)
must be, apart from constant factors, the Jacobi polynomials. These factors
can be determined by writing x = 1.
(b) The following generating function holds for ultraspherical polynomials:
P^{x)w"
D.10.6) h nX)(D n! h T(n + 2X)
J) e™{{\ -
It is different from D.7.16) and D.7.23). Setting X = \ we obtain for the
Legendre polynomials:
D.10.7) ^
n=0 n-
For the proof of D.10.6) we use D.10.3); we obtain for the left-hand side
of D.10.6):
rr(x)]«
f'
/ exp[w{x + (a;2 - l)*cos»>}] • sin2X-V dtp
which is easy to identify with the right-hand expression in D.10.6).
Concerning further formal properties of the Legendre and ultraspherical
polynomials, see Bateman Manuscript Project, vol. 1, Chapter 3 and vol. 2.
Chapter 10. Cf. also Problems and Exercises, 61-66, 69-71, 84.
D) The following remarkable identity, involving ultraspherical polynomials,
is due to Feldheim 5, p. 278:
i) r(\ -
D.10.8) 'w^ZT f^^
1 — sin

96 JACOBI POLYNOMIALS [IV]
It is an easy consequence of the generating function D.10.6). Indeed, multi-
multiplying bothsidesof D.10.8) by w" and extending the summation over n = 0,1,2, • • •
we obtain
~ (l) JO
• (w sin# sinv?) /2~*'J^^^^iw sin# sin^>) dip,
or
•x/2
D.10.9)
/'
This identity is due to Sonine (Watson 3, p. 373, A)).
E) Positivily of certain sums. The first result of this kind is due to L. Fejer
4, p. 83:
D.10.10) P0(x) + P,(x) + P2(x) + • • • + PB(x) ^0 for -lgigl.
This fact is important in the study of summability of the Laplace series. It
follows from D.8.7) by observing the identity
" sinBi> + l)g _ /sin(n +
v=0 sin^ \ sin^
Generalizations of D.10.10) are due to Fejer 12, Feldheim 5, Szego 25.
F) The formulas D.8.6) and D.8.7) of Dirichlet-Mehler can be extended to
and
D1012) /»-l.,>0.-l<x<l.
Feldheim's integral D.10.8) follows by using the quadratic transformations
D.1.5) and the integral
D.10.13)
.... .... .. . ...^
a > — 1, m > 0, — 1 < x < 1. These three integrals are all special cases of
known integrals connecting hypergeometric functions which were found by
Bateman 2. See Askey-Fitch 2 for other useful integrals. Among the conse-
consequences of these integrals is the following theorem on positive sums.

[4.10] FURTHER PROPERTIES OF JACOBI POLYNOMIALS 97
Theorem 4.10.1. // 0 ^ n ^ v and p > - l and if
D-10.14) Za^L
then
Feldheim's integral D.10.8) can be used to prove a related theorem.
Theorem 4.10.2. // v > 0, a > - 1 and if
D-10.16) I^ptw^O, -l^x^l,
n p(a+v,a+v), s
D-10.17) E °* p(,+^+>) ;; ^ 0, -l^y^l-
As Feldheim 5 pointed out, D.10.10) implies
" Pt"'aHx)
D.10.18) E p(.,a)m >0, -Ui|U>0.
Askey-Gasper 4 and Askey-Steinig 2 have obtained generalizations of D.10.18).
They proved
D-10.19) g_|_yLZ>0, -l<x<l,
for
(i) «+^^ -2, ^^0 (when a =,-2, ^=,0, assume n^ 1),
(ii) -^^a^^ + 1 («=i j8=.-i omitted),.
(iii) -)8+l ^a
(iv) -j8 + 2^a
The case a = i, 0 = — i of the sum D.10.19) is Fejer's sum F.4.3), which
is nonnegative, while the case a =3/2, 0 = — \ is equivalent to
d " sin(A:
g
It is also equivalent to
and, since the right-hand side of this inequality is the even part of D.10.18)
when a = 3/2, it is nonnegative. The positivity of the left-hand side is equivalent
to the positivity of K®\x) in A5.5.1).
G) Jacobi polynomials satisfy the addition formula

98 JACOBI POLYNOMIALS [IV
reiipsme1smd2\2 - 1)
n A'
D.10.20)
*"° m=0
0
where
Cn.i.m-.
and the limit relation
,. B + n net \ Bc
hm Cflcosy) = j ,
Icosnv?, n =,1,2, • • •,
n =,0,
is used when B = 0.
When r = 1 and a =fi this formula is the addition theorem of Gegenbauer
1 for ultraspherical polynomials. It was discovered by Sapiro 1 in the case
B = 0 and independently by Koornwinder 1 in the general case. For other
proofs see Koornwinder 2, 3, 4.
Among the special results contained in D.10.20) are
D.10.21)
a> B> — \, where
lTd)
=J0 Jo
^,r) = AaJt(\ -
and
Jo
f
J Jo
and also
D.10.22)
r1 r
J 0 J 0
When a—>/8, D.10.22) has D.10.3) as a limit. In addition to the papers listed
above, see Gasper 3, 4 and Askey 10.
Another useful formula was found by Bateman 1:
where the c*,n's are defined by

[4.10] FURTHER PROPERTIES OF JACOBI POLYNOMIALS 99
D.10.24)
The cKn can be explicitly computed by use of Rodrigues' formula and
orthogonality and they were given in Bateman 1. However, in many applica-
applications only the positivity of the ck,n is needed (see e.g. Horton 1). It can easily
be proved by induction. Bateman 3 also discovered the inverse to D.10.23):
2-i *
x+y) fx+y\
where the 6*,n's are defined by
D.10.26) P^\x) =? bk,n
This has been used by Koornwinder 3. Again the specific form of the
was not needed.
(8) Gegenbauer 6 generalized D.9.19) to obtain
[n/2]
D.10.27) P{/\x) =,? oj,,?,^!)
where
,,,„,„, r(X)(w -2k +\)T(k +» -x)r(w - k +n)
D-10-28) akn = ru)*!ru-x)r(n-*+x + i) •
Proofs are given in Hua 1 and Askey 2. This can be inverted to give
D.10.29) A -x2)"/2pW(x) =itdk.nPnU(x)(l -x2)x~1/2, M>(X-
where
D.10.30)
22 (n +2^ + 1) r(n +2M) r(n +k +X) r(* +X -
r(X -n) T{n) T{n +1) T(k + l)r(n +* +m +1) IXn +2* +2X)
See Askey 1. When'X =.1, D.10.29) reduces to D.9.22). Generalizations of
these formulas to Jacobi polynomials as well as generalizations of Problem
84 are given in Askey 4, Askey-Gasper 1, 2, Gasper 1, 2. In the general case,
the coefficients are much more complicated and one needs to obtain asymptotic
formulas and positivity when it holds. For an application of D.10.27) and
Problem 84 in which only the positivity is used see Askey-Wainger 3. The
linearization result in Problem 84 can be used to define Toeplitz operators
and much of the classical theory of Toeplitz operators and the newer work
on finite sections of such operators can be extended to these more general
operators. See Hirschman 2 and Davis-Hirschman 1 and further references
given in these papers.

CHAPTER V
LAGUERRE AND HERMITE POLYNOMIALS
Many of the properties of the polynomials with which we shall deal in this
chapter are very similar, and more or less analogous, to the properties of Jacobi
polynomials. For this reason we shall be brief and omit details, unless essen-
essential differences in statement or proof make the contrary necessary. Here, as
in the case of Jacobi polynomials, the treatment of some special problems
(zeros, extrema, and so on) is reserved for later chapters.
5.1. Elementary properties of Laguerre polynomials
A) We define the Laguerre polynomials {Lia)(x)\, for a > —1, by the
following conditions of orthogonality and normalization:
,K 1 ,, , e-xxaLla)(x)LLa)(x) dx = r(« + 1)
E.1.1) Jo
n, m = 0, 1, 2, • • •.
In addition, we require that the coefficient of xn in the polynomial Lia)(x) of
degree n have the sign ( — 1)". (This differs from condition (a) in the definition
of §2.2.) We also write LT(x) = Ln(x).
Reference is here made to Lagrange 1, Abel 1, p. 284, Tchebichef 3, pp.
506-508, and Laguerre 1, pp. 78-81 (pp. 434-437), who, however, consider
only the case a = 0. Laguerre uses the notation fn(x) = n\Ln( — x). Hilbert-
Courant A, pp. 79-80) also considers only the case a = 0; the function there
called Ln{x) is the same as n\Ln(x) in our notation. Concerning the general
case L{n\x) see Sonin 1, pp. 41-42.
We have the differential equations
xy" + (a + 1 - x)y' + ny = 0, y = Lia\x),
xz" + (x + l)g' + (n + % + 1 - f )z - 0, 2 = e-'xa*Lia\x),
E.1.2)
U" + I — ¦—— ¦+¦ -r-T- - t | U = U, U =
v" + Un + 2a + 2 - x2 + ^—^-) v =0, v = e~xV2xa+iLla)(x2).
Once again let a > — 1. Then an argument analogous to that in §4.2 B)
shows that a necessary and sufficient condition that
E.1.3) xy" + (a + 1 - x)y' + \y = 0
100

[5.1] ELEMENTARY PROPERTIES 101
have a polynomial solution is that X = n. Also, L{n\x) is the only polynomial
solution. The latter statement follows from the relation
E.1.4) u[(x)u2{x) - ui(x)u'2(x) = const.,
which holds for two arbitrary solutions u\(x), u2(x) of the third equation in
E.1.2). Incidentally, this argument furnishes slightly more: for a > — 1, the
polynomials Lia)(x) are the only solutions of E.1.3) which are analytic near
x - 0.
The analogue of Rodrigues' formula is
E.1.5) e-xxaL"na\x) = L(A)\e-'xn+a).
n! \dx/
To determine the constant factor we apply Leibniz' formula (which leads to
E.1.6)) and calculate the highest term of the right-hand member.
Further, we have the explicit representation
E.1.6) L{na\x) = ±(n + a\(^f,
,.=0 \n — vf v\
the formula
E.1.7) Ua)@) =
and the expression
E.1.8) Zia) = K ,J
for the coefficient Z^a) of xn in Z/La)(a;). As a generating function we obtain
Ua\x) + L[a\x)w + ¦ ¦ ¦ + Lla)(x)wn +¦¦¦
The following recurrence formula holds:
nLia\x) = (-x + 2n + a - ^Li'J^x) - (n + a - l)L{na22{x),
E.1.10) n = 2, 3, 4, • • • ,
Lia)(x) = 1, L[a)(x) = -x + a + 1.
For the "kernel polynomial" ifia)(a;, y) we find
r(« + l)Kl'\x, y) = ±
E>1>11) = (n + l)^ \V

102 LAGUERRE AND HERMITE POLYNOMIALS [ V ]
The special case y = 0 is particularly important:
E.1.12) x E Ua){x) = (n + a + l)Lla)(x) - (n + l)Li^(x).
>.«o
(Cf. Theorem 2.5.) Finally, by means of E.1.6), or E.1.9), we readily obtain
-^n W = Lin \X) — Ln-i \X),
E.1.14) ^Li"\x) = -Li'+f'ix) = x~l\nLi"\x) - (n + a)Lln"Mx)).
B) Theorem 5.1. Let Ja have the same meaning as in §1.71. 77ien
5 »a
E.1.15) ^
1 — w) { 1 — w
and
00 y (a) / \
E.1.16) E -, , , .< wn = ew{xw)-"nJa{2(xw)h\.
n-o F(n + a + 1)
See Sonin 1, p. 41, Wigert 1, Hille 2, Hardy 1, Kogbetliantz 12, Watson 4.
The first formula is a generalization of E.1.9) (y = 0). The second formula
is obtained from the first one by replacing w by —y~lw, y —* °o.
Direct calculation leads readily to E.1.16) on account of E.1.6). The
formula E.1.15) follows from E.1.16) when we introduce for L{na)(y) the
integral expression which results from E.4.1), and then integrate term-by-term.
Finally an integral formula involving Bessel functions (Watson 3, p. 395, A))
must be used.
5.2. Generalization
By means of E.1.6) the definition of Laguerre polynomials can be extended
to arbitrary complex values of a. No reduction in the degree ever occurs
(see E.1.8)). For n ^ 1 we have Lia)@) = 0 if and only if a = — k, k integral,
1 ^ k ^ n. In this case x = 0 is a zero of precise order k, and from E.1.6)
n! v=q \n — k — v
E.2.1)
, sk(n - k)l T (^ ,,
= { — x) —^-j—Ln-k{x).
Formula E.1.16) remains true for arbitrary real a. From here E.2.1) follows
again.

[ 5.4 ] INTEGRAL REPRESENTATIONS 103
5.3. Confluent hypergeometric series; relation between Jacobi and Laguerre
polynomials; second solution
A) In the notation of Pochhammer-Barnes, the confluent hypergeometric
series is
E.3.1) M(«;y;x) = l ±(+» ( + »*
¦ „_! 7G + 1) ••• G + V — \)v\
This is obtained from the ordinary hypergeometric series [D.21.3)] by the
limiting process
E.3.2) lim F(a, j8; y\§Tlx).
0-00
We have
E.3.3) Lia)(x) =
and using D.21.2), we obtain the following important relation between Laguerre
and Jacobi polynomials:
E.3.4) Lia)(x) = lim Pi"'ff)(l - 2p~lx).
C-»°o
This holds uniformly in every closed part of the complex x-plane. Concerning
further properties of the confluent hypergeometric functions see Whittaker-
Watson 1, Chapter XVI. Compare equation (B) in Whittaker-Watson 1,
p. 337, with our third equation in E.1.2).
B) From D.23.1), by a limiting process similar to that used in E.3.4), we
obtain as a second solution of the first equation E.1.2)
E.3.5) x-'M-n - a; 1 - a; x).
For non-integral values of a the functions E.3.3) and E.3.5) are evidently
linearly independent. The same is true if a is a negative integer less than —n,
since in this case E.3.5) is an infinite series. However, if a is an integer not
less than — n, these solutions are identical. (If — n ^ a ^ 0, use E.2.1). In
case a =¦ g,g ^ 1, it is necessary to multiply E.3.5) through by a — g before
letting a —> g.)
The representation E.3.3) makes possible an extension of the definition of
L\?\x) to arbitrary values of n.
5.4. Integral representations
Theorem 5.4. The following representation of Laguerre polynomials in terms
of Bessel functions holds:
E.4.1) e-xx"nLia)(x) = - f e-'r+a/2Ja{2(^)i}di, n = 0,1, 2, •••;«> -1.
n- Jo
For this representation, the reader may consult E. Le Roy 1, pp. 379-384,

104
LAGUERRE AND HERMITE POLYNOMIALS
Erdelyi 1. The same representation is valid if a S — 1, provided n + a > — 1.
From this, E.2.1) follows again.
Several proofs of this formula may be given. The special case n = 0, that is,
E.4.2)
e~xx"n =
e~ttal2ja[2(tx)i}dt,
can be obtained by expanding Ja(z) as in A.71.1) and integrating term-by-term.
(The formula is due to Sonin; cf. Watson 3, p. 394, D).) The general formula
then follows from calculation of the generating function of both sides of E.4.1);
here E.1.9) and E.4.2) must be used.
Fig. 6
Another proof, from a more general point of view, can be given as follows.
We shall try to satisfy the second equation E.1.2) by an integral of the form
E.4.3)
z = z(x)
= j e-'t"
-t ,n+a/2
Ja{2(tx)h] dt
with a proper path of integration. Substituting this expression in the left-hand
member of the equation mentioned, we obtain
E.4.4)
{2(te)*
In view of A.71.3) the expression in the square brackets becomes
E.4.5) {tx)hj'a[2(txf\ + (n + a/2 + 1 - t)Ja{2(tx)*\,
so that the entire expression will be
E.4.6)
at
Hence E.4.3) is a solution of E.1.2) .provided that

[5.5] HERMITE POLYNOMIALS 105
(a) the path of integration is a closed contour, and so chosen that
e~ttn+al2+1Ja{2(tx)i\ resumes its initial value, or
(b) the path of integration is an arc, and the expression in (a) vanishes at its
end-points.
For the interval 0 ^ t < + °°, condition (b) is satisfied provided n + a + 1 > 0.
Assuming first that a > — 1, we notice that the function x~a/2z is analytic near
x = 0, so that (see the remark concerning E.1.3)) z = const. e~xxa/2L<na)(x).
The constant factor can be determined by comparing the "lowest terms," that is,
the coefficients of xal2. The restriction a > — 1 can then be removed by means
of analytic continuation.
Another remarkable representation is obtained by choosing the contour as
in the figure. Condition (b) is again satisfied, and the same argument as before
yields z = const. e~xxal2L\?){x). The normalization of the integrand requires
the determination of ta at a certain point. We agree to take arg t = 0 on the
rectilinear part of the contour with $t < 0 (in its limiting position). Then
the "lowest term" becomes, if a > — 1 and a ^ 0, 1, 2, ¦ • ¦ ,
a/2 r all ("<*>
X I e-ltn+adt = * , ,. A - e~2Ha) e-ltn+adt
r(a + 1) J T(a + 1)
so that on account of E.1.7), we have
/O • \-l iT(a-J) /•@+)
E.4.7) e-*x"l2Lia)(x) = Bsm™\ e / e~l f+al2Ja{2{tx)'} dt.
Ill J+ao
This formula can be extended to arbitrary non-integral values of a.
Further integral representations can be derived from E.1.5) and E.1.9) in a
manner analogous to that used in the case of Jacobi polynomials (cf. D.4.6),
D.4.9)). We have for instance, x 9^ 0,
E.4.8) e-xxaL[a)(x) = J-. [ e^t^'it - x^1 dt,
where the contour encloses t = x, but not t = 0.
5.5. Hermite polynomials
A) These are defined by the conditions
E.5.1) f+ e~x2Hn(x)Hm(x)dx = 1ri2"n!8»m, n,m = 0,1, 2, ¦ • • .
The coefficient of xn in the nth polynomial is positive.
See the bibliography in Hille 1. Our notation agrees with that of Hille.
See also Hilbert-Courant 1, pp. 77-79 where the same notation is used. In
Polya-Szego 1 (vol. 2, pp. 94, 294, 295, problem 100), Hn(x) is written for
(-I)nBn/2n!r1#nB~i:r) in terms of our notation.

106 LAGUERRE AND HERMITE POLYNOMIALS [ V ]
B) The derivation of the following properties of Hermite polynomials presents
no difficulty:
5 W y + 2ny = 0, y = Hn(x),
\z" + Bn + 1 - x2)z = 0, z = e-xV2Hn(x),
E.5.3) e~x2Hn(x) = (-1)"
E.5.4) HMJ?AY
n! ?o ?! (n - 2k)!'
E.5.5) #2m@) = (-l)m (-^, fljm+iCO) = (-I]
E.5.6) lim x~nHn(x) = 2n,
E.5.7) 1! 2! »!
= exp
ff»(x) = 2xHn-l(x) - 2(n -
E.5.8)
n = 2, 3, 4, • ¦ • ; ffo(z) = 1, i?i(x) = 2x,
E.5.9) E <X9\)-lH,{x)H,{y) = B"+1n!)-J H»+x{x)HM ~ Hn(x)Hn+1(y)
We notice the following "individual" properties:
E.5.10) #!(*) L
From E.5.7) we obtain
E.5.11) g
and, by Cauchy's formula,
E.5.12) ^L = J-- [ v" exp Bxw - w2) dw,
n! 2m J
where the contour encloses the origin.
5.6. Relation of Hermite polynomials to those of Laguerre
A) Hermite polynomials can be entirely reduced to Laguerre polynomials
with the parameters a — d=|, for we have
E.6.1) Him(x) = (~l)m22mm\L(-i\x2)) H2m+l(x) = (-l)2m+1m!xL<*V).
These formulas are in some respects the analogues of D.1.5). Their proofs

[5.6] RELATION OF HERMITE AND LAGUERRE POLYNOMIALS 107
are similar to those given there. Note the fourth equation in E.1.2) (a = dt?),
and the second equation in E.5.2).
Combining E.6.1) with E.3.4), we obtain a representation of Hermite poly-
polynomials as limits of Jacobi, and in consequence of D.1.5), of ultraspherical
polynomials. This can be ascertained from D.7.23) and E.5.7). In fact, we
have
... .~x
E.6.2) exp Bxw - w2) = lim ( 1 - 2 - w +
x-»°o \ X X
so that
E.6.3) ^p = lim X'^P^iX^x).
From E.6.1) the explicit representation E.5.4) follows readily when we use
E.1.6). From Theorem 5.1 analogous expansions for Hn(x) can be. derived
(see Watson 5). The generating function E.5.7) follows'from E.1.16), while
the generating function arising from E.1.9) for a = ±? is of a different nature
(see Problem 24).
By using A.71.2) and E.4.1), we obtain the integral representations
E.6.4)
e~x2Hn(x) = (-l)[n/2l2n+17rH /""V'V cos Bxt) dt, n even,
h
e~x2Hn(x) = (-l)[n/212n+17rH fV'Y sin Bxt)dt, n odd.
Jo
B) Conversely, Laguerre polynomials can, to a certain extent, be reduced to
Hermite polynomials. We have (Uspensky 1, p. 604, A4))
E.6.5) L\ (x) = ———j- —-yj / A — 0 Hin(xH)dt, a> -\,
This can be readily shown by using the explicit formulas E.1.6), E.5.4), and
A.7.5).
C) We conclude these formal considerations with the following remark con-
concerning the identities D.21.7), E.1.14), and E.5.10) on Jacobi, Laguerre, and
Hermite polynomials, respectively. If the orthogonal polynomials {pn(x)\
associated with a distribution da(x) have the property that [p'n(x)\ is, save for
constant factors, a system of the same kind (that is, associated witli a certain
distribution dfi{x)), then [pn(x)} is (save for trivial linear transformations)
one of the three special systems mentioned before (classical polynomials).
(W. Hahn 3, Krall 1.) A similar statement holds if we replace [p'n(x)\ by
{pik\x)\ (Krall 2, W. Hahn 4).
Another problem of a similar nature has been considered by Bochner A).
He determines all sets of polynomials | pn{x)}, where pn(x) is of precise degree n,

108 LAGUERRE AND HERMITE POLYNOMIALS f V ]
satisfying a differential equation of the form
E.6.6) fo(x)y" +Mx)y' + [f2(x) + \}y = 0, y = pn(x), X - Xn.
Bochner obtains, in addition to the classical polynomials, certain polynomials
related to Jn+i(x), n integral, as possible solutions, as well as polynomials of
the trivial type axn + bxm, where a and b are constants.
5.7. Closure
Here we prove the analogue of Theorem 3.1.5 for Laguerre and Hermite
polynomials. The main difficulty of these cases is due to the fact that the
orthogonality interval is infinite. Using the customary notation (§1.1), we
have the following theorem:
Theorem 5.7.1. The system
E.7.1) e'xl2xal2xn, a> -1, n = 0, 1, 2, , • ¦ ,
is closed in L @, + °o); the system
E.7.2) e~xV2xn, n = 0, 1, 2, . • • ,
is closed in L2(— °o, + °o).
This statement is equivalent to the closure of the systems \e~xl2xal2L(na)(x)}
and {e~xil2Hn(x)}, respectively. Theorem 5.7.1 remains true, of course, if we
replace e~xl2 by e~x and e~x%12 by e~x\ "respectively. The idea of the following
proof is due to J. von Neumann (see Hilbert-Courant 1, pp. 81-82).
A) We start with the remark that for a > — 1 the system
E.7.3) (logjyV, n = 0,1,2,.-.,
is closed in L2@, 1). This is a consequence of Theorem 3.1.5, p = 2, since
(log (l/y))a is integrable in [0, 1].
Now let e~xl2xa/2f(x) belong to L2@, +«). Then (log (l/y))a/2f(\og A/y))
belongs to L2@, 1), and it can be approximated in mean by functions of the
form (log 0-/y))al2p{y), where p{y) is a polynomial. Thus, corresponding to
every e > 0 a polynomial p{y) can be determined such that
E.7.4) [ e'xxa{f(x) - p{e~x)Jdx < e.
Jo
Hence all that remains to be shown is that if m is a non-negative integer, there
exists for every 8 > 0 a polynomial p(x) such that
E.7.5) / e'xxa{e'mx - p(x)}2 dx < 8.
Jo
B) For this purpose we use E.1.9), writing

[5.7] CLOSURE 109
E.7.6) w = ———, to = ,
m + 1 1 — w
and choosing
E.7.7) p(x) = A - «,)"+1 E Ua)(*)wn.
n-0
Then
/ e* ^? o \ I —3r a t —mx /\>2i /i \ 2o4-2 J —x •
M\ / x I p TIP Til T I \ fiT "— I I 01) I l P T
Jo Jo
which, in view of E.1.1), is equal to
E.7.9) A - wJa+2T(a + 1) E C1 + ^
Term-by-term integration is permitted here since the series
/ e~xxa I Lia)(x) I I Ll?,\x) I dx-wn
Jo
\ f0 2 V f P 2V n+n'
< I e a;aL-Lna (^)] dx> < I e xxa[Lna> (x)] dx> wn
Uo J Uo J
00 pan
e x 1 j^n {xj 1 1 j^n' ^x; 1 ux-tT
< T
is convergent. The expression E.7.9) becomes arbitrarily small when N is
sufficiently large, and this establishes the statement.
C) Let e~xV2f(x) belong to L2(- «, + «). Then both functions
E.7.10) ,-
belong to L2@, + °o). Therefore, by the preceding result (first taking a = — \,
then a = +|), for every e > 0 there exist polynomials pi{y), pi(y) which satisfy
the inequalities
E.7.11)
<
or
E.7.12) A,
2
/o
so that
E.7.13) ( e~x2lf(x) - Pl(x2) - xP2(x2)Jda: < 2e.

110 LAGUERRE AND HERMITE POLYNOMIALS [ V J
For another proof based on the theory of integral equations, see Weyl 1, in
particular pp. 58-61, 64. See also Hamburger 1, pp. 200-205.
D) The argument used in B) leads to the following result:
Theorem 5.7.2. The system
E.7.14) e~xxa+n, a> -1, n = 0, 1, 2, • • • ,
is closed in L@, + °o); the system
E.7.15) <T*V, n = 0, 1, 2, • ¦ • ,
is closed in L( — oo, + oo).
We use again Theorem 3.1.5, p = 1. By means of E.1.1) and Schwarz's
inequality we find
E.7.16) I e~xxa | Lla\x) | dx =
J
o
For later purposes (§9.5 A)) we note the following theorem:
Theorem 5.7.3. The functions
E.7.17) fn(x) = <t>(x)xn, n = 0, 1, 2, • • • ,
where
[xa, 0 < x < 1,
E-7-18) *(*) = _, „
[ex, x ^ 1,
a > — 1, (8 arbitrary and real, form q closed system in L@, + oo).
We apply Theorem 3.1.5 with da(y) = y~l<t> (log (l/y)) dy, 0 < y < 1, and
p = 1. Furthermore, we need a bound for
/ <t>(x) I Lia\x) | dx = 0A) / e~Tx" | Lia)(x) \dx+ e~V \Lia)(x) j dx.
Jo Jo Ji
The first integral in the right-hand member is O(nal2); for the second integral,
Schwarz's inequality yields the bound
e-xxa[Lia\x)fdx\ =O(na/2).
Concerning further formal properties of the Laguerre and Hermite poly-
polynomials, see Bateman Manuscript Project, vol. 2, Chapter 10, pp. 188-190.
Cf. also Problems and Exercises, 67, 68, 72-80.
For problems related to Laguerre polynomials and applications of Laguerre
polynomials see Szego 26, Askey-Gasper 3, Askey 9, Peetre 1, and Roosenraad
1. For Hermite polynomials see de Bruijn 1.

CHAPTER VI
ZEROS OF ORTHOGONAL POLYNOMIALS
In §3.3 it was proved that the zeros of orthogonal polynomials are all real,
distinct, and lie in the interior of the orthogonality interval. We shall now
present a further and more detailed investigation of the location of these zeros.
Starting with certain theorems valid under very general conditions imposed on
the weight function, we proceed to the zeros of the classical polynomials and
point out various methods used in their investigation. A particularly important
tool in the latter connection is Sturm's theorem (§1.82), which in various cases
leads to rather exact information concerning the zeros of polynomials satisfying
certain linear differential equations of the second order.
No claim of completeness is made for the present survey on the zeros. Con-
Concerning the literature about zeros of Laguerre and Hermite polynomials, we
refer to W. Hahn's "Bericht" B).
The methods of this Chapter are quite elementary. In particular, no sys-
systematic use is made of the asymptotic properties of orthogonal polynomials of
a special and general kind (Chapters VIII and XII), from which important
information concerning zeros can also be derived.
6.1. Density of zeros
A) Theorem 6.1.1. Let da(x) be a distribution on the finite segment [a, b],
and let [pn(x)} denote the associated orthonormal set of polynomials. Let [a', b']
be a subinterval of [a, b] such that j^ d'a(x) > 0. Then if n is sufficiently large,
every polynomial pn(x) has at least one zero in [a', b'].
For the proof we shall use the Gauss-Jacobi mechanical quadrature (§3.4).
Let p(x) be an arbitrary xm which is not greater than 0 in [a, b], except possibly
in [a1, b']. Assuming that the polynomial pn(x) has no zeros xv in [a1, b'],
and taking In — 1 ^ m, we obtain
/b n
p(x) da{x) = E X,p(j\) ^ 0.
f=i
Hence, when we apply the theorem of Weierstrass (Theorem 1.3.1), it follows
that
Ja
F.1.2) / f(x)da(x) ? 0,
Ja
where f{x) is continuous in [a, b] and not greater than 0 in [a, b], except pos-
possibly in [a', b']. If we define
ill

112 ZEROS OF ORTHOGONAL POLYNOMIALS [ VI ]
@ in a ^ x ^ a' and b' ^ x < b,
F.1.3) f(x) =
{(x - a'){V - x) in,a' ^ x S V,
we reach a contradiction.
B) Theorem 6.1.2. Theorem 6.1.1 remains valid for infinite intervals [a, b]
provided the zero of the greatest modulus of pn(x) is o(n).
This remark is duo to W. Hahn A, pp. 215-217).29 If we write
max | xv — a' || b' — xv \ — Mn , v = 1, 2, • • • , n,
the assumption means that Mn = o(n2). We now choose
p(x) = [Tk[2MZ\x - a')(V -x) + I}}2, k = [n/2 - 1/4],
where Tk(x) denotes Tchebichef's polynomial A.12.3). If we next assume
that [a', b'] contains no zeros, we have
-1 ^ M~\xv - a'){b' - xv) ^ 0,
so that p(xv) ^ 1. It then follows that
/b' tb n n [b
p{x)da{x) ^ I p{x)da{x) = ^2\rp(xv) ^ ^2 Xr = I da{x).
' Ja k=1 k=1 Ja
Now T'k(x) is increasing for x ^ 1, so that Tk(? + 1) > T*(l)? = k2? for
^ > 0; thus we have in [a', b')
p(x) ^ ki{2M~\x - a')Q>' - x)}2; f p(x) da(x) > Ck'M'2,
Ja'
where C is a positive constant independent of n. Hence nM~^~ = 0A), which
is a contradiction.
Concerning the distribution of the zeros for large values of n, see Theorem
12.7.2.
6.11. Distance between consecutive zeros
Here and in the subsequent sections we consider distributions of the type
w(x) dx.
A) Theorem 6.11.1. Let w(x) be a weight function on the finite interval
[a, b], bounded from zero: w(x) Si m > 0. Let xi > x2 > ¦ ¦ ¦ > xn be the zeros
of the associated orthonormal polynomial pn(x) in decreasing order.zo On writing
F.11.1) xt = Ka + 6) + hQ> ~ a) cos e" > 0 < d, < tv, v = 1, 2, • • • , n,
29 He states (without a satisfactory proof) that if either a or 6 is finite, the subsequent
argument succeeds even with o(n2).
30 Of course, each x, depends on v and n, x, = x,n .

[6.11 ] DISTANCE BETWEEN' CONSECUTIVE ZEROS 113
we have
F.11.2) 0,,+1 -6V<K ^, v = 0, 1, 2, ¦ ¦ ¦ , w; 0o = 0, 0n+1 = ir.
flere the constant K depends only on n, a, and b.
See Krawtchouk 2; Erdos-Turan (written communication). Erdos-Turan
require the existence of /„ {w(x)}-1 dx instead of w(x) ^ n > 0. Their proof
(cf. 2) is based on the study of the distribution of the interpolation points
for which the associated, fundamental polynomials (Chapter XIV) satisfy
certain conditions. The following proof for the special case w{x) ^ n > 0
has been prepared by B. Lengyel; he eliminated all references to the theory of
interpolation from the argument of Erdos-Turan.
B) Let v be a fixed integer, 0 ^ v ^ n, and y = @, + 0,+i)/2. We then
define p{x) = p\i(a + b) + h(b — a) cos 0} by
F 113) 2P(x) = (sin±Nir±jmJm + Airi {N(y - 0)/2}
1 j W Usin{G + 0)/2}; +Wsin{G-0)/2
where N and m denote certain positive integers. The single terms in the right-
hand member represent the same trigonometric polynomial of degree m{N — 1),
taken alternately with arguments 7 + 0 and 7 — 0. Therefore, the sum is a
cosine polynomial of the same degree m(N — 1). If m(N — 1) ^ 2to — 1,
Theorem 3.4.1 can be applied.
We then have
F.11.4) @,+1 - 0,)/4 Si 7/2 < G + t)/2 ?*- @,+i - 0,)/4;
whence for all values of k, 1 ^ k ^ to,
F.11.5) 0 < @,+1 - 0y)/4 < G + ek)/2 < tv - @h-i - 0,)/4,
so that
, k = 1, 2, • ¦¦ ,n.
The same inequality, with ^ instead of <, holds for | sin G — 0*)/2 | 1, since
y - 6k\ ^ @,+1 - 6,)/2. Thus,
F.11.7) P(xk) ^ fiVsin^4^ "*\ , k = \,2,---,n.
By using C.4.1) and C.4.5), we obtain
p{x)w{x) dx ^ [N sin "+14 •) / w(x) dx.
On the other hand, the value of p(x) for 0 = 7 is not less than §, so that

114 ZEROS OF ORTHOGONAL POLYNOMIALS I VI ]
if we write x0 = \{a + b) -f \{b — a) cos y, we obtain from a later result (cf.
Theorem 7.7)
F.11.9) / P(x)w(x)dx ^ y. I P{x)dx ^ nCn-2P(x0) ^ \nCn~\
Ja Ja
where C is a positive constant depending on a and b only. Comparing F.11.8)
and F.11.9), we now have
^ (n sin 6^LZJY*"
in 6^LZJl
or
F.11.10) Sin^_-' ^ N-'l^C^r'nY^l I w(x) dx\
By substituting N —- [n/log n], m = [log n], we see that the condition
m(N — 1) ^ 2n — 1 is satisfied for large values of n, and F.11.2) follows
immediately.
The same inequality F.11.2) holds without essential change if w(x) ^
mA — x)a(l + x) , n > 0, where a and /3 are greater than —1. In this case,
the last remark in §7.71 D) must be used. The constant K in F.11.2) now
depends on n, a, b, a, and /3.
C) We note the following simple result:
Theorem 6.11.2. Let w(x) be a weight function on the interval [—1, +1],
and suppose
F.11.11) A ^ A - x2fw(x) ^ B, -l^s^
where A and B are positive constants. If xv = cos d,, 0 < dv < ?r, v = 1, 2, • • • , n,
stand for the zeros of the orthonormal polynomial pn(x) associated with w(x), in
decreasing order, we have
F.11.12) 0,+I - 0, < tZjj? ! , v = 0, 1,2, ¦¦¦,n;da = 0, 0n+i = t.
-*JL Til
This remark is due also to Erdos-Turan (written communication). The
proof can be based on Theorem 3.41.1. We denote by X, the Christoffel number
corresponding to x,. Then
(a ii i-^ AFv+i - 6,) ^ / w(cosd) sin 6d6 = / w(x) dx ^ X,
F.11.13; Je, JXl,+1
v — 0, 1, 2, • • •, n; Xo = Xn+i = 0.
On the other hand,
p(x) - picas 6) -

[6.12] VARIATION OF ZEROS WITH PARAMETER 115
is a irn_i in x = cos 6. Furthermore, p{xv) = p(cos #„) ^ 1, so that (cf. A.6.5))
" f+1 f*
X, ^ Zj 'kkp(xk) = I p(x)w{x)dx ^ B I p (cosd)dd
F.11.14) *"' ' 7°
sin {0/2}/ n
On combining this with F.11.13), the statement is seen to be true.
Erdos-Turan proved also (cf. 2) that if 0 < A ^ w(x) ^ B, -1 ^ x ^ +1,
0 < e < t/2, ?/ien, with the notations of Theorem 6.11.2,
Kl
n
<
- ev
<
K2
n
F.11.15)
provided e ^ 9, ^ ir — e. //ere i?i and i?2 depend on A, B, and e.
6.12. Variation of the zeros with a parameter
A. Markoff proved D) an important statement concerning the dependence of
the zeros of pn(x) on a parameter r which appears in the weight function
w(x) = w(x, r).
A) Theorem 6.12.1. Let w(x, r) be a weight function on the interval [a, b]
depending on a parameter r such that w(x, r) is positive and continuous for
a < x < b, ti < t < t2 . Also, assume the existence and continuity of the partial
derivative wT(x, r) for a < x < b, t\ < r < n , and the convergence of the integrals
F.12.1) / x"wT(x, t) dx, v = 0, 1, 2, ¦ ¦ ¦ , 2n - 1,
uniformly in every closed interval / ^ r ^ r" o/ ?/ie open segment rx, r2. //
i/ie zeros of pn{x) = pn(x, r) fee denoted by xx{t) > x2(t) > ¦ • > xn{r), the vth
zero xv{t) {for a fixed value of v) is an increasing function of r provided that wT/w
is an increasing function of x, a < x < fe.
The integrals B.2.1) for the moments cr[da(x) = w(x, r) dx] converge uni-
uniformly in t' ^ t ^ t", and the relations B.2.1) may be differentiated with
respect to r; v = 0, 1, 2, . • • , 2n - 1. Let a < a' < b' < b. For a' ^ x ^ fe',
t' ^ r ^ r", the function w(a;, r) has a positive minimum; whence the deter-
determinants Dn-i are uniformly bounded from zero if / ^ r ^ r" [B.2.11)].
According to B.2.6) the coefficients of pn(x), therefore also the zeros Xv{t)
(which are all distinct), possess continuous derivatives for n < r < r2.
Let p{x) be a fixed 7r2n-i ¦ Apply Theorem 3.4.1 with da{x) = mj(o;) rfx.
The Christoffel numbers X, = X,(t) are obviously functions of r with a con-
continuous derivative [C.4.3)]. Differentiating C.4.1) with respect to r, we obtain
F.12.2) / wT(x,r)p(x) dx = E X,(r)p'(a;,)a;i(T) + E ^(r)p(a;,).

116 ZEROS OF ORTHOGONAL POLYNOMIALS [ VI
Now we substitute
F.12.3) pfe) = i?=^il, whence P'(s,) = {pKz,)}1,
JO tl/y
so that
F.12.4) fwAx, t) i^l' ^
J
J X Xv
since p'(av) = 0 if n j? v. The left-hand member can be written in the form
) ) X — Xv
the second term being zero because of the orthogonality. The difference
F.12.5) (bL(x,r) - W4tA wM\ \jb^L dx>
Ja \ W{X,, T) ) X X
F 12 6)
w(x, t) w(xv, t)
has the same sign as x — xv according to the assumption. This establishes the
statement.31
B) We point out the following ©onsequence:
Theorem 6.12.2. Let w{x) and W{x) be two weight junctions on [a, b], both
positive and continuous for a < x < b. Let W(x)/w(x) be increasing. Then if
{xv\ and {Xr} denote the zeros of the corresponding orthogonal polynomials of
degree n in decreasing order, we have
F.12.7) x,< X,, v = 1, 2, • • • , n.
Defining w(x, r) = A — t)w(x) + tW(x), 0 ^ t ^ 1, we see that
F.12.8) ^M mx)~w(xl -1
w(x, t) A - t)w(x) + tW(x) 1 - t + tW(x)/w(x)
is an increasing function of x, 0 < r < 1. We also have w(x, 0) = w(x),
w(x, 1) = W{x).
Various applications of these results will be given in §6.21.
6.2. Location of the zeros of the classical polynomials
The discussion of this question, given in §3.3 for the general orthogonal
polynomials, was based on the orthogonality property. In the particular cases
called classical polynomials (§2.4) there exist various other approaches which
are interesting from the point of view of method. It is assumed that a > — 1
and /3 > — 1 in the Jacobi case, and a > — 1 in the Laguerre case; furthermore,
n ^ 2.
31 This proof does not differ essentially from the original one due to A. Markoff, although
the present arrangement is somewhat clearer.

[ 6.2 ] LOCATION OF ZEROS OF CLASSICAL POLYNOMIALS 117
A) First of all, if we are dealing with the classical polynomials, a more precise
form can be given to the argument indicated in §3.3 D). In fact, for the
polynomials in question the number of the sign variations in C.3.5) for x = a
and x = b can be readily calculated.32 We use D.1.1) and D.1.4) for Jacobi
polynomials, and E.1.7) and E.1.8) for Laguerre polynomials. In addition to
the reality and distinctness of the zeros, the statement as to their location
follows from the same argument.
In the Jacobi case the value of
n~l x" dx ( n W)
at the zeros of Plna'®(x) can also be determined by means of D.5.7) (instead of
the method in §3.3 D) generally given). In the Laguerre case, E.1.14) can be
used. The situation is especially simple for Hermite polynomials; we have
but to apply the first equation in E.5.10).
B) By Rolle's theorem the formulas D.3.1), E.1.5), E.5.3) of Rodrigues'
type furnish the statement again. We must bear in mind that the derivatives
of A - x)n+a(l + x)n+\ e~xxn, and e~x* of the orders 0, 1, 2, • ¦ • , n - 1 vanish
at a; = ±1, a; = 0, +°°, and x = ±<», respectively.
C) The statement also follows from the differential equations D.2.1), E.1.2),
and E.5.2). To this end we first show that the zeros of Jacobi polynomials
are different from —1, +1, and from one another. Differentiating D.2.1) k
times, we have
A - x2)y{k+2) + [0-a-(a + 0 + 2fc + 2)x]y{k+l)
+ [n(n + a + /3 + l)-fc(fc + a + /3+ l))yW = 0.
Were y to vanish for x —¦ +1 or x = —1, it would follow from D.2.1) that
y' = 0, whence from the equation just obtained (k = 1) y" = 0, and so on;
that is, y = 0. (The coefficient of y^k+l) is different from zero for a; = ±1.)
Therefore, each of the zeros of Pia'0)(x) is different from ±1. Moreover, the
zeros are all simple since D.2.1) combined with y = y' = 0, a; ^ ±1, yields
A — x2)y" = 0, or y" = 0. Using the equation for y', we find y'" = 0 in
the same way, and so on; that is, y = 0 again.
Similarly, we show that the zeros of Lia)(x) are simple and different from 0,
and the zeros of Hn(x) are simple.
We next apply the following theorem due to Laguerre (P61ya-Szego 1, vol. 2,
pp. 59, 244-245, problem 1.18).
Let f{x) be a irn and xa one of its simple zeros. Then any circle through the
points
F.2.1) xo and x'o = xo — 2(n — 1) ,,,-. °
/ Kxo)
32 Professor P61ya has kindly called to my attention the fact that the same can be done
for the general orthogonal polynomials by using the determinant representation B.2.6).

118 ZEROS OF ORTHOGONAL POLYNOMIALS [ VI ]
contains some zeros of f(x) in both domains bounded by it, unless all the zeros
lie on the circumference of this circle. The same is true if a straight line replaces
the circle.33
The proof is as follows. Let f(x) = (x — xo)g(x), and let xx, x2, • • • , xn~i
denote the zeros of g(x). Then g(x0) = f'(x0) and g'(x0) = \f"(xo), so that
F22) _JL_ + _!_+... + _J_ i
x x x x t z g(x) f'
JL + _!_+.. + _J_
x0 - xi xo - x2 .to - zn_i g(x0) f'(x0)
Hence F.2.1) becomes
F.2.3)
( + + ) ,
n — 1 \x0 — X\ xo — x2 Xo — xn-i/ xo — x0
The linear transformation X = (x0. — a;) carries the points xx , x2, • • • , xn-i,
and Xo into certain points Xx, X2, • • • , Xn_i and Xo . Then we have
F.2.4) —Lj (Xt + X2 + • ¦ • + Z»_i) = Xo',
so that any straight line through Xo separates the points Xi , X2, • • • , Xn_i
from one another, unless they all lie on this straight line. Referred back to the
x-plane, this yields the theorem.
For the Jacobi polynomials y = Pia'0)(x) we obtain from D.2.1)
so that
2(n - 1)
F.2.6) xo = xo -
a+1 /3+ 1
1 — x0 1 + Xo
Let Xo be a zero of y with the greatest imaginary part. If there were any non-
real zeros, we should have S(xo) > 0, and
whence $(x'o) > S(x0). Therefore x'o lies in the half-plane $(x) >
and a circle can be drawn through x0 and x'o which contains no zeros. (The
zeros cannot all lie on this circle since they would then have to coincide with
Xo, an impossibility.) Hence all the zeros are real. Now let x0 be the greatest
zero, xo 9^ ±1. If we had x0 > 1, F.2.6) would give x'o > x0. Considering
an arbitrary circle through x0 and x'o , we are again led to a contradiction.
In the case of Laguerre polynomials,
33 If /"(zo) = 0, we have x'o = «, and a straight line through x0 must be considered.

[6.2] LOCATION OF ZEROS OF CLASSICAL POLYNOMIALS 119
F.2.8) xo = x0 — ;—-.
From 3(z0) > 0 we obtain, as before, $(xo) > S(x0). From x0 < 0 we find
' 34
34
In the case of Hermite polynomials,
/ n —
F.2.9) x'0 = x0
x0
so that S(x'o) > 3(xo) if 3 (a*) > 0.
D) Laguerre's theorem also furnishes certain bounds for the zeros. Let X\
be the largest and xn the smallest of the zeros ofP{na<l>)(x). Then (a + 1) / A - xt)
- @ + 1)/A + zi) > 0 and (cf. F.2.6))
F.2.10) -1<***-±l~?±l<*'
1 - Zi 1 + Xi
so that
F 2 11) a±l *±i > 2(w ~ 1) > /3 ~ a + 2n - 2
1-Xl l + x1 1 + Xl ' /3 + a
or for |8 ^ a
F.2.12) Xl>!LZ_1.
w + a
In the ultraspherical case xn = — Xi , so that
te9i<rt a+ X oc + l n-l ^ ( n - 1
F213) > X> [
F.2.13) r > __, Xl
(If n = 2, the sign > is to be replaced by =.) This bound is better than the
preceding one. Both bounds have the form 1 — (a + l)/n + O(n~2). Simi-
Similarly, an upper bound can be obtained for xn .
An analogous argument yields the bounds
F.2.14) xo > 2n + a - 1, x0 > \\{n - 1)}*
for the largest zeros x0 of Lia)(x) and of Hn(x), respectively. (For n = 2 the
second inequality becomes an equation.) These are very rough estimates
(Theorem 6.32).
E) Another proof of the reality and simplicity of the zeros (also furnishing
a < xv < b) can likewise be based on the differential equation by using the
considerations of §6.7. It must be remembered that the polynomials in ques-
question are the only polynomial solutions of the corresponding differential equa-
equations (see §4.2 B), §5.1 A)).
34 From F.2.8) we also find that the least zero is <a + 1.

120 ZEROS OF ORTHOGONAL POLYNOMIALS [ VI ]
F) In this connection, we refer to a very elementary method due to Laguerre
C), which furnishes certain upper bounds for the zeros of the classical poly-
polynomials. Since g"(x0) = if"fa), we have from F.2.2)
i + i , ... , i J±\^
(xo — XiJ (x0 - x2J (x0 — xn-iJ \dxj g(x0)
F.2.15)
{/fa)}2 W() _ 3{/"fa)}2 - 4f'(xo)f'"(xo)
and accor-ding to Cauchy's inequality
(n - 1) ?
— XpJ \y~l Xo — X,
or
F.2.16) 3(n - 2)if"(x0)}2 - 4(n - l)f'{xo)f'"(xo) ^ 0.
This condition is necessary for each zero of a polynomial with real and distinct
zeros.
In the case of Legendre polynomials,
A - xl)f"(xo) = 2xof'(x0),
A - xl)f"(x0) = ixof"(x0) -{n- l)(n + 2)f'(x0)
_2-n-n2+F + w+ n*)
— g
1 — x0
so that
2)xl
3(n - 2Lx2o - 4(n - 1)[2 -n-n+{<o + n + n2)xl] ^ 0;
whence
F.2.17) -I* |s (»-
1
1 \n{n2 +
The "true" constant, as n —> oo, in the second term is ji/2 = 2.891592 • • •
(instead of 5/2), where jx is the least positive zero of Jo(x) (see F.3.15)).
In the case of Hermite polynomials, from E.5.2)
/"(so) = 2xof'(xo), f'"(xo) = 2Bx1 - n + l)/'fa),
3(n - 2)xo - 2(n - l)Bx20 - n
whence
F.2.18) | xo | ^ ^Fajr = Bn + D* - g Bn

[ 6.21 ] ZEROS OF CLASSICAL POLYNOMIALS 121
This bound is better than that obtained by Sturm's method (cf. §6.31 D)),
although according to F.32.5) the "true" order of the second term is n~1/6
6.21. Inequalities for the zeros of the classical polynomials
A. Markoff's theorem (§6.12) furnishes several remarkable inequalities for
the zeros of the classical polynomials.
A) In discussing the zeros of Jacobi polynomials, we again enumerate the
zeros xv = cos dv in decreasing order:
F.21.1) +1 > xi > xt > ... > xn> -1; 0 < 0i < 02 < • • • < 0B < tv.
Theorem 6.21.1. Let {x, = x,(a, $)} denote the zeros of the Jacobi polynomial
Pn"'^(x) in decreasing order. Then
F.21.2)
In the ultraspherical case a
F.21.3)
?¦'<<>,
9a
= /3 we have
dxv
9^ *
V -
v = 1,
= 1,
2, •
2,
, [n/2]
Applying Theorem 6.12.1 to w(x, T) = A — x)a(l + x)p with a = r, p fixed,
or |8 = t, a fixed, we obtain the inequalities F.21.2) (cf. A. Markoff 4; Stieltjes
6, p. 76). In the first case we have, in fact, wT/w = log(l — x), which is a
decreasing function of x. The proof is similar in the second case.
The inequality F.21.3) for the ultraspherical case is due to Stieltjes F, p. 77).
For the negative zeros the opposite inequality holds. This inequality does
not follow directly from Theorem 6.12.1, since for w(x, r) = A — x2)T the ratio
wT/w = log(l — x2) is not monotonic. However, it follows immediately from
F.21.2) by using D.1.5).
[The proof of Stieltjes for F.21.2) and F.21.3) is entirely different from that
of Markoff and is based on. the differential equation (see below §6.22). Markoff
also attempts a direct approach to F.21.3) through a general theorem, but his
proof is incorrect. In his notation D, p. 181) the function
fl
is equal to ?/(log 1/A — if))'1 in the ultraspherical case. This function ap-
approaches + OT as y —* +0 and — <» for y —* — 0. Therefore, although/'(y) < 0,
nothing can be said about the sign of the ratio (f(y) — f(xi))/(y — xi). Inci-
Incidentally, the condition dV(y, ?)/d? > 0 of Markoff is not satisfied in the ultra-
ultraspherical case at y = e — 0:
In the general case, f(y) > 0 for y > e, and f(y) < 0 for y < e. This fact is
compatible with the decreasing property only if the denominator of f(y) becomes

122 ZEROS OF ORTHOGONAL POLYNOMIALS [ VI ]
0 as y —> e. The function f(y) is, however, in any case discontinuous at y = e,
so that in the general1 case the same criticism applies as in the special case
mentioned before.]
Apparently, Stieltjes was in possession of the general theorem of §6.12 (see
6, p. 79, section 5, and the remark on p. 88).
B) Theorem 6.21.2. Let the parameters a and /3 of the Jacobi polynomial
P(na'^(x) be subject to the conditions
F.21.4) -\ ^ a ^ +i -* ^ |8 ^
Then we have for the zeros {notation as before)
F-21'5) ?TTl'-s''ssST-ir
with equality only in the special cases a = — \, /3 = -\-\ and a = +4, /3 = — \,
respectively.
For Legendre polynomials, that is, for a = /3 = 0, this result is due to Bruns
A). The general case is due to A. Markoff and Stieltjes. For the proof we
observe that according to F.21.2) the maximum and minimum of xv — cos 0,
are attained in the special cases mentioned above. Now we use D.1.8).
C) Theorem 6.21.3. In the ultraspherical case
F.21.6) -i^a = /3^+i
we have the inequalities
F.21.7) (v - i) - ^ 6V ^ v -4- , v = 1, 2, • • •, [n/2],
with the equality sign valid only in the special cases a = /3 = — \ and a = /3 = -\-\,
respectively.
The first proof of these inequalities, which are more precise than the corre-
corresponding inequalities F.21.5) of "Bruns's type," is due to Stieltjes F). Mar-
Markoff's proof is not correct (see above). For the proof of Stieltjes see §6.22.
Refer also to §6.3 B) and C). Corresponding inequalities for the negative
zeros can readily be obtained from the symmetry relation xv + xn+i-* = 0.
We base our proof on Theorem 6.21.1. According to F.21.3), the maximum
and minimum of xv, v :§ [n/2], are attained if a = /3 = — ^ and a = /3 = +4>
respectively. Now the zeros of the polynomials D.1.7) are
F.21.8) cos (v — \) - and' cos v ——-, v = 1, 2, • • •, n.
n n + 1
D) In the case of Laguerre polynomials we have w(x, r) = e~xxa, with a = r,
and wr/w = log x increasing. Hence the zeros of Laguerre polynomials are

[6.22] PROOF OF STIELTJES 123
increasing functions of the parameter a, a > — 1. Thus, on account of E.6.1),
we obtain the following theorem:
Theorem 6.21.4. If
F.21.9) -i ^ a ^ +i
the zeros x, of L(na)(x), arranged in increasing order, have the bounds
F.21.10) ? ^ xv ^ vl ¦
Here ?„ and y, denote the vth positive zeros of the Hermite polynomials Hzn(x) and
#2n+i(x), respectively.
6.22. Proof of Stieltjes for the monotonic variation of the zeros of the
classical polynomials
Stieltjes F, pp. 73-77) gives a proof of Theorem 6.21.1 along the following
lines. Substituting x = xv in D.2.1), we have
I yl +1a +* +1 ? + x = _ J + ... + x
Ji V Ji Xy 1 Ji %y l" 1 Xy X\ Xy Xfi
F.22.1)
1 a±l
2 x 1
2 x, - 1 2 xv + 1
Differentiation of this equation with respect to a yields
1 (d_h _ ^l) 4. ! ^' _ ^.A 4- ...
(xv-XiJ\aa da) (Xv - X2J\da da)
F.22.2)
Xv - Xny\da da) 2 (x, - IJ da
4" 1 dXy 1 1
, 1
2
2 (Xv + IJ 9a 2 Xv - 1
or
F.22.3) Sfl./f'-—, p = ],2,..
where
1 , , 1 , 1 ,
a" (x xy ^ ^ (x xy ^ (x xy "*"'''
(xv x,y (x, x^y (x, x,+1y
F.22.4)
l 1 , 1 «+ 1 . 1 |8+ 1
^ (x, - xnJ ^ 2 (x, - IJ "^ 2 (x, + IJ'
and
F.22.5) aVM = aMV = - —-—^, v ^ m-
(x x;

124 ZEROS OF ORTHOGONAL POLYNOMIALS [ VI
The matrix (a,M) is positive definite since
n n
F.22.6)
1 ? ^^Y lyj « + l fl + 1 \ 2
2 ,^,^...n\x, - x,/ ^ 2 ?i\(x, - lJ^ (x,+ IJ
Stieltjes now uses the following theorem: If A = (aVM) is a positive definite
matrix with a,M < 0, v ^ m, then the reciprocal matrix D) has only positive
elements.
[We can assume that avr = 1, v = 1, 2, • • • , n, so that K = E — L, where E
is the unit form, and the coefficients of L are non-negative. Then the absolute
value of L is less than 1 HE =1, and the reciprocal form of K can be writ-
written as follows:
F.22.7) OK) = E + L + L2 + LZ+-...
All the forms of the right-hand member have non-negative coefficients, and the
coefficients of E + L are also positive.36]
By virtue of this theorem, the statement follows immediately from F.22.3).
The proof for the second inequality of F.21.2) is similar. The ultraspherical
case F.21.3) can either be treated by means of D.1.5), or directly handled (cf.
Stieltjes, loc. cit.). The same method applies to Laguerre polynomials (Theo-
(Theorem 6.21.4). In this case we have
1 4 4 1 , 1 ,
+ ^ * ^
=
a'y (X, - X:J
with Bxy)~1 on the right-hand side of the equations corresponding to F.22.3).
6.3. Sturm's method; Jacobi polynomials
Sturm's method (see §1.82) leads very simply to certain inequalities for the
zeros of Jacobi polynomials (see Szego 20, Buell 1). In this way we not only
confirm some of the results of §6.21, but we are also able to improve them to a
considerable degree. We assume in this section that
F.3.1) ~\^a^ +\, •-$??? +*,
35 This argument is different from that of Stieltjes, which is likewise very simple and
elementary. The present argument, however, furnishes similar theorems in more compli-
complicated cases.

[6.3] STURM'S METHOD; JACOBI POLYNOMIALS 125
excluding, in general, the case a = /32 = J; we arrange the zeros xv = cos 0, of
P(na'e)(x) in decreasing order:
F.3.2) +1 > xi > xo > ¦ ¦ ¦ > Xn > -1; 0 < 6Y < 92 < . ¦ • < 6n < v.
A) Theorem 6.3.1. Under the conditions mentioned we have
F.3.3) ».-^<r+(+T<, +
TTiis ftoZds /or a2 = /32 = | mYA Me = s^n instead of <. Here we define
f ° */ « > -i f t i/ ^8 > -*,
F.3.4) 0o = anrf dn+1 =
[-0: 7/ a = -J, [2t - 0n z/ 0 = -i
We notice that for a = /3 = — §, a = j3 = +2, « = — 0 = |, a = —/3 = — ?
F.3.5) »'=("-4)-» v^+i>
2>
IT V — h
respectively.
Inequality F.3.3) follows immediately from Theorem 1.82.1 by comparing
D.24.2) with
We consider the solution t; = sin {(n + (a + /3 + l)/2) @ — 0^)}. For a> —\,
the condition corresponding to A.82.2) is satisfied at x0 = do = 0 (and similarly
for /3 > -| at Zo = 0n+i = t).
B) Theorem 6.3.2. Under the conditions mentioned we have
whereas in the ultraspherical case a = C = \ — \
The bounds F.3.7) follow from F.3.3) by addition and by using D.1.3) (see
Buell 1, pp. 311-312). In case a = — \, the factor v of the upper bound can be
replaced by v — \, while if /3 = —\, the factor v + (a + /3 — l)/2 of the lower
bound can be replaced by v + (a + /3)/2. In the cases F.3.5) the same bounds
hold, at times with = replacing <. These inequalities are more precise than
F.21.5) provided a + /3 > 0. In the case of Legendre polynomials (a = /3 =*= 0)
they are identical with F.21.5), that is, with the inequalities of Bruns A).
The inequality F.3.8) holds only for the zeros 0 < 0, < t/2; it becomes an

126 ZEROS OF ORTHOGONAL POLYNOMIALS [ VI
equation for a = /3 = — ? or a = /3 = +?• It is more precise than the lower
estimate of F.21.7).
For the proof of F.3.8) we observe that according to F.3.3), the sequence
F.3.9) 6l = 6, - v-±?H^.Vt „ = 0, 1, 2, • • •, [(n + l)/2],
is decreasing. Now for n odd, we have 0(n+i)/2 = 0. For n even it suffices to
show that 0n/2 > 0. This follows from F.3.3), since 0n/2 + 6nn+\ = t.
A similar argument can be used to improve the left inequality in the general
case F.3.7) provided the zeros considered all lie in a certain preassigned part
0 ^ 0, ^ c of the interval [0, t\. For further proofs of F.3.8) see E) and
§6.6 B).
C) In this connection we prove the following theorem:
Theorem 6.3.3. Let n 2? 2. Under the condition —\<a = @< -\-\ the
sequence
F.3.10) 00 , 01 , 02 , • • • , 0[n/2]+l
of the zeros of Pi°'o)(cos 0) is convex, that is, 0V — 0v_i is increasing.
This follows by applying Theorem 1.82.2 to D.7.11), 0 < X < 1. In fact,
the coefficient of u in D.7.11) decreases monotonically. In the cases a = ?t =±\
the differences 0V — 0v._i are constant. Here again 60 = 0 if a > —\, while
0o = — 0i if a = — \. If n is even, the last term of F.3.10) lies in [t/2, t],
and then A.82.4) is used. For a > — \ condition A.82.5) is satisfied.
From Theorem 6.3.3 a similar convex property for the sequence {xv\ can
easily be derived (cf. Hille 4).
By means of Theorem 6.3.3 the upper estimate of F.21.7), more precisely
F.3.11) 6'<^T~\V> -h^cc< +\;v= 1,2, •••,[n/2],
can be proved in another way (Szego 20, pp. 5-6, 8). For, let — | < a < +•§.
The sequence
F.3.12) e" = 0v ^-, t, v = 0, 1, 2, • • • , [n/2] + 1,
n + 1
is convex; it therefore attains its maximum either for v = 0 or for v = [n/2] + 1.
Now do = 0[n/2]+i = 0 if n is odd. If n is even, we must bear in mind that
0n/2 + 0n/2+l = 0.
D) Finally, by means of Sturm's method, we derive certain inequalities which
involve the zeros of Bessel functions. In some respects these are more precise
than the preceding inequalities, although not so simple.
Theorem 6.3.4. Let a = & = \ — \, 0<X< 1. Denoting byji<j2<J3 < ¦ ••
the positive zeros of Bessel's function Ja(x), we have

[6.31 ] LAGUERRE AND HERMITE POLYNOMIALS 127
F.3.13) 0v<_iL_, , = 1,2,3, ...,n.
For X = 0 and X = 1 the sign < in F.3.13) has to be replaced by the sign =.
The statement follows by comparing D.7.11) with (see A.8.9))
F.3.14) d-^A
The estimate F.3.13) of 6y is the best possible in the sense that for a fixed v
and for n arbitrary the factor j, cannot be replaced by a smaller one since"
(Theorem 8.1.2)
F.3.15) lim n6v = lim ndvn = jr.
n-*oo n-*oo
Incidentally, for 0 < 0, ^ t/2, a similar lower bound for 0v can be obtained,
namely,
F.3.16) 6, > j\{(n + XJ + fc\(l - X)}"*,
where k is a positive numerical constant (Szego 20, p. 9). Then F.3.15) follows
from F.3.13) and F.3.16).
E) Finally, another remarkable property of the zeros 0, = 6vn of Pix)(cos 6)
can be proved by a proper application of Sturm's theorem. On substituting
0 = f/(n + X) in D.7.11) we obtain
(R 9 17^ dU 4_ /i 4_ X(l ~ X) \
F.3.17) - + |1 + ———w-TJ-]j u = 0.
If 0 < X < 1, then (n + XJ sin2 \%/(n + X) 1 increases with n provided ? is fixed
and 0 < ? < (n + X)t. Thus (n + \Nvn increases with n if v is fixed.86 From
these considerations the estimate F.3.13) follows again if F.3.15) is known.
Indeed, (n + XHvn < limn_oo (n + XHvn = jy.
As another application we can give also a new proof of F.3.8) since
F.3.18) (n + XH,n S: Bv - 1 + XH,,2>,_i = B? - 1 + X)t/2
for n ^ 2v - 1. Cf. Problem 32.
6.31. Sturm's method; Laguerre and Hennite polynomials
Suppose a > — 1.
A) Theorem 6.31.1. Let xv = xvn = xvn(a), v = 1,2, ¦ ¦ • , n, be the ze^os
of L^ix) in increasing order. Then
'6.31.1) *->n
Here > /ias the same meaning as in Theorem 6.3.4.
" By the separation theorem (Theorem 3.3.2) 0,n decreases as n increases for fixed v.

128 ZEROS OF ORTHOGONAL POLYNOMIALS [ VI
When we compare the third equation in E.1.2) with
F.31.2) U
"
+
\ x 4a;
which has the solution
kJa J2xJ (n + ^^
F.31.3) U
(cf. A.8.10)), the statement follows immediately. Condition A.82.2), x' = 0,
is satisfied in the present case.
An upper bound of x, of a similar kind can alsq be easily obtained. Let «
be a positive constant such that u < 4n + 2 (a + 1). Compare the same
equation E.1.2) as before with
F.31.4) „<< + /« + («+ i;/2 - -/4 LpVj, ,
( a; 4a;2 J
where 0 < x ^ «. Then
F'31'5) * <
n + (. +()/2
if the expression on the right-hand side is not greater than u. (For a fixed v, this
is the case, provided n is large enough.) The constant (j,/2J in the inequalities
F.31.1) and F.31.5) is the best possible in the sense explained in §6.3 D). For
a fixed v, we have, for the zero x, = x,n ,
C6.31.6) lim nx,n = 0",/2J.
The same results can be obtained by use of A.8.9) and the fourth equation
in E.1.2). Condition A.82.2), x' = 0, is again satisfied.
B) Both equations E.1.2) mentioned furnish upper bounds for the zeros
if we use Theorem 1.82.3 and take into account the fact that the correspond-
corresponding solutions vanish at x = + =c. The bound which is obtained from the fourth
equation is slightly better. It is given in the following theorem:
Theorem 6.31.2. The largest zero of L(na)(x) satisfies the inequality
F.31.7) xn < 2n + a + 1 + {Bn + a + IJ + | - a\h ^ 4n.
C) Introducing x = {n + (a + l)/2}~1^ in the third equation E.1.2),
we obtain
Since the coefficient of u increases with n, it follows that, for a fixed v, the
expression

[6.31 1 LAGUERRE AND HERMITE POLYNOMIALS 129
F.31.9) \n + (a + l)/2}z, = \n + (a + l)/2}_,n
decreases with increasing n. The limit as n —> <x> is 0V2J (according to
F.31.6)). An interesting consequence of this decreasing property is
F.31.10) \n + (a + l)/2}z,n ^ {„+(« + l)/2}z,,, n = v, p + 1, ¦ ¦ ¦ .
Applying F.31.7) and recalling F.31.1), we obtain the following result:
Theorem 6.31.3. Let a> - 1 ;for the zeros x,n ofL(na)(x), arranged in increas-
increasing order, the following estimates hold:
F.31.11) 2v + a + 1 + {Bv + a + IJ + j - q2}*
n + (a + l)/2
" = 1, 2, ••• , n;n = 1, 2, ... .
/n particular, for the least zero x\n we have
re 3112) °"l/2J < r < («+ i)(« + 3) _ 2 „
F3112) <Xlns -^~^r+r' n ~ h2>3> ¦ ¦ ¦ •
re 3112)
F.31.12) __
Here jv has the same meaning as in Theorem 6.3.4.
If v is large, (j.,/2J ^ 7rV/4 (cf. A.71.7)), while the coefficient in the right-
hand member of F.31.11) is ^Av'. For v = 1 we do not need F.31.7) since
xu can be calculated explicitly. In fact, xn = a + 1; whence F.31.12) follows.
E. R. Neumann B, p. 26) obtains for a = 0 an inequality similar to F.31.11)
in a different manner. He finds in this case
F.31.13) _,„ = Cvn ~~~- > p= 1,2, ... ,n;n = 1,2,3,-¦-,
where I < Cvn < 4. In view of the well-known estimate /„ > (v — %)tt (cf.
(8.1.4) and Problem 32), we can derive the following inequalities of the type
mentioned from F.31.11):
F.31.14) Ctt/16J < C,n < 4.
(The upper bound 4 can not be diminished; cf. (8.9.15); also Problem 33.)
W. Halm A, pp. 228-238) generalizes and extends Neumann's method to arbi-
arbitrary real values of a.
A part of these results is more precise than those occurring in the literature
(see W. Halm 2, pp. 228-230).
D) The corresponding considerations for Hermite polynomials are very
simple. To begin with, the second equation E,5.2) /urnishes the upper bound
Bn + 1)* for the zeros. This is not as good as the bound F.2.18). Further-
Furthermore, assuming n 2: 2, we obtain the convexity of the sequence of the zeros

130
ZEROS OF ORTHOGONAL POLYNOMIALS
VI
F.31.15)
Xln < X2n
of Hn(x), where xOn =-- 0 if n is odd, and xOn =-xinifn is even. In all cases,
x\n , xin , ¦ ¦ ¦ denote the positive zeros in increasing order. (See W. Hahn 1,
p. 244.)
On comparing the same equation with Z" + Bn + l)Z = 0, we find37
,. _ i
F.31.16)
!Bn + 1)* '
v
1.2.
, [n/2].
(This follows also from F.31.1) for a = ±f.) Now let w be a fixed positive
number, « < Bn + 1)*. Then
F.31.17)
xvn < •
Bn + 1
Ti
lBn+
7T,
= 1,2,3, ¦••,[n/2],
provided the right-hand members are not greater than «. For a fixed i> we
see that the constants (y — %)t and vtc are the best possible.
By introducing x = Bn + 1)~*?, the differential equation mentioned is
transformed into
F.31.18)
Bn
= 0,
z = exp {-Bn
The coefficient of z increases with n; hence (for fixed i>) Bn + l)*a;vn decreases
as n increases. Therefore, we have (cf. C)) Bn + l)*:c,n ^ Dj/ + l)Ja;v,2v or
Di> + 3)Ja;v,2v+i, respectively. Thus
F.31.19)
Bn
Bn + 1)*"
<
*Cyn
<
4v
kBn
+
+
+
1
r
iy
,=
, [n/2].
For the least positive zero xXn , we obtain (x12 = 2~*, a;i3 = C/2)J)
t/2 ] / 5/2 Y
< \2n+l/'
i\2n + 1/ '
F.31.20)
Bn + 1)*
Bn
n ^ 2.
37 In this and subsequent formulas the upper lino corresponds to the ease n oven, while
the lower lino to the case n odd.

16.32 1 THE LARGEST ZEROS 131
The upper bounds are more precise than those resulting from F.31.19) for
v = 1.
From F.31.20) we can derive bounds for the minimum distance dn between
consecutive zeros. The convex property mentioned above easily furnishes
dn '= Xin — xon , that is, dn = 2xin if n is even, and dn = xln if n is odd. It
follows that
F.31.21)
In every case we have
F.31.22)
Bn
T
+ 1)*
T
C dn ^ '
<rfn <
f 10*
Bn+ I)*'
B1/2)*
B1/2)*
n > 2.
Concerning the extensive literature on this subject, we refer to Laguerre B,
p. 105), Korous (I), Wiman (I), A. Brauer (I), Hille D), and Winston (I).
Hille obtains the same lower bounds as in F.31.20) and (by a suitable choice
of co in F.31.17)) the upper bounds
t/2
Bn + I)*
F.31.23) xln<\ , r , ^ NnJW
These bounds are better than those resulting from F.31.20), except for
n ^ 6. The results of the other authors are less precise than the preceding
inequalities.38
6.32. Sturm's method; the largest zeros of Laguerre and Hermite polynomials
(I) Let a > — I, and enumerate the zeros xv = x,n of Lna)(x) or Hn(x) in
decreasing order:
F.32.1) xi > x2 > x3 > ¦ ¦ ¦ .
Our purpose is to derive inequalities and asymptotic relations for xin , as well
as for x,n for fixed values of v as n —> oo.
Theorem 6.32. Let ix < i2 < i3 < • • • be the real zeros of Airy's function
A(x) (§l.8l, i\ > 0). // | a | ^ \, a > — I, the following inequalities hold for the
zeros [xv\ of L(na)(x):
F.32.2) x\ < (in + 2a + 2)* - 6~}Dn + 2a + 2)"*i,,
38 An exception is the lower bound in F.31.20) for the special values n = 2 and n = 3
in which Wiman's expression furnishes the exact values. For n = 3 the upper bound of
Wiman is the same as in F.31.20).

132 ZEROS OF ORTHOGONAL POLYNOMIALS [ VI ]
whereas for the zeros \x,} of Hn(x):
F.32.3) x, < Bn + 1)} - 6~}Bn + I)"**,.
Furthermore, we have for a fixed v
F.32.4) x\ = {An + 2a + 2)} - 6~}Dn + 2a + 2)~k\i? + *„},
F.32.5) xv = Bn + 1)} - 6~}Bn + l)"*{i, + en},
in the Laguerre and Hermite cases, respectively, where limn_oo en = 0.
These remarkable results possess an extended literature. (Zernike 1, W.
Hahn 1, p. 227. See also Korous 1, Bottema 1, Van Veen 1, and Spencer 1.')
In what follows, Sturm's method is used to prove F.32.3). A similar argument
can be applied for the proof of F.32.2) (cf. the fourth equation in E.1.2)).
Formulas F.32.5.) and F.32.4) follow from a certain asymptotic expansion of
Hermite polynomials due to Plancherel and Rotach as well as from correspond-
corresponding expansions for Laguerre polynomials; F.32.4) holds for arbitrary real a.
We discuss these formulas in Chapter VIII (of. §8.9 C)). They show that the
constant i, in F.32.2) and F.32.3) is the best possible if v is fixed and n is
arbitrary.
We notice that the expressions
{Dn + 2a + 2)* - 6~}Dn + 2a + 2)~iii}\
F.32.6)
Bn + 1)* - 6~}Bn + 1)~*A
are upper bounds for the zeros of Lia)(x) and Hn{x), respectively, \ a\ ^ \,
a > — 1. Here the constant
F.32.7) 6~*ti = 1.85575 •••
cannot be replaced by a smaller one. These bounds are more precise than
those previously given.
Alternative forms of F.32.4) and F.32.5) are
x\ = Dn)* - 6~}z;Dn)"J + o(n~*),
F.32.8) ii!
xv = Bny - 6~iiyBn) i + oin'*), n *-» «>,
re.spectively. Also, the first formula can be written as follows:
F.32.9) xv = 4n - 2-6~}^Dn)} + o(n}).
B) Writing hn = Bn + 1)}, we substitute x = hn — ? in the second equation
E.5.2) and obtain
F.32.10) ^ + Bhn? - f)z = 0.
Next we compare this equation with

t 6.32 | THE LARGEST ZEROS 133
F.32.11) d^ + 2hnZZ = 0,
which has the solution Z = ,4 {F/u)*?}; here A(x) is Airy's function defined
in §1.81. We can then apply Theorem 1.82.1 in [- a>, + a>]; condition A.82.2)
is satisfied at ? = - <x> (cf. the last remark in §1.81). Thus we have F hn)~hi,
< hn — x,, which establishes F.32.3).
C) We add a sketch of a direct proof of F.32.5) based on Sturm's method.
Let the real variable ? be subject to the condition | ? | 2= 2hnen , where 0 < en < 1;
we shall dispose of en later. Then
ffi,212x 2h > ,2> r—~€n) if *-0'
[2hnZ(l + en) if ? S 0.
We now compare F.32.10) with
F.32.13) ^| + 2/in?(l db e«)f = 0,
a?2
where the signs + and — correspond to ? ^ 0 and ? ^ 0, respectively. Using
the notation A.81.1), we consider, for — 2hnen ^ ? ^ 0, the solution
F.32.14)
with
F.32.15) Xn =
This solution vanishes for ? = — 2hnen . On the other hand, for 0 ^ ? ^ 2hnen,
we shall consider the solution
F.32.16)
At ? = 0 it has the same value and the same derivative as F.32.14) on account
of A.81.1). According to Sturm's theorem, Hn(hn — ?) oscillates more rapidly
in the interval — 2hnen S ? ^ + 2hnen than the function f = f (?) represented by
F.32.14) and F.32.16).
The only negative zero of f(?) is ? = — 2hnen . We now calculate the positive
zeros of F.32.16), that is, the values of ? for which
H-Xn)'
If en is small and Xn large, the right-hand member is nearly —1, and the pth
zero in question is, for a given v, near to Fhn)~\l — en)~}zF . If Xn is large
and positive, we obtain from A.81.3) and A.81.5)

134 ZEROS OF ORTHOGONAL POLYNOMIALS [ VI ]
F-32-18) 4F23 + : - S s 3'exp I -4(x"/3)!«¦
Now let j>0 be a fixed positive integer and e an arbitrary positive number.
We choose
F.32.19) en = K4% Xn = 2-6§(l + en)§a. > 2.6}w,
where w is a fixed positive number, so large that 2-6'w > iVo + 1, and ;the left-
hand member of F.32.18) is less than e. For sufficiently small values of e,
and as n —> oo, the first v0 zeros of F.32.17) then have the form
F.32.20) F/inr}(l - en)~\i, + 8), v = 1,2, ¦•¦ ,v0,
where | 5 | is arbitrarily small with e. (At any rate let | S \ be less than 1.)
We see that for large n, the expressions F.32.20) are less than 2hn en. Hence,
when we apply Sturm's theorem in — 2hnen S ? S +2hnen , we have
F.32.21) hn - x, < F/u)~§(l - en)~}(^ + 8).
This latter relation combined with F.32.3) establishes the statement F.32.5).
6.4. Theorem of Polya-Szego on trigonometric polynomials with
monotonic coefficients
Theorem 6.4. Let do > ai > ¦ ¦ ¦ > am > 0. Then the functions
f(t) = do cos mt + ax cos(m — l)t + • • • + am-i cos t + am ,
F.4.1) g(t) = a0 cos(m + %)t + ai cos(m — %)t + • • • + am-i cosC</2)
+ am cosi.t/2),
have only real and simple zeros; there is, respectively, exactly one zero in each of
the intervals \
F.4.2) ——j x < t < —-—| x and —-~ x < t < —t~]t>
yu = 1, 2, • • • , 2m, a?id /u = 1, 2, ¦ • • , 2m + 1, respectively.
The first part of the statement is due to Polya C, p. 359); his proof qses the
principle of argument (Theorem 1.91.1). The following proof furnishes Polya's
result again and, in addition, the inequalities F.4.2) (Szego 20, pp. 9-11). It
is based on Fejer's fundamental theorem (Fejer 1; see Polya-Szego 1, vol. 2,
pp. 78, 269, problem 17) which asserts that the sine polynomials
an(t) = sin(</2) + sinC*/2) + • • • + sin(n + \)t,
F.4.3)
n = 0, 1, 2, • • ¦ ; 0 < t < 2x,
are non-negative. ;
If f(t) and g(t) denote the conjugate functions of f(t) and g{t), respectively,

[ 6.5 ] FEJER'S GENERALIZATION 135
we have
(.0.4.4; —^Je jjyj + z/^JJ = — !je \g(t) + ig{t)\ = Zj^smin + §)<,
which is positive for 0 < t < 2x, according to Abel's transformation A.11.4).
Therefore,
,„ . fit) sin(m + |)< - 7(<) cos(m + \)t > 0,
F.4.5)
git) sin (to + 1)J - jj(O cos(w + 1)< > 0, 0 < t < 2x;
whence
F.4.6)
This shows the existence of at least one zero in each of the intervals F[4.2).
On the other hand, the functions F.4.1) cannot have more than 2m and 2ml + 1
zeros, respectively, in [0, 2x].
6.5. Fejer's generalization of Legendre polynomials
A) Starting from the representation D.9.3) of Legendre polynomials, Feje'r
(9) defines the "Legendre polynomials Fn(x) associated with a given sequence
ao, ai, <X2, ¦ • • "in the following way:
/^(cos 9) = 2a0an cos nd + 2aian_i cos (n — 2)9 +
F.5.1) J2a(n_i)/2a(n+i)/2 cos^, if n odd,
[a2n/2, if n even.
The classical Legendre polynomials Pn(x) are obtained if
F.5.2) ao = go= 1; an = gn = -^ Bn2~ 1}, n= 1,2,3, •••,
the ultraspherical polynomials P^(a;) if (§4.9 D))
F.5.3) ao = V, «»
Various properties, well-known in these special cases, can be extended to the
general polynomials Fn(x) by imposing proper restrictions on the sequence (an|.
These restrictions concern certain properties of monotony and asymptotic
behavior.
B) Theorem 6.5.1. The zeros of Fn(x) are real and simple and lie in the inter-
interval — 1 < x < + 1, provided an > 0 and the sequence
F.5.4) ai/ao , /0:1 , • • • , an/an_i , • • • ';

136 ZEROS OF ORTHOGONAL POLYNOMIALS { VIJ
is increasing. More precisely, each interval
nTl % < n~+~i T' " = 1> 2, • • •, n,
contains exactly one zero of Fn(cos 6).
See Szego 20, pp. 15-17. Under the condition mentioned the coefficients of
F.5.1) are decreasing, and the statement follows immediately from Theorem
6.4 if we write n = 2m or n = 2m + 1, according as n is even or odd, and 26 = t.
The condition in question is satisfied for Legendre polynomials and in the
ultraspherical case for 0 < X < 1.
The inequalities F.5.5) are not so precise as the Bruns inequalities F.21.5).
However, they hold for a comparatively general class of polynomials.
C) Theorem 6.5.2. Let the sequence {«„}, a« > 0, be completely monotonic,
that is, for all the differences™
/k\ /k\
A™ =: n I l/v .4-11™ . _1_ { 1^ n >H
(C K R\ " " 1 1 I "n+1 T I o I "n+2 • ' ' T~ I, i-J <*n-f k ~ U,
VD.o.o; \l/ \2/ '
k, n = 0, 1, 2, • • •.
Then the zeros xv = cos 6,, 0 < 9P < r, of Fn{x) are not only real and lie in
[—1, +1], but they also satisfy the inequalities F.21.7) of Stieltjes:
F.5.7) (v — j) - s 0v s v —¦—-, v ~ I, 2, • ¦ •, [n/2\.
n n + 1
Here the signs of equality hold if and only if Fn(x) is Tchebichef's polynomial of
the first (see below) or of the second kind, respectively.
See Feje"r 17, pp. 311-312. According to an important theorem of Hausdorff
A) the class of completely monotonic sequences {an}, an > 0, is identical with
the class of sequences which can be represented in the form
F.5.8) an= I fda{t), n = 0,1,2, ••-,
Jo
where a(t) is a non-decreasing function, not constant, with 2a{t) — a(t + 0)
-f a(t — 0) for 0 < t < 1. For sequences of this kind the condition of Theorem
6.5.1 is satisfied (Sch war z's inequality). The ultraspherical polynomials P(n\x),
0 < X < 1 are obtained [A.7.2)] if
F-5.9) an = (n + * ~ X) = x sin Xx jV^U - tTxdt, n = 0,1, 2,
Tchebichef's polynomials of the first kind are a limiting case of F.5.9) since
39 This definition of the differences of various orders is not the same as in B.8.4).

[6.5]
F.5.10)
so that
F.5.11)
FEJl
n
AT ION J37
1 i
f-'dt = -, n = 1,2,3,
lim X ^(cosfl) ^ sosnfl,
\->o
n = 1, 2, 3,
Tchebichef's polynomials of the second kind arise if an = rn, 0 < r <; 1, that is,
a@ has only one point of increase in 0 < t ^ 1.
FejeVs argument is as follows:
F.5.12) Fn(cos 6) = [ [ ( ? tkun~k cos (w - 2k)e\da(t) da{u).
Jo Jo [k~>o J
An elementary transformation of the integrand gives
~ ^)(tn+1 ~ un+i) 2tu(t* + u")sin20 sin (n + 1)8
^ + ^ no + iT_
so that
F.5.14)
Fn(cos 6) = An(d) cos nd + Bn(d)
where An@) and .Bn@) are positive functions in 0 < 9 < x provided a(<) has
at least two points of increase. From this it follows that
sgn Fn< cos (v - ?) - > = —sgn Fn < cos v
F.5.15)
v = 1, 2, • • •, [n/2],
which establishes the statement.
The last part of this argument is similar to that used in the proof of
Theorem 6.4.
D) Fejer considers B0, pp. 40-45) another remarkable generalization of
Legendre polynomials. He starts from the representation D.9.5). Let
j3m I 0, and
F.5.16)
Gn(cos 6) = 0o sin (n + 1N + ft sin (n + 3N + • • •
sin (n + 2m + 1H + • • •.
This series converges for 0 < 6 < x (see §4.9 B)). Legendre polynomials are a
special case, as well as the more general functions (sin 0JX-IP^ (cos 6),
X > 0, X 7^ 1, 2, 3, ¦ • ¦ (see D.9.22)). In these cases the sequence /?m = /ix>
(using the notation of D.9.22)) is completely monotonic. In fact, we have
-fm+1 -

138 ZEROS OF ORTHOGONAL POLYNOMIALS [ VI ]
The sequence {ym\ is completely monotonic and, on account of a well-known
formula, we have
k /i\
(,0.0.18; A Jm — A \Jm — Jm+i\ — Z^l )Ajm A 7m+(,.
Hence the statement follows by induction.40
Fejer shows (loc. cit.) that Gn(cos 0) has at least one zero in each interval
F.5.19) (v-h)~ <0 <»-?—, * = 1,2, •••,[n/2],
provided
F.5.20) 0m ^ 0, A/3m ^ 0, A2;3m ^ 0, A^ ^ 0, m = 0, 1, 2, ¦ • • .
His proof is based on the positiveness of certain special trigonometric poly-
polynomials. We shall prove the following theorem:
Theorem 6.5.3. The junction (?n(cos 6) has at least one zero in each interval
F.5.19) provided }/?„,} is a completely monotonic sequence.
This condition is more restrictive than that of Fej?r. The proof is, how-
however, very simple. Using F.5.8), with /?„ and I3(t) in place of an and a(t),
respectively, we obtain
\1t
O lm=0
G«(cos e) = \ \1ttm sin (n + 2m +
l
F-5.21)
-i-72sm fl oos nd + / i—^— no i .2sm
+ t2 J 1 2< 20 + <2
fo 1 — 2t cos 20 + t2 Jo 1 — 2< cos 20 +
From this point the statement follows in the same way as in the proof of
Theorem 6.5.2.
6.6. Recapitulation; additional remarks on ultraspherical polynomials
A) We have obtained the following inequalities for the zeros x, = cos dv ,
0 < dv < x, (arranged in decreasing order) of the ultraspherical polynomial
Pia'a)(x), a = X - h, provided 0 < X < 1:
(a) Inequalities F.5.5), derived from the representation of Pi"'a)(cos 0) as a
cosine polynomial:
F.6.1) V-^~ x < 0, < *-J-f T, „ = l, 2, ...,n.
n + 1 n+ 1
(b) Inequalities of the Bruns type:
F.6.2) JL^ !
40 For X = 1/2 this follows directly from D.9.9).

6.6] RECAPITULATION; ADDITIONAL REMARKS 139
F.6.3) 'JL^Zi^K-L.., ,-1,2,
*.
Inequalities F.6.2) follow from F.21.2), which was proved by A. Markoff and
by Stieltjes in two different ways (cf. F.21.5)); F.6.3) is a special case of the
more general inequalities F.3.7) proved by Sturm's method. The inequalities
F.6.2) are more precise than those of F.6.1) and those of F.6.3) with X < \]
the opposite is true if X > \.
(c) Inequalities of Stieltjes' type:
F.6.4) V^J x < 6. < ^-x x, „ = 1, 2, • • •, [n/2].
These follow from F.21.3) and were proved by Stieltjes. They can also be
readily derived from F.21.2) (which is due to A. Markoff and to Stieltjes).
Feje> obtains them from D.9.19) or D.9.22) (cf. Theorems 6.5.2 and 6.5.3).
An alternative proof for the upper bound is due to Szego (Sturm's method,
§6.3 C)). The upper bound is better than that in each of the preceding in-
inequalities; the lower bound is better than that in F.6.2), and is better than
that in F.6.3) provided X < \.
(d) Szego's lower bound:
F.6.5) B, > " " A ~ X)/2 T, ,, = 1,2,..., [n/2].
This follows by Sturm's method in two different ways (cf. 6.3 B) and E)).
For a third method see B) below. This lower bound is more precise than
any of the preceding ones.
B) By combining the integral representation D.82.3) with the argument
used in the proofs of Theorems 6.4, 6.5.2, 6.5.3,
(e) Feje> obtains A9, p. 208)
F.6.6) '-^nr1»< * < '4rir *¦- = >¦ >¦¦¦¦¦ w
The lower bound is the same as in F.6.5). The upper bound is less or greater
than that in F.6.4) according as X < -? or X > ;}.
For the proof we substitute the bound in F.6.5) for 9 in D.82.3), 0 < d < x/2,
and find that
F.6.7) sgn PLX)(cos d) = (-1)' sgn ^^'"'""(l - te2ie)^}.
It happens that the last sign is constant if t varies in 0 < t < 1. Indeed, the
argument of the expression in the braces lies between 0 and — X(x/2 — 0).
Thus F.6.7) becomes (-1)'+1.
Upon substituting the upper bound of F.6.6) in D.82.3), we have
F.6.8) sgn P^X)(cos d) = {-!)" sgn 2Tx
which establishes the statement.

140 ZEROS OF ORTHOGONAL POLYNOMIALS [ VI ]
To recapitulate: the lower bound F.6.5) is the best of all lower bounds given
here; while in the case of the upper bounds, either F.6.4) or F.6.6) is best
according as X > § or X < f. Here we did not refer to inequalities which in-
involve zeros of Bessel functions.
6.7. Electrostatic interpretation of the zeros of the classical polynomials
Stieltjes gave D, 5; 6, pp. 75-76; cf., also, Schur 1) a very interesting deriva-
derivation of the differential equations of the classical polynomials, which is closely
connected with the calculation of the discriminant of these polynomials (cf.
§6.71) and can be interpreted as a problem of electrostatic equilibrium.
A) Problem. Let p and q be two given positive numbers. If n unit "masses,"
n > 2, at the variable points X\, x2, x3, ¦ ¦ ¦ , xn in the interval [ —1, +1] and
the fixed masses p and q at -\-\ and —1, respectively, are considered, for what posi-
position of the points X\, x%, Xz, • • ¦ , xn does the expression
F.7.1) T(xr,x2, ¦¦¦,xn) = T{x) = II A - xr)'(l + **)' II 1^-xJ
e<j»
become a. maximum?
Obviously, log G1) can be interpreted as the energy of the system of electro-
electrostatic masses just defined. They exert repulsive forces according to the law
of logarithmic potential. The maximum position corresponds to the condi-
condition of electrostatic equilibrium. A maximum exists because T is a continuous
function of Xi , x2 , ¦ ¦ ¦ , xn for — 1 ^ x, S +1, v = 1, 2, • • • , n. It is clear
that in the maximum position the xv are each different from ±1 and from one
another. In addition, this position is uniquely determined. To show this, let
us suppose that (cf. Popoviciu 2, p. 74)
+ 1 > xi > xi > ¦ ¦¦ > xn > -1,
F-7-2)
+ 1 > Xi > X2 > ¦¦¦ > Xn > "I
are two positions of this kind; we write
F.7.3) y, = (X, + x,)/2, v = 1, 2, • • • , n.
Then
F.7.4)
Xy
1
—
db
Xn
y»
+
2
All
/
1 ±a
•J1
All
|1
|Xi.
±
—
/
XM
Xu Xu
so that T(y) ^ {T(x)}i{T(x')}i, the equality sign being taken if and only if
xy = x'y . This establishes the uniqueness.
Theorem 6.7.1. Let p > 0, q > 0, and let {xv}, -1 ^ xv ^ +1, be a system
of values for which the expression F.7.1) becomes a maximum. Then the {xv)
are the zeros of the Jacobi polynomial Pna^\x), where a = 2p - 1, P = 2q - 1.

['6.7 ] ELECTROSTATIC INTERPRETATION OF ZEROS 141
From this fact the uniqueness of the maximum position follows again. For
a maximum we have the conditions dT/(dxv) = 0, or
+ + +
Xy X\ Xy Xy—\ Xy "~~
Xy ' Xfi Xy 1 Xy \~ 1
If we introduce the polynomial f(x) = (x — Xi)(x - x2) ¦ ¦ ¦ (x - xn), this
becomes
F.7.6) l^
2 f (Xy) Xy—l Xy + 1
or
A - xl)f"(xy) + \2q - 2p - Bg + 2p)Xy}f'(Xy) = 0.
The last equation means that A - x2)f"(x) +{p-a-(a + P + 2)x}f'(x)
is a xn which vanishes for all the zeros of f(x); whence this expression is equal
to const. f(x). By comparing the terms in xn we obtain for the constant factor
the value —n(n + a + j8 + 1). The resulting differential equation reduces to
D.2.1), so that according to Theorem 4.2.2, f(x) must be a constant multiple
of Pia-*\x).
See also Problem 37.
B) The zeros of Laguerre and Hermite polynomials admit a similar inter-
interpretation.
Theorem 6.7.2. Let us consider the positive mass p at the fixed point x — 0
and unit masses at the variable points X\, Xi, ¦ • ¦ , xn in the interval [0, + °° ]
such that, the "centroid" satisfies
F.7.7) n-\xx + x2+ ... +xn) g K,
where K is a preassigned positive number. Then the maximum of
F.7.8) U(xi, x2) ¦ ¦ ¦ , xn) = fl tf II I x, - x, |
* —1 e,j»—1,2,- ¦ -,n
is attained if and only if the \xv] are the zeros of the Laguerre polynomial Ln" (ex),
where a = 2p — 1, and c = K~l{n + a).
Theorem 6.7.3. Let us consider a unit mass at each of the variable points
xx, X2 , • • • , xn in the interval [— °°, + °°] such that the "moment of inertia"
satisfies
F.7.9) n'\x\ + xl + • • • + xl) ^ L,
where L is a preassigned positive number. Then the maximum of

142 ZEROS OF ORTHOGONAL POLYNOMIALS [ VI ]
F.7.10) V(Xl,x2, ¦¦¦ ,xn) = II \x,-x,\
is attained if and only if the \xv\ are the zeros of the Hermite polynomial Hn(c'x),
d = {2LY\n - l)h.
The existence and uniqueness of the position of maximum is clear in both
cases. The corresponding xv are all different from one another; in the first
case they are positive. It is clear, furthermore, that for the maximum position
the sign of equality holds in F.7.7) and F.7.9). Hence, if p is a proper
"multiplier," we have
. J*>V **-l ^c »*2 JLy Jbn Xv ll
F.7.11)
= 0
in the first case, and
F.7.12)
f"{xv) - ^ xvf'{xp) = 0
in the second. In both cases we have written
fix) = (x - xx)(x - x2) ¦ ¦• (x ~ xn).
If we replace x by ex, with c a proper constant factor, these conditions can easily
be reduced to the first equation in E.1.2) and to the first equation in E.5.2),
respectively.- Therefore,
f{x) = const. L(n\cx), a = 2p — 1, c = 2pn~1,
in the first case, while f{x) = const. Hn{c'x), c' = Bp/n)* in the second case.
The constants c and d can be determined from the conditions F.7.7) and
F.7.9), in which the equality signs now hold. We observe that according to
E.1.6) the sum of the zeros of L^ix) is equal to n(n + a); according to E.5.4)
the sum of the squares of the zeros of Hn{x) is equal to n(n — l)/2.
Cf. also Problem 38.
6.71. Discriminants of the classical polynomials
The maximum problems treated in the preceding section are closely related
to the calculation of the discriminants of the classical polynomials (Hilbert 1,
Stieltjes 4, 5). The following method is due to I. Schur B) (cf. Popoviciu 2).
A) Let {pn(x)\ be a sequence of polynomials satisfying the recurrence
formula

[x6.71
DISCRIMINANTS OF CLASSICAL POLYNOMIALS
143
F.71.1)
Pn{x) = (anx + bn)Pn^(x) - cnPn_i{x), n = 2, 3, 4, • • • ;
po(x) = 1; pi(x) = axx + 6i.
We suppose that ancn ^ 0. Denoting by {xvn) the zeros of pn(x), we show that
F.71.2) An = ft Pn_,{xvn) = (_!)-<»-"/» ft {ar^c-1}, n = I, 2, 3, • • • .
Suppose n ^ 2. The coefficient of xn~l in pn_i(a;) is.aia2 • • • an_i , so that
- xi,B_i)(a;w - x2,n-i) • • • (a;,n - xB_i|B_i)
F.71.3)
c—1
• • an-i)n / \
F.71.4)
n-l
Using the recurrence formula, we obtain
A» = das • • • an_i-a1n~n(-cn)n-1 II p»_»(^,n-i)
= (- 1)"-1 ax 02 • • • an_! • an-ncrJ An_!,
which establishes the statement.
B) Theorem 6.71. The discriminants of P(na^(x), L(na)(x), Hn(x) are
F.71.5) D™ = 2-("-1) ft /-2n+2(, + a)"-\v + py-\n + v + a + /3)n-,
c=l
F.71.6) ?>„"' =
F.71.7) Z>n =
We start from the familiar expression (cf., for instance, 0. Perron 4, vol. 1,
p. 259, A2), A3); p. 260, A6))
F.71.8)
where l(na<® has the same meaning as in D.21.6), and \xvn) denotes the zeros
of Pl"l0)(x). The discriminants Dna) and Dn admit a similar representation.
According to the first formula in D.5.7), we have

144 ZEROS OP ORTHOGONAL POLYNOMIALS [ VI
F.71.9) A - x2) ?. (P^fa)} = ^n + «"" + PJ p<-jf fa) if
aa; 2n + a + /3
so that
F.71.10)
The last factor can be calculated by means of F.71.2), so that on account of
D.21.6), D.1.1), D.T.4), and D.5.1), we obtain F.71.5).
The expression F.71.6) can be calculated in the same way by using E.1.14),
E.1.8), E.1.7), and E.1.10); or even more simply, from F.71.5) by using the
limiting process of E.3.4). Indeed, if [xvn = xvn{fi)\ denotes the zeros of
P(na'®{x) in decreasing order, we have, for fixed v and n,
F.71.11) lim/3(l -x,n) = 2f,»,
where {?„„} are the zeros of L(n\x) in increasing order. Therefore,
/)(«) = G(«>Jn-2 TT (t _ t J
c,M=l,2,- • -,n
^R7i to\ = "» ) lim (/3/2)
which establishes the statement.
The discriminant Dn can also be obtained either directly, or from F.71.5)
by using E.6.3), or from F.71.6) by using E.6.1). The first method is the
simplest. By using E.5.6), E.5.10), E.5.8), and F.71.2), we find F.71.7).
6.72. Distribution of the zeros of the general Jacobi polynomials
A) Let a and /3 be arbitrary real numbers, n ^ 1, and let P(n'®{x) denote
the generalized Jacobi polynomials defined in §4.22. Then F.71.5) still holds.
It follows from D.1.1) or D.21.2) that x = +1 is a zero of Pia-0)(x) if and
only if
F.72.1) a = -1, -2, •• • , -n.

6.72
ZEROS OF JACOB! POLYNOMIALS
145
(The multiplicity of this zero is \a\; cf. D.22.2).) Similarly, x = -1 is a
zero if and only if (cf. D.1.4))
F.72.2) 0 = -1, -2, ... , -n.
Finally, it follows from D.21.6) that x = °o is a zero, if and only if
F.72.3) n + a + 0 = -1, -2, .
-n.
If such values of a and 0 are excluded, the zeros of P^'^ix) are different from
±1 and =o; in addition, F.71.5) shows that they are distinct. (This follows
also from D.2.1); cf. §6.2 C).) Let Ni, N2, N3 be the number of zeros in
— 1 < x < +1, — =o < x < —1, and +1 < x < + oo, respectively. We shall
now determine these numbers as functions of a and 0.
Hilbert A) calculated the number Ni + N2 + N3 of the real zeros. A remark
of Stieltjes E, p. 444) indicates that he obtained the numbers Nx, N2, N3 three
years before Hilbert's paper. The later results of Klein concerning the number
of the zeros of the general hypergeometric function A, pp. 562-567) readily
lead to these numbers. (Cf. also Shibata 1, Fujiwara 1, Sen-Rangachariar 1.)
By use of Klein's .symbol
' . 0 if m^O,
F.72.4) E(u) = ¦ [u] if u > 0, u non-integral,
u - 1 if u = 1, 2, 3, ¦•• ,
we can formulate the following theorem:
Theoeem 6.72. Let a, /3 be arbitrary real values, and set
X = X(a,P) = E{$(\2n + « + P + M - |«l - \P
F.72.5) Y = Y(a,l3) = #{J(-| 2n + a + 0 + 1 | + | | |
Z = Z(a, fl) = E{\{-\ 2n + a + /3 + l|
a | -
1I,
+ 1)},
If we exclude the cases F.72.1), F.72.2), and F.72.3), the numbers of the zeros of
P(naJ)(x) in -1 < x < +1, —oo < x < — 1, +1 < x < +oo, respectively, are
F.72.6) JVi = Ni(a, 0) =
if (-
F.72.7) N2 = N2(a, 0) =
2[X/2] + 1 if (-l)nl
2KFH-D/2] if Bn+:+f>)(ny)>».
wm+i if Bn+:+%n+/)<o,

146
ZEROS OF ORTHOGONAL POLYNOMIALS
VI
F.72.8) N, = N3(a, j8) =
if
We notice that the numbers 2[{X + l)/2], 2[Z/2] + 1 can be characterized,
respectively, as the even or odd of the numbers X and X + 1, so that iVi
is either X or X + 1. We also see that the conditions in F.72.6) are
equivalent to P?° n(l)Pln"lp)(-l) > 0 or < 0, respectively. For instance, if
sgn PiaJ)(l)P(naJ\-l) = (-1)* or (~1)*+1, then ^(a, |3)=IorI+l,
respectively. Similar remarks hold for N2 and N3 .
E(n + 1 + a)
E(n + 1 + a + /$)
E(-n) =
O
E(n + 1) = n
0
a
E(n
Fig. 7
?¦(-«- a) =
It is sufficient to calculate Ni. On account of D.22.1) we obtain Nz by
replacing a by — 2n — a — /3 — 1 in F.72.6), while Ns is obtained from N2 by
interchanging a and /3. The following proof is based on the continuity of the
zeros as functions of a and /3.
B) For convenience, we introduce the notation M(a, /3) for the function of
the right-hand member of F.72.6) so that we must prove Ni(a, ?) = M(a, /3).
If the point (a, /3) varies, the function Ni(a, /3) can change only if (a, ?) crosses
one of the straight lines F.72.1) or F.72.2). We first show that the function
M(a, /3) has the same property.
An easy calculation furnishes the value of X(a, /3) in the seven regions
bounded by the a- and /3-axes and by the straight line 2n + a + /3+l = 0
(see figure 7). The only discontinuities of the function E{u) are at the points

[ 6.72 ] ZEROS OF JACOBI POLYNOMIALS 147
u = 1, 2, 3, • • • ; hence, apart from the straight lines F.72.1) and F.72.2), a
jump of the function M(a, 0) is possible only in the triangle where X(a, 0) =
E(n + 1 +a + /3)ifn + l+a + /3 coincides with an integer k, 1 ^ k ^ n.
Let a = ao and /3 = /30 be negative non-integers, n + 1 + a0 + A> = k, and
let the integers p and g be chosen such that
F.72.9) -p < ao < -V + 1, -q < po < -q + 1, 1 ^ p ^ n, 1 ^ q ^ n.
Then we necessarily have k = n + 2-p — q. We investigate Mia, 0) for
n + l + a + /3 = A;db€, 0<e<l, where | a - a0 \ and | /3 - /3o | are suffi-
sufficiently small. Obviously,
furthermore, X(a, /3) = E(n + 1 + « + /3) = ^(A; db e) = k or A; - 1, respec-
respectively. In other words k = Xork = X + l, respectively. Then (see the
remark concerning Theorem 6.72) M(a, /3) = k in both cases, so that no change
occurs in M(a, /3).
C) We now show that when (a, /3) crosses one of the lines F.72.1) or F.72.2),
the jumps in Ni(a, /3) and M(a, /3) are the same. This will prove the statement
F.72.6), since, for a > 0, 0 > 0, we have #,(«, 0) = n and X(a, 0) =
Mia, 0) = n.
Because of the symmetry in a and 0, it is sufficient to consider the case
a = — A; db e, e > 0, 1 ^ k ^ n, P not an integer, and to discuss the location of
the zeros near x = +1.
From D.21.2) we obtain for a = — k + e
F72io) + * + 1) •••(« + n) n W
= co(e) + Cl(e)(x - 1) + ... + Cfc_1(e)(a; - I)* + (x - l)k
+ ck+1(e)(x - l)k+1 + ... +-cn(e)(x - l)n.
Here the coefficients are real rational functions of e, regular for e = 0, and
F.72.11) Co(O) = c,@) = ... = Ck-^0) = 0.
Furthermore,
4(W(a+D(« +2) •••(q + fc).
By means of a simple consideration from the theory of analytic functions (see
below), we now obtain the following result. If 5 is an arbitrarily small positive
number E < sin ir/k for k > 1), then for sufficiently small values of e > 0, the

148 ZEROS OF ORTHOGONAL POLYNOMIALS [ VI ]
function Pna> (z), a = — k + e, has exactly k distinct zeros in the neighborhood
of a; = +1. More precisely, if Vi, V2, vs, • • • , Vk denote the numbers satis-
satisfying the equation
F.72.13) sgn c'0@) + rf = 0,
these roots lie in the circles
F.72.14) x = 1 + (e | c?@) |I/fc(,, + z), | z | < 6, v = 1, 2, ¦ ¦ ¦ , k.
By replacing e by -e (that is, for a = —k — e), the same result holds with
the circles
F.72.15) x = 1 + (e | cJ(O) |)lM(^ + z), | z | < 6,
where f i, f2, • • • , f * are the roots of
F.72.16) -sgn co(O) + f* = 0.
The root corresponding to a real ??„ or f, is obviously real.
To prove the previous statement we introduce x = 1 + (e | c'0@) \ )Uhy in
F.72.10) and obtain
+ ~ ci'CO) +•••+;, ^@) + • • • + Cl(e)(e | ^@) \)llk y
F.72.17) + • • • + U«)(« I co'(O) D^-^y-1 + € I c'0@) I /
+i + ... + Cn(€)(€ 1 Co'(O) |)n/y = 0.
If this be divided through by e j co(O) | , the "principal terms" are sgn co(O)
+ y°. Now we can apply Theorem 1.91.2 (Rouche"'s theorem); whence the state-
statement follows immediately.
We are interested especially in the number of real zeros x < +1 near x — +1.
From the preceding result we see that this number increases or decreases by
one unit if we replace e by — e, according as ( —l)*Co(O) is positive or negative.
With reference to F.72.12) this is equivalent to the condition that
tO<o or >o'
respectively.
D) On the other hand, we discuss the jump of M{a, /3) as (a, /3) crosses
the line a = — k with /3 non-integral. First suppose /3 > 0, so that Z(a, /3)
= E(n + 1 + a). For a = ~*k ± e, e > 0, we have
in + i - k,
F.72.18) X(a, /3) = E(n + 1 - k db e) =
\n - k,
respectively, and

6.72 ] ZEROS OF JACOBI POLYNOMIALS 149
a)(n
F.72.19) sgn (-l)"(n + a)(n + ?) = sgn (_i> (
respectively, which in both cases is (-l)*(otl/J>. Hence (cf. the remark to
Theorem 6.72) M(a, /3) changes from n + 1 - k to n ~ k, a loss of one unit.
In this case
Now let /3 be negative and non-integral. Then n + l+« + /3is non-
integral near a = ~k, so that X(a, /3) remains constant in this neighborhood.
More precisely,
(n+1 - k + \fi] if n + 1 - k + 0 > 0,
F.72.20) X(a, 0) =
I 0 ifn+i_& + /3<o.
In the first case we have for a = — A; ± e, e > 0,
F.72.21)
respectively, which are (_i)jr<a'«+1 an(j (_i)^(a.«; respectively. Hence (cf.
the remark to Theorem 6.72), M(a, /3) changes from X{a, /3) + 1 to X{a, /3),
a loss of one unit. In this case, again
Let us now consider the second case in F.72.20). Then
F.72.22) g(r^
j or (
respectively, for a = — A; ± e, e > 0. For instance, if
M(a, 0) changes from 0 to 1, a gain of one unit. The opposite is true if
This establishes Theorem 6.22.

150 ZEROS OF ORTHOGONAL POLYNOMIALS [ VI ]
6.73. Distribution of the zeros of the general Laguerre polynomials
The discriminant formula F.71.6) is valid for general Laguerre polynomials
L(na){x), a arbitrary and real, n ^ 1. On account of E.1.7), x = 0 is a zero of
L{n\x) when and only when
F.73.1) a = -1, -2, ... , -n.
(Its multiplicity is | a |; cf. E.2.1).) If such values of a are excluded, the zeros
of Lia)(x) are finite and different from 0; in addition, we conclude from F.71.6)
(or from the first differential equation E.1.2)) that they are distinct. Let
ni(a) and rh(a) denote the number of positive and negative zeros, respectively.
By using Theorem 1.91.3 (Hurwitz's theorem) and E.3.4), we see that if /3 is
large, P\?'®(x) has at least ni(a) zeros in [ — 1, +1], at least nz(a) zeros in
[+1, +°°], and at least n — ni(a) — n2(a) zeros which are not real. There-
Therefore, using the notation of the preyious section, we see that ni{a) = Ni(a, /3)
and ni{a) = N3(a, /3) if /3 is large; that is,
F.73.2) ni(a) = lim iVi(a, /3); ^(a) = lim N3(a, /3).
Obviously, if a > —1, then ni{a) = n, and n2(a) = 0.
Now suppose a < — 1, a t^ —2; —3, • • • , —n. From F.72.5) we obtain
in + [a] + 1 if a > -n,
lim X(a, /3) = E(n + a + 1) = \
0-+00 [ 0 if a < —n,
since the argument of E in the formula for X(a, /3) is not a positive integer;
furthermore,
limZ(a, 0) = E{-n) = 0.
Now
a > — n,
a < — n.
Thus, Ni(a, /3) = n + [a] + 1 in the first case, and Ni(a, /3) = 0 in the second.
Therefore,
in + [a}+ 1 if a > -n,
F.73.3) rh(a) =
[ 0 if a < — n.
Furthermore,
F.73.4) nx(a) = 0 or 1,
according as
. , / nrt -X- rv \
0.
n

16.8] THEOREM OF HEINE-STIELTJES 151
Theorem 6.73. Let a be an arbitrary real number, a ?± — 1, — 2, • • ¦ , — n.
The number of the positive zeros of L^ix) is n if a > — 1; it is n + [a] + 1 if
— n < a < — 1; it is 0 if a < — n. The number of the negative zeros is 0 or 1.
This result can also be obtained by a direct method similar to that used in
§6.72. In fact, the numbers n^a), n2{a) can change only if a passes one of the
integers —1, —2, • • • , — n. If a decreases through an odd value of this kind,
a positive zero is lost and a negative zero gained. If the passed value is even,
a positive and a negative zero are lost. For a < — n, there are no real zeros if n
is even and one real zero (which is negative) if n is odd. (Compare Lawton 1,
W. Hahn 1.)
6.8. Polynomials which satisfy a second-order linear homogeneous differential
equation with polynomial coefficients; theorem of Heine-Stieltjes
Heine C, vol. 1, pp. 472-479) has studied the following problem:
Peoblem. Let A(x) and B(x) be given polynomials of degrees p + 1 and p,
respectively. To determine a polynomial C{x) of degree p — 1 such that the
differential equation
F.8.1) A(x) 0 + 2B(x) d? + C(x)y = 0
has a solution which is a polynomial of a preassigned degree n.
Heine asserts that, in general, there are exactly
_ _ (n + p — Is
F.8.2)
a =
determinations of C(x) of this kind.
The hypergeometric equations D.2.1) and D.21.1) are of this type with
p = 1. Lame" functions satisfy an equation of the same type with p ^ 2.
These cases were the starting points of Heine's investigations on this subject.
Stieltjes C) discusses only a special case of F.8.1) which, however, is of
primary importance. He obtains the following result:
Theoeem 6.8. Let A(x) and B(x) be given polynomials of precise degree
p + 1 and p, respectively, and let the highest coefficients of A(x) and B{x) have the
same sign. If the zeros of A(x) and B{x) are real, distinct, and alternating with
one another, there are exactly <x polynomials C(x) of degree p — 1 such that the
differential equation F.8.1) has a solution which is a polynomial of degree n.
Here <r has the meaning F.8.2).
The proof of Stieltjes uses a part of Heine's assertion, namely, that <r is an
upper bound for the number of polynomials C(x) in question. He obtains,
however, not only the existence but also a characterization of the a solutions
mentioned, in the following way: The n zeros of these solutions are distributed

152 ZEROS OF ORTHOGONAL POLYNOMIALS [ VI ]
in all possible ways in the p intervals defined by the p + 1 zeros of A (x). (The
number of such distributions is obviously a.) The solutions are obtained by
means of a maximum problem similar to those treated in §6.7.
The following proof of Stieltjes' theorem uses the latter's idea based on a
maximum problem, but Heine's elimination process (which furnishes the upper
bound a) is replaced by certain elementary considerations related to Sturm's
theorem (see §1.82). Our proof is, consequently, independent of Heine's work.
6.81. Preliminary remarks
We assume with Stieltjes that
F.81.1) A(x) = (x — ao)(x — ai) ¦ • ¦ (x — ap), a0 < ai < • • ¦ < aP)
and
F.81.2) ^=_*L_ + -^L_ + ... + _??_, Pv>0,v = 0,1,2, ¦¦-,?.
A (x) x — a0 x — ai x — ap
This is equivalent to the assumption that the zeros of A(x) alternate with
those of B(x) and that the highest coefficients of A(x) and B(x) have the
same sign.
Let C(x) be a given polynomial. Then F.8.1) cannot have two polynomial
solutions y and z linearly independent of each other. Otherwise, we should
have for x ?± av
A(x)(y'z - yz'Y + 2B(x)(y'z - yz') = 0,
F.81.3) ( [2B(x) 1 A -2
y'z — yz' = const, exp \ — / , dx\ = const. 11 \x — av\ p".
{ J A{x) ) ,~o
Since the last product approaches » as x —* a,, this leads to a contradiction
unless y'z — yz' = 0.
Now let y be a polynomial solution, y f? 0. We show that y ?* 0 at x = av.
If the contrary were true, substitution of x = a, in F.8.1) would give y' = 0.
Differentiation of F.8.1) k times results in a differential equation of order
k + 2 in y, which has the form
A(x)y(k+2) + {kA'(x) + 2B{x)\y{k+1) + ... = 0;
the further coefficients are again polynomials. Now from F.81.2) we see that
B{av) = PA'(av): so that kA'(a,) + 2B{av) ^ 0. Thus y = y' = y" = ¦ • ¦ =
yw = o would imply ya+1) =0. In a similar way we can show that all the
zeros of y are distinct.
Next we prove that the zeros of y lie in the interval [a0, ap]. Upon writing
y = f(x) = const, (x — xi)(x — x2) ¦ • • (x — xn), we have, according to F.8.1),
F.81.4) A(xk)f"(xk) + 2B{xk)f'{xk) = 0;
or, using F.2.2) and F.81.2),

[6.82] A MAXIMUM PROBLEM 153
F.81.5)
i p
+ _-L_ + ? _?_ = o.
We assume that some of the zeros of f(x) lie outside of [a0, ap]. Let xk be one
of these zeros such that the segment [a0, ap] and the remaining zeros all lie in a
closed half-plane with xk as a boundary point, the segment itself lying in the
open interior of the half-plane. Then the complex vectors
Xfc ~~* Xm j Xk ~~ Q>y
lie in an angle not greater than x for all values of v and m,m 9* k, and the same
is true of the reciprocals of these vectors. The vectors Xk — av are directed into
the interior of this angle. But then we see that F.81.5) is a contradiction.
(For this argument, see Polya 1.)
Let Wi, n2, • • • , np be the number of zeros of y in [a0, a,\], [ai, a2], • • • ,
[ap-i, ap], respectively. Then we say that y is of the type {n} , n2, • ¦ • , np).
Here wi + n2 + n3 + • • • + np = n, and c represents the number of all possible
types. It is our intention to show the existence of exactly one polynomial
solution of each type, corresponding to a different determinations of the poly-
polynomial C(x) of degree p — 1.
6.82. A maximum problem
Following Stieltjes (loc. cit.), we first show that a polynomial solution of each
type exists.
Let Xi, x2, • • • , xn be variable points, each different from the a, and dis-
distributed in [ao, ap] so that in each interval [a»_i, a,,] there lies a certain pre-
assigned number, say n,, of these points, ]>^=i nv = n. Letting each Xk vary
in a fixed interval, we consider the maximum of the product
W = TT T — n Pl> TT I rt — r
The existence and positiveness of this maximum are clear. In the maximum
position, the points Xk are different from the a, and from one another, and they
are of a preassigned type. Furthermore, we have dW/dxk = 0; whence F.81.5)
again follows. If f(x) denotes the polynomial with the zeros xk, the latter
equation means that A{x)f'{x) + 2B(x)f'(x) vanishes for x = xk and hence is
divisible by f(x). If this ratio is denoted by —C(x), F.8.1) follows. It is
clear that log (W), apart from the constant terms p^p* log \ali — av\~ , n 9* v,
is the "energy" of the system of masses p, concentrated at a, and of unit masses
concentrated at the xk. (Cf. §6.7 A).)
Incidentally, the argument used in §6.7 A) shows that the system {xk} with
the maximum property is uniquely determined. This is not the same as the

154 ZEROS OF ORTHOGONAL POLYNOMIALS [ VI ]
uniqueness of the solution of a given type since F.81.5) is not equivalent to the
maximum property. It is, however, easy to show (cf. F.22.6)) that F.81.5)
is equivalent to a relative maximum of W.
6.83. Uniqueness
Let C(x), y and D(x), z be two solutions of our problem of the same type,
C(x) f? D(x). Suppose both polynomials y and z have positive highest coeffi-
coefficients. By combining F.8.1) with the corresponding equation for z, we find
the relation
F.83.1) * (y'z - yz>) + 2 ± _*_ (y'z - yz>) + §^ yz = 0.
ax „_<> x — a, A(x)
Introducing H = YJl»-o \x — av |2p', we obtain for x ?± av
F.83.2) l^±Z^l
Suppose the function {D(x) - C{x)\/A{x) is non-negative in the fixed interval
av-i < x < av. Then between two consecutive zeros a and /3 of y, a < /3, in this
interval, z must change its sign at least once. Otherwise, yz would be perma-
permanently positive or negative, and therefore, H(y'z — yz') increasing or decreasing
in a < x < /3. For x = a + «, « > 0, however, the last expression has the
same sign as y'z or yz, and for x = /3 — «, « > 0, the same sign as y'z or — yz.
Hence it passes from positive to negative values if yz > 0, and from negative
to positive values if yz < 0. Either case contradicts the property of monotony
of H(y'z — yz') mentioned before.
Under the previous assumption concerning {D(x) — C{x)}/A (x), the function
z must change its sign also in [aP-i, y] and in [8, a*] if y and S are, respectively,
the first and last zeros of y in [a,_i, a,]. Otherwise, y and z would each have
a constant sign in those intervals; moreover, y and z would have the same
sign in a given interval because they are of the same type. Then H(y'z — yz')
would be an increasing function. It vanishes for x —* av-i + 0 and also for x —>
a, — 0, so that y'z — yz' must be positive in [a,_i, y] and negative in [S, a,].
Now for x = y, we have sgn (y'z — yz') = sgn y'z, which is the same as sgn (y'y)
at x = y — «,« > 0, that is, negative. This is a contradiction. The same argu-
argument can be used at x = 8.
Finally, we remark, under the assumption made about {D(x) — C(x)\/A(x),
that the polynomial z must vanish in [a,,_i, a,,], even if y has no zeros there.
Otherwise, because of F.83.2), the function H(y'z — yz') would be monotonic.
This is impossible since H vanishes for x = a,_i and for x = av.
To recapitulate: this argument would furnish at least one more zero for z
in [c_i, a,] than for y, which is impossible. Therefore, \D{x) — C(x)}/A(x)
must be negative in some points of a,,_i < x < av. By interchanging C(x), y
with D(x), z, it is seen that the same function must also be positive somewhere
in C-i < x < av ; whence D(x) — C(x) must have at least one variation of sign

[ 6.9 ] LEGENDRE FUNCTIONS OF SECOND KIND 155
in [a_i, oj. Since this is true for v = 1, 2, •.. , p, the function D(x) - C(x)
has at least p variations of sign in [a0, ap]. However, this is impossible since
D(x) — C(x) is of degree p — 1.
6.9. Zeros of Legendre functions of the second kind; generalization
A) In connection with Fejer's second generalization of Legendre poly-
polynomials (§6.5 D)) we consider the function
#n(cos 0) = /30 cos (n + 1H + ft cos (n + 3H
F.9.1)
+ • • • + j8» cos (n + 2m + .1H + • • • .
Here /3m J, 0, so that the series converges for 0 < 0 < x. This is the conjugate
series of the function Gn (cos 0) defined by F.5.16). The function Qn(cos 0)
of D.9.16) is a special case; then the sequence /3m is given by D.9.5) and is
completely monotonic. Now we prove the following theorem:
Theorem 6.9.1. Let 0m > 0 and let {/3m} be a completely monotonic sequence.
The function Hn (cos 0) definedby F.9.1) has at least one zero in each of the intervals
F.9.2) —J-i x < 0 < !L±i x, v = 0, 1, 2, • • • , n.
n + i n + f
More precisely, Hn (cos 0) has an odd number of zeros in each of these inter-
intervals. For the proof we again use Hausdorff's representation and obtain for
F.9.1)
]C tn cos (n + 2m + 1N}dp(t)
JO {m->0 )
F.9.3) ri
_ / cos (n + 1H - t cos (n - 1H
" Jo 1 - It cos 20 + P
where @(t) is a function of the same type as a(t) in F.5.8). Upon substituting
S = v ¦ x and 0 = V . \ w,
n + I n + i
we find
f(-l)"{cos @/2) - t cos C0/2)),
F.9.4) cos (n + 1H - i cos (n - 1H =
l(-l)"+1{sin {6/2) +t sin C0/2)},
respectively. Both expressions in the braces are positive; this establishes the
statement. (For v = 0 and v = n we must take into account that
lim Hn(cos6) = (-l)n+1 lim Hn(cos6) > 0
0-.-fO 8-+T—0
«o'3'» is divergent).)

156 ZEROS OF ORTHOGONAL POLYNOMIALS [ VI ]
B) Theorem 6.9.2. The function Qn(cos 6) (§4.62 C)) has exactly n + 1
zeros in 0 < 6 < x, which lie in the intervals F.9.2).
In this special case of Theorem 6.9.1 there cannot be more than one zero
in each of the intervals F.9.2); otherwise, Pn(cos 0) would have more than n
zeros by Sturm's theorem.
Inequalities F.9.2) for the zeros of Qn(cos 0) are due to Stieltjes (8, p. 252),
whose proof is, however, different from that just given. Fejer obtains B0, pp.
51-52) less precise inequalities in a manner similar to that used above; his
conditions concerning the sequence {j3m}, however, are more general.
C) Theorem 6.9.3. Legendrc's function of the second kind Qn{x) (§4.61 A))
has no zeros in the complex plane cut along the segment [ — 1, +1], except x = oo,
which is a zero of multiplicity n + 1.
This theorem is due to Hermite C) and Stieltjes (9) (cf. also Hermite-
Stieltjes 1, vol. 2, pp. 80-104, no. 267-274). The following argument is a slight
modification of the second proof of Stieltjes.
-1 +1
Fig. 8
We start from D.62.10). This function Qn(x) has n + 1 zero§ in the interior
of [—1, +1], which alternate with the zeros of Pn(x) (cf. Theorem 6.9.2). Let
+ 1 = xo > Xi > ¦ • • > xn > a;B+i = — 1 denote the zeros of A — x2)Pn(x) in
decreasing order. Then
F.9.5) sgnQ.(*,) = (-1)', v = 0, 1, 2, •.. , n + 1.
Furthermore, Qn(x) is a solution of D.2.1), a = j3 = 0, so that Qn(x)/Pn(x)
is increasing in the interval —1 S x S +1 (cf. D.2.6)); it becomes infinite at
each xy .
The curve in the figure encircles the points x = =t 1 and avoids the zeros of
Pn{x) by means of semi-circles. Now we shall show that the variation of
arg {Qn{x)/Pn{x)\ along this curve is equal to 2xBn + 1). Then, according
to Theorem 1.91.1 (principle of argument), the function Qn{x)/Pn(x) has
exactly 2n + 1 zeros exterior to this curve. But since it has a zero of multi-
multiplicity In + 1 at x = *, the statement will follow.
Let e be a sufficiently small positive number. As x encircles +1 in the nega-
negative (clockwise) direction from 1 + «to 1 - «, the variation of the argument in
question is approximately (cf. D.62.7)) the variation of arg {log l/(x - 1)},
a quantity which tends to 0 with «. If z describes the segment from x, - e
to xy+1 + «, v = 0, 1, 2, ¦ • ¦ , n, along the "lower border" of [-1, +1], we have

[6.10 J FURTHER RESULTS 157
n\X IV) Zx , \?n\X)
KS>'™> y = P(x) ' = 2 + PlFY
Then */ describes a straight line 32/ = ir/2 in the direction of decreasing abscissas.
The variation of the argument of y is -j-x.
In the neighborhood of xv, j» = 1, 2, ¦ • • , n, the function Qn(x)/Pn(x) differs
only by a bounded term from
F.9.7) " " , where " < 0.
•* n\Xv) X Xy *-n\Xp)
Therefore, the half-circle in the lower half-plane around x, will be carried over
into a curve which approximates a large semi-circle in the lower half-plane; the
argument of y then increases by +x.
Finally, if x = — 1 is encircled in the negative sense, from — 1 + eto— 1 — «,
the variation in arg y is again a quantity which tends to zero with «.
To recapitulate: While x moves along the lower border from 1 + «to — 1 — «,
the total increase in the argument of y is (n + l)x + nx = Bn + l)x. The
same is true on the upper border as x varies from — 1 — « to 1 + e. This es-
establishes the statement.
6.10. Further results
A) Let {dm} denote the zeros of Pn(cos0) in the interval [0, x], ordered in
an increasing way. Turan A) proved that the sequence xVJi — xv,n_u where
xvn = cos dm) is increasing as v runs from 1 to [%(n — 1)]. Szego proved (in a
correspondence with Turan, 1946) the same fact for the differences 0*,,,_i — 6vn.
Cf. Szego-Turan 1.
B) Concerning the topics treated in §§6.8-6.83, see also Makai 3.
C) The argument of §6.9 C) leads to a more general result concerning the
number of the zeros of Qn(x) — aPn(x) in the complex plane cut along the
segment [ — 1, +1] where a is a given complex constant (Hermite and Stieltjes,
loc. cit.). This number is again = 2n + 1 if -x/2 < 3a < +*"/2, and =n if
Indeed, in the case —x/2 < 3« < +*72 no essential change in the reasoning
is needed. Now let 3& > +*72. If x describes the segment from xv — e to
x,+i + «, v = 0, 1, 2, • • •, n, along the lower border of [ — 1, +1], the variation
of the argument of y — a is — x. There is no change in the contribution of the
semi-circles around xv and of the whole upper border. Thus the total increase
in the argument of y — a is — (n + 1 )x + nx + (n + 1 )x + nx = 2nx.
Let 3a = +x/2. On the segment from xv — e to xv+i + e there is a unique
point $ such that Qn (?)/Pn (?) = 9?a. We must use an indentation of the contour
into the lower half-plane and take into account that for S > 0
Qn(x - i0)\ _ (Qn{x - i'0)

158 ZEROS OF ORTHOGONAL POLYNOMIALS [ VI ]
the imaginary part of the right-hand side is x/2 - S' < x/2 where 8' > 0. The
argument is similar if $a = —x/2.
D) Makai-Turan A) have proved the following theorem of the Picard-
Landau type. Let Hn(z) be Hermite's polynomial. There exists a positive
(absolute) constant A such that every equation of the form
H0(z)+Hl(z)+yHn(z) =0
has a solution in the strip |3z|^A; n ^ 2, y arbitrary real or complex.
The exact value of A has been determined by Schmeisser 1. The extremal
polynomials are of third degree. The corresponding problems for quadrinomials
and more general equations remain open.
E) Szego 20 improved the right-hand estimates in Theorem 6.4 when
2a0 — Oj > Oj — a2 si a2 — a3 ^ • • • ^ an_i — an = «n = 0. Both sides can be im-
improved when Bk -l)aA_! ^ 2kak > 0, k ^ 1. See Askey-Steinig 1.
F) The results mentioned in §6.10 A) have been extended to ultraspherical
polynomials in Szego-Turan 1.
G) For the positive 0-zeros of the Legendre polynomials Pn(cos6) and
also for the positive zeros of the Hermite and Laguerre polynomials, written
in increasing order, the second differences of the respective sequences of con-
consecutive zeros are all positive, as an immediate consequence of the Sturm
theory (cf. Theorem 6.3.3). In L.Lorch-P.Szego 1, 2, it is conjectured that
all higher differences are also positive, but this remains unresolved. Sub-
Substantiating numerical evidence is cited there and in Davis-Rabinowitz 1. The
latter present also similar evidence connected with P,J(cos0). See also Lorch-
Muldoon-P.Szego 1, 2.
(8) If irn(x) is an arbitrary polynomial of degree less than or equal to n
and if
*¦„(*) = b0L0(x) -\ h bnLn(x)
is its Laguerre expansion, then the number of sign changes of irn(x) for x > 0
is at least as great as the number of sign changes of the sequence of the differences
See Turan 2.

CHAPTER VII
INEQUALITIES
No inequalities, except trivial ones, are known for general orthogonal poly-
polynomials. However, inequalities involving an unspecified constant can easily
be derived under certain conditions concerning the weight function w(x).
Still more precise estimates can be obtained if w(x) is monotonic, and a great
number of special inequalities follow from this added restriction.
Another very extensive class of inequalities can be derived for the classical
orthogonal polynomials, and in the present chapter we intend to enumerate and
compare the various methods used to obtain these inequalities. Aside from inte-
integral and series representations, the main tool is differential equations. As regards
the latter, we remark that there is a special method for deriving inequalities for
the solutions of certain differential equations (cf. Theorem 7.31.1). In recent
years this method has been used in several special problems (not only for poly-
polynomials), with slight variations, primarily by G. N. Watson and S. Bernstein;
it originated, however, in an idea of Sonin.41
At the end of this chapter we use the above mentioned inequalities in dealing
with certain extremum problems which involve polynomials of a fixed degree.
The selection of the material treated in this chapter has been influenced by the
needs of later chapters, especially by those of Chapters IX, XIV, and XV.
Historically, the major part of the inequalities for classical polynomials arose
from the discussion of the corresponding expansion problems.
We shall postpone till Chapter VIII the asymptotic calculation of certain
maxima (which can also be expressed in terms of inequalities), since they
require more intricate asymptotic considerations. However, we have found it
necessary in the present chapter to anticipate certain asymptotic results of
Chapter VIII.
7.1. Rough bounds for orthogonal polynomials
In this section we make essential use of the representation of positive func-
functions discussed in §10.2. However, this does not play a r&le in the further
course of Chapter VII.
A) Let w{x) be a weight function on the interval [-1, +1] for which the
integral
i A - z2r* log w(x) dx
exists in Lebesgue's sense. (This implies that w(x) cannot vanish on a whole
41 Cf. Sonin 2, pp. 23-24. I owe this reference to Professor J. A. Shohat.
159

160 INEQUALITIES [ VII ]
segment.) Let D(f; z) = D(z) be the analytic function associated with /@) =
w(cos 0) | sin 0 | in the sense of §10.2 B).
The conformal representation x = \{z + z) maps the unit circle | z \ < 1
(or | 2 | > 1) onto the z-plane cut along the segment —1 ^. x ^ + 1. For
z = e*9 we have x = cos 0. (Cf. §1.9.)
Theorem 7.1.1. Let [pn(x)} be the orthonormal set of polynomials associated
with a weight function w(x), —l^x^ + 1, for which G.1.1) exists; then
G.1.2) |
where x = \{z + z) is an arbitrary point of the cut plane.
For, (cf. A0.2.9))
r+i
1 = / \Pn{x)\2 w(x) dx-
G.1.3) =lj {pn(cosd)}2w(cos6)\sind\dd
4 J-r
- r x f+r \h -1)} nl2i o\2 - ie
r—1-0 2 J-r
Now if f(z) = E">-o Cm2"* ^S regular in | z | < 1, we have, according to Cauchy's
inequality,
G.1.4) |/B)]2^ E|cm|2E|22m| = (l I^
m-=0 m—0 r-*l—0
This establishes the statement.
The bound in G.1.2) becomes infinite if x lies on the segment [ —1, +1];
for all other values of x it furnishes a first appraisal of the magnitude of pn(x)
under a rather general condition. This information is comparatively precise
because we shall prove, (cf. Theorem 12.1.2) that, for a fixed x, the left-hand
member of G.1.2) tends to 2~* as n —* ». It is rather remarkable that no use of
the orthogonal property has been made in deriving G.1.2); only the normaliza-
normalization of pn(x) is employed.
B) Theorem 7.1.2. Let w(x) be bounded from zero, that is, w(x) ^ y. > 0.
Then, if x is not on the segment [—1, +1],
G.1.5) \pn(x)\ < A\x+ {x2 - 1)M",
where {x2 — 1)* is chosen so that \ x + (x2 — 1)* | > 1. The constant A depends
on x and n but not on n; A is uniformly bounded in the exterior of any closed curve
which contains [—1, +1] in its interior.
We have in this case | D(z) \ > \ M(l - z2)/2 |* (cf. A0.2.10)), so that from
G.1.2)

7.1 ] BOUNDS FOR ORTHOGONAL POLYNOMIALS 161
G.1.6) | Ml ~ *2)/2 |* | Vn{x)zn I < A - I z I 2y\ I z I < l,
follows.
C) In the same case w(x) ^ m > 0, we can also easily obtain bounds for
pn(x) on the orthogonality interval -1 ^ x S +1 itself. In fact, if | z \ < 1,
we have, in view of G.1.6), (ttm/2)*A - | z |2)* | pn(x)zn \ < A - | z |2)~*- Let
x be on the segment [-1, +1], x = \{z + z), where \z\ = 1. Then pn(x)zn
is a x2n in z, and we have for — 1 ^ x ^ + 1, | z \ = 1, r < 1,
I Vn(x) I = | pn(x)zn I < max |
I1
= r~2n max | pn(?)f" | < r~2n(irn/2)^A - r2).
iri
Here $ = |(f + f). On putting r2 = 1 - l/n, for n ^ 2, -1 ^ x ^ +1,
we have
G.1.7) | pn(x) | <
For —1 < a; < +1, we can reduce the exponent in G.1.7) from 1 to % (cf.
G.71.28)).
D) The same elementary method gives an idea of the magnitude of Jacobi's
polynomial if n is large. In this case w(x) = A — x)a{\ + x)?, a > -1,
0 > - 1 and (cf. A0.2.13))
G.1.8) D(z) = 2-(a+^+1)/2(i - Z)a+\l + zf+\
Then by G.1.2)
ir2 | j z | t + 2 j | ^^^^ | < A | 2 |)j
X = h{z + Z'1), I 2 I < 1.
Now assume —1 ^ x ^ +1; we obtain, as in C),
G.1.10) | pn(x) | < Cr~2"(l - r2)-* max | 1 - f p" | 1 + f |~/*~*,
lfl=»-
where C depends only on a and C, and 0 < r < 1. We choose again r2 = 1 — l/n.
Discussing the right-hand member of G.1.10) for | f | = r, 9?(f) ^ 0, and for
| f | = r, 9HD ^ 0, we obtain
G.1.11) | pn{x) | < Cv»«<«+i./»+i.i>j -1^x^+1.
Here C depends only on a and /3. The "true" exponent is max(a + |, /3 + |, 0)
(cf. G.32.2)).
For later purposes we give a formulation of G.1.9) in terms of the Jacobi
polynomials Pna'^(x) (of. D.3.4)). If x is exterior to the segment [—1, +1],
we have
G.1.12) P[a^ (x) = n~hO{\ z D, x=\{z + z'1), \ z \ < 1 ;
a > - 1,C > - l,n—> oo.

1G2 INEQUALITIES [ VII ]
This holds uniformly in the exterior of any curve containing the segment
[ — 1, -f 1] in its interior. Inequality G.1.12) follows, of course, immediately
from the asymptotic formula (8.21.9).
E) The following theorem is rather useful in obtaining bounds for ortho-
normal polynomials:
Theorem 7.1.3. Let w{x) and w(x) be two weight functions on the segment
[—1, +1], w(x)/w(x) = k(x). Assume k(x) ^ k > 0, and let k(x) satisfy the
Lipschitz condition
G.1.13) | &(zi) - k(x2) | < X | X! ~ x21.
If \'pn(x)) and \pn(x)} are the orthonormal polynomials associated with w(x)
and w{x), respectively, we have
G.1.14) | pn{x) \^k~h\ pn(x) | + XAT»{ | pn(x) | + | pn_x(x) |}.
This theorem is due to Korous C). The proof follows from the identity
(C.2.3))
P»@sX) Pv{x)pv{t)>w{t) dt
^pn(x) + [pn(t)\f, Mx)M))m(i ?n
kn J-i U=o J \ k{x
k k _
^p(x) +
_ 1 f+1
~ rj- pn(t){pn(x)pn-.i(t) - pn-i(x)pn(t)\
kn Kkx) J-i
x — t
where kn has the same meaning as in B.2.15), and kn has the corresponding
meaning for pn{x). Now, according to Schwarz's inequality
fCn
fCn
Pn(t)pn-l(t)w(t) dt
tpn-x{t)Pn{t)w{t) dt g f I pn^(t) I I Pn(t) | W{t) dt^
J
-1
which establishes G.1.14).
We mention two important special- cases which follow immediately from
G.1.14) by use of the bounds of the Legendre and Tchebichef polynomials (con-
(concerning the firjst case, see G.21.1) and G.3.8)):
(a) If tt'(.r) is positive and satisfies a Lipschitz condition | w(xi) — w(x2) \
< \\ .i'i — x21, we have

[7.2] MONOTONIC WEIGHT FUNCTIONS 163
(Art,
G.1.15) |pB(a:)| <
[A'(l - x)~\ -I<x<+1.
Here the positive constants A and A' are independent of x and n.
(b) If w{x) = A — x2)~*k(x) where k(x) is positive and satisfies a Lipschitz
condition | k(xi) — k(x2) | < X | Xi — x2 |, we have
G.1.16) \pn(x)\<A, -I<z<+1,
where A is independent of x and n.
For other elementary considerations of a similar nature, see Shohat 4, pp. 165-
166 and Jackson 6, pp. 893-898. See, also, §7.71 F).
7.2. Monotonic weight functions'
Theorem 7.2, Let w(x) be a weight function which is non-decreasing in the
interval [a, b], b finite. If {pn(x)\ is the set of the corresponding orthogonal poly-
polynomials, the functions \w(x)} | pn{x) | attain their maximum in [a, b]for x = b.
See Szego 3. A corresponding statement holds for any subinterval [xo, b]
of [a, b] where w{x) is non-decreasing.
The proof is based on the identity
w(b)\pn(b)}2 - w(x)\pn(x)}2 = 2 jTw(t)pn(t)p'n(t) dt+ jf \pn(t)}2dw(t),
which follows from A.4.4). It suffices to show that this expression is non-
negative in a ^ x ^ b. Denoting by Xi < x2 < '• ¦ • < xn the zeros of pn(x)
in increasing order, we have pn(t)p'n(t) > 0 for t > xn and pn{t)p'^{t) < 0 for
t < Xi. Therefore, the statement is trivial for xn ^ x ^ b and follows from
w{t)pn{t)p'n{t) dt = \ w(t)pn(t)p'n(t) dt - / w(t)pn(t)pn(t) dt
Ja Ja
= - I w(t)pn(t)p'n(t)dt
for a ^ x ^ Xi. Here we used the monotony of w{x) only in x ^ t ^ b.
Now let xv ^ x ^ x,+i., v = 1, 2, • ¦ • , n — 1, n ^ 2. If we introduce the
new weight function
W(x) = w(x)\(x - x1)(x - x2) ••• (x - x,)\2,
the corresponding orthogonal polynomial of degree n — v will be, save for a
positive constant factor,
Q (x) =
qn-KX) (X-Xl)(x-X2)...(x-Xy)>
with the zeros xy+i, x,+2, • ¦ • , xn . In fact, if p(x) is an arbitrary 7rn_x_i ,

164 INEQUALITIES [ VII ]
f
b
W{x)qn-,{x)p{x)dx = I w{x)pn{x){x - Xi)(x - x2) • ¦ • (x - x,)p(x) dx = 0.
J
But W(x) increases monotonically for x, ^ x ^ b, so that the preceding argu-
argument furnishes, for a;, ^ x ^ x,.rl ,
W(b){qn_Xb)\2 ~ W{x){qn^{x)\2 = w(b){pn(b)\2 - w{x){pn(x)\2 ^ 0.
In addition we conclude that the equation w(b){pn(b)\2 = w{?) {pn(?) j2 holds
if and only if ? < xi and io(i) vanishes in [a, ?] (this condition has, of course,
no significance if a = ?), and is step-wise constant in [?, &]. Furthermore, pn(i)
must vanish at the points of increase of w(t). (The weight function w{x) cannot
vanish for x ^ xi .)
7.21. Applications
On putting a = —\,b = +1, and w{x) = 1, we obtain the important in-
inequality
G.21.1) \Pn(x) |^1, -1 g i g +1,
for the Legendre polynomials Pn(a;), Pn(l) = 1. If n > 0, the equality sign
holds only for a; = ±1.
Another interesting case is a = —1,6 = +1, and w{x) = \ x \2k, k > 0. On
account of D.1.6) we have \x\k\ P^k~h\2x2 - 1) | ^ Pf'k^\l) = 1, so that
G.21.2) {A - x)/2}a/2+i | Pia'°\x) |^1, -1 ^ x ^ +1, a ^ -\.
The equality sign holds only for x = — 1. In the interval 0 ^ a; ^ 1 this is,
for large n, less precise than the first inequality G.32.6).
In case a = 0, b = 4- oo, and w(x) = e~x, we obtain for the Laguerre poly-
polynomials
G.21.3) e~x!2 \Ln{x) | ^ 1, x ^ 0,
the equality sign holding only for x = 0 if n > 0.
With regard to these special cases, see Szego 2 and 3.
7.3. Legendre polynomials
A second proof of G.21.1) and various other important inequalities can be
obtained by means of the differential equation for Legendre polynomials.
A) Theorem 7.3.1. Let n ^ 2. The successive relative maxima of \ Pn(%) \,
when x decreases from 1 to 0, form a decreasing sequence. More precisely, if
Mi , M2, • • • , M[«/2] denote these maxima corresponding to decreasing values of x,
we have
G.3.1) 1 > Mi > M2 > ••• > M[»/2].
From this G.21.1) follows again.

[ 7.3 ] LEGENDRE POLYNOMIALS 165
If ft is even, we have
- \P(n\\ - 1-3 ••• (n - 1)
Mn/2 - \Pn{0) = — .
2-4 • • • n
For the proof let
G.3.2) n(n + l)f(x) = n{n + l){Pn(x)\2 + A - x2){P'n{x)\2.
Then we have/(a;) = {Pn(x) j2 if P^(a) = 0, or if x = ±1. Therefore,
G.3.3) max {Pn(x)\2 ^ max /(x).
-]gxg+l -lgxg+1
Now, on account of D.2.1),
n(n + l)f'(x) = 2P'n(x){n(n + l)Pn(x) - xP'n(x) + A - x2)K{x)\
= 2P'n{x).xP'n{x) = 2x{P'n{x)\2,
so that f{x) is1 decreasing for x < 0 and increasing for a; > 0. This establishes
the statement.
B) Theorem 7.3.2. Let n ^ 2. 77ie successive relative maxima of
(sin 0)* | Pn(cos 0) | when 6 increases form 0 to tt/2, /orw an increasing sequence.
From D.24.2) we obtain for a = j8 = 0
<22u
k 6),
G.3.5) "e
4>(B) = Bsin0)-2+ (n + iJ.
Introducing
G.3.6) f(e) = \uF)}2 + +(e){W(e)}2, 4,@) =-
we have
G.3.7) f'F) = 2u'{6)\u(d)
Now ^@) is an increasing function in [0, tt/2], so that/'@) > 0, and/@) is also
increasing. But/@) = \uF)}2 if u'@) = 0; this proves the theorem.
C) An important, application of Theorem 7.3.2 is
Theorem 7.3.3. We have
G.3.8) (sin 0)* | Pn (cos 0) | < B/ttIw"', 0 <| 0 ^ tt.
//ere i/ie constant B/x) cannot be replaced by a smaller one.
The first proof of an inequality of the type G.3.8), with a constant A instead
of B/7r)*, is duo to Sticltjcs (8, p. 241). Further proofs have been given by
Gronwall A, p. 221) and Fejer (9, pp. 289-291). The following proof is due to
S. Bernstein B, p. 236); it was the first leading to the precise constant B/ir).
Let n be even. From Theorem 7.3.2

1G6 INEQUALITIES [ VII ]
G-3.9) (sin 0)* | Pn(cos 6) \ ^ \ Pn@) |, 0 ^ 6 ^ *-,
with the sign of equality holding if 6 = tt/2. Now let n be odd. Then for
0 ^ 6 ^ 7T
(sineI |Pn(cos0) | < max {/(fl)}*
G.3.10)
42
where fF) is defined by G.3.6). Using the notation D.9.2), we have
I J°»@) | =gn/2
I P'n@) | = (n +
according as n is even or odd. (In the second case we can use D.7.31).) Now
W + (n + JJ}"*(n + 1)* < n~*; therefore G.3.8) follows.
That B/7r) is the best possible constant is easily seen by considering.! Pn@) |,
n even. Besides this we have (cf. G.32.9), a = /3 = 0)
G.3.12) max (sin 6)h \ Pn (cos 6) | ^ B/tt)S2~j, n -> » .
7.31. Theorem of Sonin; Bessel functions
The argument used in the preceding section can be generalized in various
ways. For instance, the following important theorem holds.
Theorem 7.31.1. Let y = y{x) satisfy the differential equation
G.31.1) y" + 4>(x)y = 0,
where <j> (x) is a positive function having a continuous derivative of a constant sign
in xo < a; < Xo. Then the successive relative maxima of \ y \ , as x increases
from Xo to Xo ,form an increasing or decreasing sequence according as 4>(x) decreases
or increases.i3
If we write
G.31.2) f(x) = {y(x)\2 + Mz)rV(s)}2 = {y(x)\2 + ^{x)\y'{x)\\
we have, in fact,/(x) = {y(x)\2 if y'(x) = 0, and
G.31.3) f(x) = 2y'(x){y(x) + t(x)y"(x) + W(x)y'(x)\ = t'(x){y'(x)}2.
That is, sgn/'(x) = — sgn 4>'{x); whence the statement follows.
42 The sequence {mll2gm\ is increasing, so that mmgm < limm^oomil2gm = tt".
43 Professor P61ya has kindly pointed out to me the following generalization of this
theorem: Let y(x) satisfy the differential equation
\k(x)y'}' + 4>(z)y = 0,
where k(x) > 0, <t>(x) > 0, and both functions k(x), <t>(x) have a continuous derivative. Then
the relative maxima of \ y \ form an increasing or decreasing sequence according as k(x)<t>(x)
is decreasing or increasing. This was obtained independently by Butlewski 1.

[ 7.32 ] JACOBI POLYNOMIALS 167
As an illustration, we apply this result to A.8.9) for k = 1, x > 0. In this
case we have
G.31.4) 4>(x) = l t
Xi
so that 4>(x) is decreasing if h? < \ and increasing if a > \. In the latter
case x must be taken so large that <j>(x) > 0. We then conclude that the
relative maxima of z*| Ja(x) | form an increasing sequence if a2 < \, and a
decreasing sequence if a > \. In the first case x > 0, and in the second
X > (a2 - 1)\
According to A.71.7), there are infinitely many such maxima, and they
tend to B/71-)*. Thus, we have the theorem:
Theorem 7.31.2. If Ja(x) denotes Bessel's function of order a, we have
i f B/tt)* if-\<a^h,
G.31.5) sup{s»|J.(*)|} = »
«&° [ finite and > B/V)* if a > \.
Fora = dbj we can use formulas A.71.2). Fora < — \ we have xVa(a;) —> <»
as a; —> +0 [A.71.1)]. The second statement holds in this case provided the
least upper bound in question is taken in an arbitrary interval [xo, + °° ] with
xo > 0.
See Szego 17, pp. 40-41, and compare similar theorems in Watson 3, pp.
488-489. See also §7.8.
7.32 Jacobi polynomials
A) The consideration of §7.3 can easily be extended to ultraspherical poly-
polynomials. A treatment of this case will be given in §7.33. First, however,
we discuss general Jacobi polynomials Pl?'®(x). In applying the previous
methods to Pi"'0)(x), the main difficulty lies in the fact that there is no special
point x = I interior to [ — 1, +1] at which P(n"'p)(x) and its derivative have
values which are easily calculated, as in the case of the ultraspherical, and
especially of the Legendre, polynomials at x = 0. This is the reason that here
we must anticipate certain comparatively simple results from Chapter VIII.
These are the following:
(a) The formula of the Mehler-Heine type [(8.1.1)],
lim n~aPia'e) (cos -) = (z/2)~aJa(z),
n-»oo \ n/
which holds uniformly for | z \ ^ R, R fixed.
44 After completing the manuscript I received a paper of Korous C) in which some of
the results of §7.32 are derived by use of the differential equation of Jacobi polyno-
polynomials but without using the asymptotic formulas (8.1.1), (8.21.10).

168
INEQUALITIES
VII
(b) Darboux's formula [(8.21.10)],
Pi"^(cos B) = n+kifl) cos (NO + 7) +
N = n + (a + 0 + l)/2, 7 = -(a +
e ^ 0 ^ tt - e,
Here e is a fixed positive number; the bound for the error term holds uniformly.
In both cases, a and 0 are arbitrary real numbers.
Concerning the results of this section and subsequent ones, cf. Kogbetliantz
19, p. 125, S. Bernstein 2, and Szego 17.
B) Theorem 7.32.1. Let a > -1, 0 > -1,
G.32.1) *
We have
a + 0 + 1
'n + gN
max
G.32.2)
•e x' is one of
prpV)i~n
maximum points nearest xo
~*
= max (a, 0) ^ — \,
= max (a, 0) < — |.
The symbols ~ refer to the limiting procedure n —> <x>. In the second case
— 1 < Xo < +1; then we use (8.21.10). The subsequent argument furnishes
the more precise result that the maximum of | P(na'®(x) | in the interval 0 ^ x ^ 1
is of order nmax(a>~1/2). A similar result holds for - 1 ^ x ^ 0.
For the proof we generalize the argument of §7.3 A) as follows. Let n ^ 1,
and let
n(n + a + 0 +
G 32 3)
Then by using D.2.1), we obtain
G.32.4) n(n + a + 0 + Df'(x) = 2{a - 0 + (a + 0
Thus/'(x) can change its sign only at x = xo.
We see that the condition —1 < x0 < +1 is equivalent to (a + |)@ + 2) > 0.
Now let a > — 5 and 0 > — |. Then the sequence formed by the relative
maxima of | P(na'^(x) | in — 1 ^ x ^ Xo and by the value of this function at
x = —1, is decreasing, while the sequence of the maxima in Xo ^ x ^ +1 and
of the value of the function at a; = +1, is increasing. Therefore, | P(na'fi\x) \
attains its maximum in [ — 1, +1] at one of the end-points.

[ 7.32 ] JACOBI POLYNOMIALS 169
In case a ^ - % and -1 < /3 ^ - 5, the linear function a - /3 + (a + 0 + l)x
is non-negative, so that the sequence of relative maxima in question is increasing
in [-1, +1], save for the case a = /3 = -^, in which it is stationary; the situ-
situation is opposite in the case /3 ^ — ^, — 1 < a ^ — ^. Finally, let — 1 < a < — \,
— 1 < /3 < — \, so that again — 1 < x0 < +1; then the sequence of maxima is
increasing in [— 1, Xo] and decreasing in [x0, +1], so that the absolute maximum
of I Pl?'e)(x) I in [ — 1, +1] is attained at the point of maximum nearest x0 on
the left or on the right.
See Problem 39.
C) Theorem 7.32.2. Let a and /3 be arbitrary and real, and c a fixed positive
constant, n —> °°. Then
G.32.5)
{0(n") -•/ n ^ „ ^- -_-i
See S: Bernstein 2, pp. 225-232 where Sonin's theorem is applied, but where
the proof is perhaps slightly more complicated than ours. Szego A7, p. 77)
uses the asymptotic formula (8.21.17) which is, of course, a more complicated
tool than (8.1.1) and (8.21.10) used below.
The bounds in G.32.5) are precise as regards their orders in n. They follow
also, as mentioned, from the more complicated asymptotic formula (8.21.17)
of "Hilb's type." By use of D.1.3) we can obtain similar bounds for the
intervals tt/2 ^ 0 ^ tt.
We notice the useful inequalities
, m ieo(rr),
G.32.6) Pi"'ft(cos B) = 0 < 6 ^ tt/2, a ^
[O(na),
- J;
G.32.7) Pi"•*(cose) =O(n~h), . 0 < 6 ^ tt/2, a ^ -?
?,
which follow from G.32.5). Concerning the second bound in G.32.6), and
concerning G.32.7), see §7.32 B).
We observe that 6~ n ~ n" if 6 ~ ri~l; thus it suffices to prove G.32.5)
for a special value of c. Apply Theorem 7.31.1 with [D.24.2)]
x = 6,y = un{6) = ( sin 2J ( cos |J P^'W 0),
G-328)
4 sin ^ 4 cos -
First, let 5 = 8(a, /3) be a fixed positive number, sufficiently small. Then
is positive and decreasing in 0 < 0 ^ 5 if a.2 < \. It is positive and increasing
in kn~l ^ 0 ^ 5 if a > |; here /c is a fixed number, k > (a2 — \)h, and n is

170 INEQUALITIES [ VII ]
sufficiently large. Thus in both cases, the function <p(d) is positive and monotonic
in /en ^6^8, where k = k(a, /3), 5 = 8(a, /3), and n is sufficiently large. The
same holds for a2 = \,&2 9± \. Therefore, the sequences of the relative maxima of
| un{6) | in the interval /en ^ 6 gL 8 are increasing and decreasing, respectively,
for large n, according as a2 < \ or a > \. According to (8.1.1) and (8.21.10),
we find in both cases un{6) = 0(n~4). (We have (sin 0/2)a+i(cos 6/2)M
~ 6a+i.) This furnishes the first part of G.32.5) with c = k. The second part
follows immediately from (8.1.1).
In the case a2 = K2 = -j, excluded before, <j>F) is constant. Then we know
Pia>/3)(cos 6) explicitly (cf. D.1.7), D.1.8)).
D) Theorem 7.32.3. Let unF) have the same meaning as in G.32.8), and
Mn = max | unF) \ when 0 < 6 ^ v/2. We have
G.32.9) lim n*Mn = 1 -*
»-°° [finite and > tt if a > \.
Here /3 is greater than — 1.
Cf. S. Bernstein 2, pp. 225-232; Szego 17, pp. 79-80. Cf. Theorem 7.31.2.
The preceding argument needs only a slight modification. We have to discuss
the maximum of n* | un(d) | for 5 ^ 6 ^ tt/2 if — \ g a ^ +?, and for 6 = n~xz,
0 ^ z ^ c, if a > |. The first is ^ tt"* as n —> <x> (according to (8.21.10));
the second is
(according to (8.1.1)). For sufficiently large c this is independent of c and
greater than tt"* (Theorem 7.31.2).
E) Finally, as an application of D.21.7) we point out the following generali-
generalization of Theorem 7.32.2:
Theorem 7.32.4. Let a and /3 be arbitrary and real, and c a fixed positive
constant, n —> °° . Then
G.32.10) {(^yP»"'")(a;)} „=} 2k+a)H % ^ <e<"-i
From this we find, uniformly in x, — 1 ^ x ^ +1,
G.32.11) -f ) Pi"lft(x) = OU9), ? = max BA; + a, 2k + /3, A; - ?).
7.33. Ultraspherical polynomials
In the ultraspherical case the preceding considerations can be simplified.
A) By the same argument as that in §7.32 B) we find that

7.33] ULTRASPHERICAL POLYNOMIALS 171
n(n4-2X)/(z) = n(n+ 2X){PlX)(a;)}2 + (l - x2) \~
is increasing or decreasing in 0 ^ x ^ 1 according as X > 0 or X < 0, X non-
integral; we assume n > 0 in the first and n > ~ 2X in the second case. Thus
we obtain the following theorem:
Theokem 7.33.1. We have
Un + 2X - l\ if X > 0,
G.33.1) _maxjPLx)(a;)| = V n )
{ | P^ix') | if X < 0, X non-integral.
Here x' is one of the two maximum points nearest 0 if n is odd; x' = 0 if n is even.
In the first case D.7.3) has been used (cf. also Theorem 7.4.1). In the second
case we obtain, if n is even,
G.33.2) max | P^x)(a;) | = | P^x)@) | = | (n/2
whereas, for n odd,
max | P(nX)(x) | <
G.33.3)
= |2X|{n(n + 2X)}
-i
X + (n - l)/2\
(n - l)/2 ,/
Both bounds G.33.2) and G.33.3) are ^ 21-x | r(X) l^ as n -> oo ; the first
bound is attained for a; = 0, the second bound is precise in the asymptotic sense.
B) By use of D.7.11) we obtain in a manner similar to that in §7.3 B), C),
the following:
Theokem 7.33.2. Let 0 < X < 1. Then we have for 0 ^ 6 ^ t
(sin0)x|PlX)(cos0)|
G.33.4) f an/2 if n is even,
= j{X(l - X) + (n + XJp*(n + l)a(n+i)/2 t/n is
and
G.33.5) (sin 0)x | P^X)(cos 0)\ < 21~x{r(X)rV~1.
Here the constant 21~x{r(X)}~1 cannot be replaced by a smaller one; an has the
same meaning, as in D.9.21).
In G.33.4) the sign of equality holds only for even n and 6 = tt/2. Now
a, ^ {r(X)}-V~\ and an < {r(X)}~1nx~1 (since jn^On} is increasing45); more-

172 INEQUALITIES [ VII ]
over, {X(l - X) + (n + XJ}~J(w + 1)* < n"~\ 45 so that equation G.33.5)
follows.
Less precise (but more general) inequalities can be obtained from the general
result G.32.5) for a = j8 = X -|. We have •
?w)r:?
G.33.6) { Oin*-1), 0 ^
en
;
X arbitrary and real, X ^ 0, — 1, — 2, • ¦ ¦ ; c > 0.
C) We point out an interesting special case of G.33.6), namely, X = f
(cf. Szego 16). We have P'n(x) = P^liix) (cf. D.7.14)), so that
\e~lO{nh), en'1 ^ 9 ^ ir/2,
G.33.7) Pi (cos*)
[ O(n) ,0^0^ cn~\
The first bound can be used for the whole interval 0 < 0 ^ u-/2 [cf. G.32.6)].
According to G.33.1) the inequality
G.33.8) | P'n(x) | ^ n(n + l)/2, -1
holds, the equality sign being taken if n = 0, 1, or n > 1, x = =fcl.
By using the first identity in D.7.27), we find
G.33.9) A - x2)P'n(x) = ^t
-f- l
Thus we conclude from the first bound in G.33.7) (which now holds for
0 < d ^ tt/2, cf. the previous remark) the following:
Theorem 7.33.3. If Pn(x) denotes Legendre's polynomial, we have for 0 < 9 <ir
G.33.10) P»_i(cos 6) - Pn+i(cos 6) = (sin 0)*0(n~*).
The bound of the factor 0{ri~*) is independent of 6.
This result, without the factor (sin 0I, is due to Stieltjes (cf. Hermite-Stieltjes
1, vol. 2, pp. 174-177; Feje> 9, pp. 295-298). The present form of the theorem
is implied in the previous more general results of S. Bernstein, Kogbetliantz,
and Szego; cf. Szego 16.
7.34. Bounds for integrals involving Jacobi polynomials
Theorem 7.34. Let a, /3, y. be real numbers each greater than — 1. Then as
n — > oo (concerning the second part of the statement see below)
45 In view of the concavity of log x, we have
A - X) log (n - 1) + X log n < log (n + X - 1),
A - X) log(n») f Xlog \(n+ IJ) < log (A - X)n2 + X(n + lJj.

: 7.34]
INTEGRALS INVOLVING JACOBI POLYNOMIALS
173
G.34.1)
— x)
n * log n,
2m
2m
2m
< a
= a
> a
-I
-b
— 3
2"
See Szego 17, pp. 84-86, where the existence of the limits of the corresponding
ratios is proved, and the limits are calculated. The proof of the second part
of G.34.1) requires a more complicated apparatus [(8.21.18)]; here we prove
only that
G.34.2) un = nh f A - xY \ Pia>l>)(x) | dx = O(log n), and un -> «.
Jo
This is sufficient for later purposes (cf. §9.41 E)).
We use G.32.5); in fact, we have
/i r*72
A - xY I Pn"'P)(x) I dx = 0A) / 02M+11 PnaJ)(cos
Jo
G.34.3)
de
"*
O(n"*){O(l)
If 2ju — a + | == 0, the last term must be replaced by O(log n).
On the other hand,
G.34.4) / (l-xY\PiaJ)(x)
o
A /
Jo
I
< a -
T/2
/4
where A is a proper positive constant. According to (8.1.1), the first bound is
f1 2 +1 a -a -1 a-2 -2
Jo
According to (8.21.10), the second bound is
fr/2
~ n~h I I cos (Nd + 7) I dd ~ n~K
J r/i
In case 2a = a — I, a > — |, we have
A - xY I Pia>l>\x) \dx> A
Jo
fl)
where w is a fixed positive number, and A' is independent of n and w. From
this inequality, according to (8.1.1),

174 INEQUALITIES [ VII ]
* T A - xf \ Pia^(x) \ dx ^ 2" A' /" 2"»
lim inf n* T A - xf \ Pia^(x) \ dx ^ 2" A' / 2"» \ Ja(z) \ dz
n-»oo JO Jo
follows. The last integral becomes arbitrarily large with w; this furnishes the
second part of G.34.2).
7.4. Fejer's generalization of Legendre polynomials
A) Theorem 7.4.1. Let {«„} be a sequence with positive terms. Then the
"Legendre polynomials" Fn(x) F.5.1) associated with the sequence {«„} satisfy
the inequalities
G.4.1) \Fn(x) | ^ Fn(l), -1^*^+1.
The sign of equality holds only if n = 0, or n > 0 with x = — 1 or x = +1.
This leads to a new proof for G.21.1) and for the first part of G.33.1).
B) The inequalities G.3.8) and G.33.5), of the Stieltjes type, can likewise
be extended to the polynomials Fn(x), though with certain larger constants.
We prove the following:
0, A2an = an - 2an+i + an+2 > 0,
n = 0, 1, 2,
G.
Theorem
.4.2)
and
G
.4.3)
7.4.
> o,
2. Let
Aan = an — an+i
f{z) = ao + a* H
) anzn + • • • .
Then for the "Legendre polynomials" Fn(x) associated with the sequence {<*„},
we have
G.4.4) | Fn(cos 6) | ^ 4a[(n+1)/2]| f(e2ie) \, 0. < 6 < tt) n = 0, 1, 2, • • • .
See Feje'r 9, pp. 291-295; Szcgo 11, p. 179. Szego obtains a larger bound
under a more restrictive condition. The inequality G.4.4) and the present
proof are new.
Under the conditions G.4.2) the function/(z) is regular for | z \ < 1 and con-
continuous for | z | ^ 1, | z — 1 | ^ 5, where 5 is an arbitrarily small positive number.
Indeed, we see that limn_«,an = a ^ 0 exists. If a = 0, we use a well-known
case of Abel's inequality A.11.6); if a > 0, we write an = (an — a) + a.
Now from F.5.1),
[n/2] [n/2]
G.4.5) Fn(cos 9) = z~nn E' «*«„-*«* + znn E' «*«»-*«"*, z = eie,
k-0 A-0
where the sign E' indicates that for even n the last term k = n/2 has to be
multiplied by \. Hence,
[n/2]
G.4.6) I Fn(cos 6) | ^ 2
El k
OLkOLn-UZ
k-0
z = e
2t«

[7.5] RECAPITULATION 175
By virtue of A.11.6) we obtain
G.4.7) \Fn(cos 0)| ^ 2am max
k-y
m = n- [n/2], z = e™.
Now, writing
G.4.8) pn(z) = anzn + an+1Zn+1 + an+2zn+2 + • • • ,
we have, according to a theorem of Fej?r-Szego A),
G.4.9) |/(z) | ^ | Pl(z) | ^ | P2(z) | ^ • • • , | 2 | ? 1, 2 ^ 1.
But
, . [^V k (p,(z) - ^{Pn/2(z) + pn/2+i(z)}, »even,
G.4.10) 2_, a*2* =
*-' [py(z) - p(n+i)/2(z), n odd,
so that the modulus of this sum is in both cases not greater than 2| f(z) \ ;
whence G.4.4) follows.
We give here a sketch of the proof of G.4.9). It suffices to show that | /(z) |
^ | Pl(z) | , or | /(z) | ^ | /(z) - a0 |, or 3? [/(z)] ^ ao/2 for | 2 | < 1. Now
limn_Mo;n = a ^ 0 exists. If/(z) is replaced by/(z) — a(l — z), it is seen
that we can assume a = 0 from the start. Then for | z | < 1
2
/(z) = ? A2an{n + 1 + nz + (n - l)z2 + • • • + zn\
71=0
- I E A2an(n + 1) + E A2an{(n + l)/2 + nz + (n - l)z2 + • • • + zn\
oo
=: W2 + E A2an{ (n + l)/2 + nz + (n - l)z2 + • • • + zn}.
n=-0
But for a real 0
+ l)/2 + we' + (n — l)e ' + • • • + ent }
n
= E (i + cos 0 + cos 20 + • • • + cos j/0)
+ |H 1
sin (y + j)g = 1 /sin{(n + l)g/2}\
2 sin {0/2! 2\ sin {0/2} /
(cf. Fejcr 1; see also Polya-Szego 1, vol. 2, pp. 78, 269, problem 17).
Fejer (9, pp. 295-298) also gives an extension of Stieltjes' theorem on
P*-.i(x) - Pn+i(x) (cf. Theorem 7.33.3) to the polynomials Fn(x).
7.5. Recapitulation
In the last sections we gave various derivations of the important inequality
G.21.1):

176 INEQUALITIES [ VII ]
(a) from the orthogonal property, by use of the general Theorem 7.2;
(b) from the differential equation (§7.3 A));
(c) from the trigonometric representation D.9.3).
Moreover it follows also
(d) from Laplace's integral representation D.8.10); in fact, if — 1 ^ x ^ -f-1,
| x + (x2 — II cos <j> I = I x -f- i(l — a;2M cos <j>]
= [x2 + A - x2) cos2 4>)h < [x2 + A - x2)}h = 1.
7.6. Laguerre and Hermite polynomials
A) Theorem 7.6.1. Let a be arbitrary and real. The sequence formed by the
relative maxima of | Lia)(x) | and by the value of this function at x = 0, is decreasing
for x < a + 21 and increasing for x > a + h The successive relative maxima of
I Hn(x) | form a decreasing sequence for x ^ 0, and an increasing sequence for
x S: 0.
Indeed, the function
G.6.1) nlL^fx)}2 + xl—L^U:
[dx
is decreasing for x < a + \ and increasing for x > a + §. The function
G.6.2) 2n{Hn(x)\2 + {tfU*)}2
is decreasing for x < 0 and increasing for x > 0. Both statements follow by
differentiation as in §7.3 A); we use the first differential equation in E.1.2)
and E.5.2), respectively.
B) Theorem 7.6.2. Let a be an arbitrary real number. The successive relative
maxima of
G fi 1\ ,,-*/2T("+1)/2 i l}a)(x\ I and p~x'2t"/2+1 I L(a)(r} I
form, an increasing sequence provided x > Xo- In the first case
0 ifa^l,
G-6.4) * = «' - 1 ., 2 . .
>+!T+i %ja > L
7n Z/ie second case
( Va ~~ I/* %] a. ^ %.
In the first case we take n so large that 2w + a + 1 > 0.
Sonin's theorem 7.31.1 applies to the functions u and v occurring, respectively,
in the third and fourth equations of E.1.2). The larger zero yn of the coefficient

[7.6] LAGUERRE AND HERMITE POLYNOMIALS 177
of u is an upper bound for the zeros of u (cf. Theorems 1.82.3 and 6.31.2).46
The differential equation shows also that x = yn is the last point of inflexion of u;
thus Y« is at the same time an upper bound for the points at which the relative
extrema of u are attained. Similarly, if y'n denotes the larger zero of 4n +
2a + 2 - x + (| - a2)x~\ we find in ly'n)h an upper bound for the points at
which v attains its relative extrema. (The bound of Theorem 6.31.2 is yI.)
Now if x0 is chosen according to G.6.4) and G.6.5), we have
1 - a2
n +
3.6)
(« +
X
4n
l)/2 .
+ 2a'
1
+
2
— a
4a:2
2 - x +
i > o
a;
> o,
n
+
(«
a
-1
+
.2
l)/2
i
a:2
1
c
for x
2
a <
for Xo
2
— a
2a:3
o < a:
0
< X
<
<
respectively. This establishes the statement.
Theorem 7.6.3. The successive relative maxima of
G.6.7) e~xV2 \Hn(x) |
form an increasing sequence for x ^ 0.
Here the second equation in E.5.2) can be used.
C) The bounds analogous to G.32.5) are readily obtained by means of the
asymptotic formulas (8.1.8) and (8.22.1), which correspond to (8.1.1) and
(8.21.10), respectively. It is convenient to use the fourth equation in E.1.2).
Then 4n + 2a -f* 2 — x + (I — a )x~l is positive and decreasing in 0 < x ^ 5
if a2 ^ \; it is positive and increasing in kri~l ^ x ^ S provided a2 > \, k >
(a2 — i)/4 and n is sufficiently large. Here 5 = 8(a) is a sufficiently small posi-
positive constant. Therefore, as in §7.32 C), we obtain the following:
Theorem 7.6.4. Let a be arbitrary and real, c and w fixed positive constants,
and let n —* °o. Then
[ af a/2-l0(na/2-*) if cn~l ^ x ^ «,
G.6.8) Lia)(x) =
I O(na) if 0 ^ x ^ cn~\
These bounds are precise as regards their orders in n; they follow also from
the more complicated formula (8.22.4) of Hilb's type.
For a ^ — \, both bounds hold in both intervals, that is,
G.6.9) Lia)(x) = \ 0<x^u,a^-h
{ O(na),
40 If this coefficient is constantly negative for x > xa, \u\ has no zeros and no maxima.
Similarly for v. These cases can be excluded.

178 INEQUALITIES [ VII ]
On the other hand,
G.6.10) Lia\x) = O(na/2-}), 0 ^ x ^ «, a ^ ~ \.
And generally, with a arbitrary and real,
G.6.11) Lia\x) = 0{na), a = max (\a - \, a), 0 ^ x S «.
Finally, we obtain the following analogue of Theorem 7.32.3:
Theorem 7.6.5. Let Mn = max0<tgo e~xl2xal2+i \ L(na\x) \. Then
G.6.12) lim n-a/2+iMn =
> tt ?Ja >
The proof is very similar to that in §7.32 D); of course, G.6.12) also follows
directly from the deeper formula (8.22.4) combined with G.31.5).
7.7. Problem of Lukacs
A) This problem (Lukacs 1) deals with a more precise form of the mean-value
theorem
G.7.1) A g ?-L
where A = mina?*?i> f(x), B = msLXa^x^b f(x), provided f{x) is restricted to the
set of all 7rn with a fixed value of n.
Theorem 7.7. Letf(x) be an arbitrary wn with the minimum A and maximum B
in [a, b]; then
G.7.2) A + B-^^ ^ -— fbf(x)dx SB - ~^,
rn b - a Ja Tn
where
( (m + IJ ifn = 2m,
G.7.3)
I (to + 1)(to + 2) ifn = 2m + 1.
!T/ie number rn cannot be replaced by a smaller one.
This result is the analogue of an older theorem due to Feje"r which deals
with the analogous question for trigonometric polynomials of a fixed degree n,
b — a = 2w. In this case rn = n + 1. The proof of FejeYs theorem can
be based on Theorem 1.2.1; however, various other methods have been used
(cf. P61ya-Szego 1, vol. 2, pp. 83, 277-279, problem 50).
It suffices to prove the first inequality of G.7.2); the second one follows when
we replace f(x) by — fix). In addition, there is no loss in generality in assuming
A = 0. It is readily seen then that rn is the greatest possible value of f{x) if x
varies in [a, b], and f(x) ranges over the class of the wn which are non-negative in

[7.7]
[a, b] and satisfy
G.7.4)
PROBLEM OP LUKACS
f(x)dx = 1.
179
b ~ a
Let now max f(b) = Mn for the same set of irn; we shall prove that rn = Mn.
It is clear that Tn ^ Mn . On the other hand, if x0 is an arbitrary point in [a, b],
we see, by means of a linear transformation, that
G.7.5) f(x0) ^ Mn
x0 - a
f(x) dx, f(xo) ^ Mn j
b — x0
f(x) dx.
Multiplying the first inequality by x0 - a, the second by b - x0, and adding, we
find
G.7.6)
1 fb
^ Mnf-^— / f(x)dx = Mn.
o a ja
B) First method of calculating Mn . See P61ya-Szego 1, vol. 2, pp. 96, 297,
problem 108.
Assuming a = — 1, b = + 1, we use Theorem 1.21.1 and represent the poly-
polynomials in A.21.1) as linear combinations of certain convenient polynomials.
For arbitrary and real«,, vy (subject only to the normalization condition G.7.8)),
we write
G.7.7) f(x) = I
if n = 2m,
>«o
if n = 2m + 1.
Because of the orthogonality of Jacobi polynomials, we have, in the notation of
D.3.3),
G.7.8)
But
G.7.9)
2 =
+i
(x) dx =
c=-0
e
m
E, A,0) 2
fly Uy
c—0
ly Vy
/"(I) =
if n = 2m,
if n = 2m + 1.
if n = 2m
if n = 2m + 1,
so that

180
INEQUALITIES
VII
G.7.10)
Af» =
r-0
c=o
ifn-2m,
which is attained for ut = const. {^0'0>}~1Pi°'0>(l)J v, = 0, and u, = 0, v, =
const. {^°'1)}~1 P^M)A), v = 0, 1, 2, ... , m, respectively. The corresponding
polynomials are obviously the "kernel" polynomials Kt?'fi)(x) (cf. D.5.3)) for the
cases a = /3 = 0 and a ~ 0, /3 = 1, that is, constant multiples of P^'0)(x) and
P^l\x). In other words, we have f(x) = const. {P?'0)(x)}2, and /(«) =
const. A + x) {P^1'1^)}2, respectively. From G.7.10) we now obtain G.7.3)
by direct calculation or, more easily, by mathematical induction, reasoning from
m to m + 1-
C) Second method of calculating Mn. See Lukacs, loc. cit.47 This method is
based on certain mechanical quadrature formulas related to considerations
similar to those in §3.4.
Let n = 2m, x0 = 1, and let the zeros of A — x) Pn'°\x) be denoted by
Xo, Xi, ¦ • • ,xm. Iff(x) is a 7rn and L(x) the Lagrange interpolation polynomial
of degree m which coincides with f(x) at x0, Xi, • • • , xm , then
/(*) - L(x) = A - x) P'y\x)p{x),
where p(a;) is a proper 7rm_i. Therefore,
G.7.11) C f(x) dx - f1 L{x) dx = r1 A - x)P^\x)p{x) dx = 0,
so that as in C.4.1)
G.7.12)
+i
/
2 y-
where the coefficients X, do not depend on f(x). Upon writing
(^ if v > 0,
G.7.13) f(x) = U^j
I i^1 if , = 0,
we show as in §3.4 B) that the numbers X, are positive. Now in view off(x) ^ 0
and of G.7.4), we obtain from G.7.12)
G.7.14) 1 ^ Xo/A), /(I) ^ X^1;
this is the precise bound of /(I), attained when an only when f{xv) = 0, v =
1-, 2, • • • , m; that is, when f(x) = const. {P^i0)(x)\2. In order to find Xo, it is
convenient to write f{x) = y P?'°\x) in G.7.12) (cf. C.4.3)), where 7 is a con-
47 Lukdcs uses this second method in 1; however, the first method was also in his possession
(cf. 1, p. 296).

[ 7.71 ] GENERALIZATIONS; APPLICATIONS 181
stant; then, on account of D.5.3), a = /3 = 0,
=i r p«*\S) dx=_z_ r
2 J-i m + 1 J-i
-i 2 J-i m + 1 J-i m
Consequently, X^1 = y Pill0)(l) = (m + IJ.
Now let n = 2m + 1, x0 = 1, xm+x = — 1, and denote the zeros of the poly-
polynomial A - x2)P?A)(x) by rc0, a* , • ¦ ¦ , xm+l. The same argument as before
leads to
1 r+i
G-7.15) ^ / /(*) dx = E XJK>, X, > 0,
where /(a;) is an arbitrary irn . The maximum in question is again XiT1, which is
attained when and only when f(xy) = 0, v = 1, 2, • •. , m + 1; that is, f(x) =
const. A + x){P?'1\x) }2. In order to find Xo, we write f(x) = y A + x) P?'l)(x),
so that on account of D.5.3), a = 0, /3 = 1,
G.7.16)
1/ ./\, y i i - \ -,(i,i
x)P{
/—I
m+2j_i v ' ' m K~"*~ m+2 '
and we obtain
Xo = 27(m + l) = (m + l)(m + 2).
7.71. Generalizations; applications
A) Let da(x) be an arbitrary distribution on the finite or infinite segment
[a, b], {pn{x)} the associated set of orthonormal polynomials, and x0 an arbitrary
but fixed point. Then if p(x) is an arbitrary irm with
G.71.1) j\p{x)\2da{x) = 1,
we have
G.71.2) \p(xo)\2^~~
c=0
with the sign of equality if and only if p(x) = const. ]>3"=o py(x0)py(x). This
was proved in §3.1 C).
In certain special cases we can calculate the maximum of (or some upper
bounds for) the right-hand member of G.71.2) if x0 runs over a certain interval.
The bounds obtained hold uniformly in that interval for the set of all p(x)
which are wm and satisfy G.71.1).
In what follows we consider various distributions of the form da(x) = w{x) dx;
p(x) denotes an arbitrary irm which satisfies G.71.1).
B) Let a = - 1, b = + 1, w(x) = 1.

182 INEQUALITIES [ VII
Theorem 7.71.1. Let p{x) be an arbitrary irm subject to the condition
G.71.3) j^ \p{x)fdx = 1.
Then we have for — 1 ^ x0 ^ + 1
( 2~*(m + 1),
G.71.4) |p(x,)| ^] _ i
[A(l - x0) *m\
Here A is an absolute constant.
The first bound is precise; it is attained for x0 = =fc 1 if p(x) is a proper irm.
These inequalities follow from G.71.2) when we use G.21.1) and G.3.8).
Now let a = - 1, b = + 1, w(x) = A - x)a A + x)".
Theorem 7.71.2. Let a > — 1, /3 > - 1, and p(z) an arbitrary irm subject to
the condition
G.71.5) jT1 A - x)a(l + xf
G.71.6) P(cos 6) =\ ~
[O(ma+1) if O^d^cmT1.
Here c is an arbitrary but fixed positive number, and the constants in the O-terms
depend only on a, /3, and c. Similar bounds hold in the interval [ir/2, ir].
For the proof we notice that in this case pn(x) ~ n*P(na'^(x) [D.3.3)], and
according to G.32.5),
e)}2 = E vO(v2a) + ? ve
= o(e~2a-2) + e-*-lo{
if cmT1 ^ 6 S t/2. For the same sum we obtain the bound ]>3^=i vO{v2a) =
O(m2a+2) if 0 ^ 6 S cm'1.
By use of G.32.2) certain precise bounds can likewise be derived.
Let a = 0, b = + », w(x) = e~x. Then, according to G.21.3),
G.71.7) e-x21 P(xo) | ^ (m + 1)J, x0 ^ 0,
This bound is attained if x0 = 0 and p(x) is a proper irn .
The case a = 0, b = +°o, «>(z) = e^x", a > — 1, can be treated by a
method similar to that used in the Jacobi case discGssed above (cf. G.6.8)).
C) Now let [a, b] be a finite interval and fix) an arbitrary irn which is non-
negative in [a, b] and satisfies the condition

f 7.71 ]
GENERALIZATIONS; APPLICATIONS
183
G.71.8)
(x)w(x)dx = 1.
We intend to determine the maximum of \f(x0) |, where x0 is a fixed real or
complex value.
By Theorem 1.21.1 we can write
c=-0
2 fm-\ \ 2
(x ~ a)(b ~ x) { 23 vvqv{x)
y=0
r^ I2 (m ^2
if n = 2m,
{x ~ a) < 23 u,r,(x) > + F - x) { 23 v,Sy{x) > if h ~ 1m + 1,
G.71.9) f(x) = -
where {pr(x)}, {qy(x)}, \ry(x)}, {sy(x)\ are the orthonormal sets of polynomials
associated with the weight functions
G.71.10) w(x), (x ~ a)(b - x)w{x), (x - a)w(x), (b ~ x)w(x), a ^ x S b,
respectively. The third and fourth sets are special cases (x0 = a, x0 ~ b) of
the "kernel" polynomials (Theorem 3.1.4); the second set can be calculated by
means of Theorem 2.5. In both cases, n = 2w and n = 2m + 1, we have for
the real numbers u,, vv
/b mm
f(x)w(x) dx = 23 ul + 23 v\ = 1, vm = 0 if n = 2m,
<¦=<) >¦="¦<)
so that according to Cauchy's inequality
G.71.12)
max
max
2,\xo — a\\b — xo
m-1
— a
|2, '| 6 — ar0
Sy(x0)
if n = 2m,
if n = 2m + 1.
In case a ^ x0 ^ fe, the absolute value signs can be omitted. The right-hand
side of G.71.12) represents the maximum required.
D) Let o = - 1, b = + 1, w(x) = A - x)a(l + xf, a and 0 > - 1. Then
the four sets of polynomials mentioned in C) are, respectively, constant mul-
multiples of
G.71.13) {P<"»(*)}, {
In the special case x0 = 1, for the maximum of /(I) we obtain (cf. D.5.3)):
G.71.14)
23
>—0
if n = 2m,
if n = 2m + 1.
Therefore, the following theorem holds:

184
INEQUALITIES
VII
Theokem 7.71.3. Letf{x) be an arbitrary irn which is non-negative in[— 1, + 1]
and satisfies the condition
G.71.15)
Then
G.71.16)
r+i
/ f(x)(l - x)a(l + xHdx = 1, a > - 1, 0 > - 1.
2-a-
T(m + a + 2)r(m + a + |8 + 2)
^ + i)r(« + 2)r(m + i)r(m +18"+
r(m + a + 2)r(m + a + |8 + 3)
r(a + i)r(a + 2)r(m + i)r(m -\
These bounds are precise; both are ~ m2a+2 as m —» oo.
2)
if ft = 2m,
if n = 2m + 1.
Cf. Polya-Szego 1, vol. 2, pp. 96-97, 298, problem 110. Upon permuting a
and /3, we obtain the corresponding bounds for/(— 1). In general, we find,
under the same conditions as in Theorem 7.71.3 (cf. Theorem 7.71.2),
( 6~ia~lO{m) if cnT1 ^ 6 ^ tt/2,
G.71.17) /(cos^) = j 0 i0
1 O(m2a+2) if 0 ^ 0 ^ cm~\
The bounds for /(cos 0) are similar in ir/2 ^ 0 ^ ir. Further, a bound of the
form 0{mc) holds uniformly in 0 ^ 0 ^ ir, where c = max Ba + 2, 2/3 + 2, 1).
The constants of all these O-terms depend only on a, /3, and c.
E) By means of Theorem 1.21.2 we can treat the following problem. Let
f(x) be an arbitrary irn , non-negative for x ^ 0, and
G.71.18)
What is max /@)?
We write (cf. E.1.1))
e~xxaf(x)dx = 1,
a > - 1.
G.71.19) f{x) =
<'t")
-i
LT\x)
m—\
E
V,^T(a + 2)
if ft = 2w,
E«,{r(«
+ a + 1
-1
(x)
if n = 2m + 1,

[7.71] GENERALIZATIONS; APPLICATIONS 185
where the complex numbers u,, vv satisfy the condition
G.71.20) EK|2+?k|2=l.
(In the second sum vm = 0 if n = 2m.) Now in both cases we have (cf. E.1.7))
/@) =
G.71.21) g §{r(a
and this is the required maximum.
If a = 0, we obtain
G.71.22) /@) ^ [n/2] + 1, [" e~xf(x)dx = 1,
provided /(x) is a irn , non-negative for x ^ 0. We can readily prove the more
general inequality
G.71.23) e-'f(x) S [n/2] + 1,
where f(x) is subject to the same condition as in G.71.22). To this end, let x0
be an arbitrary positive number. Applying G.71.22) to
fix + xo)l I e~xf(x + xo) dx\
which satisfies the required condition, we find
/(*>){( e-'fix + xo)dx\ S [n/2] + 1.
Whence
l)e'Xo f e~xf(x + xo)dx
Jo
= ([n/2] + 1) r e~xf(x)dx ^ [n/2] +1.
See also Problem 42.
F) Certain bounds for the orthonormal polynomials {pn(z)\ can be derived
from the preceding results, provided the weight function w(x) satisfies an
inequality of the type
G.71.24) w{x) ^m>0, a S x < b;

18(
or
G.
or
G-
71
71
.25)
.26)
w(z) ^ M1
(x - a)
INEQUALITIES
a(b - xf, a < x
w(x) ^ fix",
<~b,
a >
X
-1,
>0,
[
/3 ^*
a >
VII]
- 1;
_ !
In the first and second cases a and b are finite.
For instance, under the condition G.71.24) we obtain
G.71.27) f {pn(x)\*dx ^ /T1 f \Pn(x)}2w(x)dx = /T1.
Ja Ja
Consequently, according to Theorem 7.71.1,
f Am,
G-71.28) |p»(zo)| < • , . a < x0 < b.
{A'[(xo - o)F - xo)rimi,
Here A and A' are positive constants depending only on a, b and m-
A similar argument applies to the cases G.71.25) and G.71.26).
If we assume that w(x) satisfies a Lipschitz condition, the bounds m and mi
in G.71.28) can be replaced by m* and 1, respectively. (Cf. G.1.15).)
G) Finally, by use of Theorem 7.32.4, we obtain the following generalization
of G.71.28). Let w(x) ^ m > 0, a = - 1, b = + 1, and k ^ 0, an integer.
Then
(*w x i(dTO(n),
G.71.29) p^ (cos0) = 0 < 6 < tt.
1 O(n ),
The bounds of the O-terms depend only on /* and k.
For the proof we use an argument similar to that used in proving Theorem
7.71.2.
7.72. A problem of Tchebichef
A) Problem: Let w(x) be a weight junction on the interval [a, b], and let W(x)
be a given real-valued function, defined on the same interval, and for which the
integrals
G.72.1) (" W(x)xhdx, k =0, 1, 2, ... ,n,
Ja
exist. Let f(x) be an arbitrary polynomial of fixed degree n, not identically zero,
and non-negative in a g x S b. To determine the maximum and the minimum of
the ratio
rb rb
G.72.2) / f(x)W(x)dx: / f(x)w(x)dx.
Ja Ja
See Tchebichef 7. First let a and b be finite. By using the representation
G.71.9) again, we easily find that the quantities in question are the maximum

[ 7.72 J A PROBLEM OF TCHEB1CHEF 187
and minimum of the following quadratic forms in \uy} and \vv\:
G.72.3)
/ \ Z, urpAx) \ W(x) dx + / {T, vvqv{x) (i-o)F-
if n = 2m,
I vL u,r,{x» (x - a)W(x) dx + / IJ^ Vt8t(x) > (b - x)W(x) dx
if n = 2m + 1,
under the condition X!™=o w^ + ZXo w* = 1. (In the first case vm = 0.) Here
{p.-(aO)i {?»(«)}, {^B)}, {s»(z)} have the same meaning as in G.71.9).
Let now a be finite and b = + 00. Then we have to consider the maximum
and minimum of the form
/co rin/2] \2 r«> f((n-l)/2] ^2
{ Z m^P^U) ^(z) rfa: + / <^ Z v,qv{x) \ (x - a)W(x) dx
under the condition X) w^ + Z) ^ = !• Here (p,(a;)} and {^(z)} are the ortho-
normal sets associated with w(x) and (x — a)w(x), respectively, x ^ a.
In case a = — °o,6 — + oo,we must consider the form
/•+co r[n/2] -\2 [n/2]
G.72.5) Z E
,,=,0
where \pv(x)} is associated with w(a;) in [— oOj + oo].
Thus, in all these cases, the problem in question is reduced to the determina-
determination of the greatest and least characteristic values of a certain quadratic form.
In dealing with the sum of two quadratic forms in [u,\ and \v,), respectively,
we determine the greatest characteristic value of the single forms, and the
greater of these values is the maximum in question. A similar remark applies
to the minimum. The actual application of this method is difficult, however,
and certain mechanical quadrature formulas (see below) are often preferable.
Similar considerations apply if the integrals G.72.2) are replaced by Stieltjes
integrals.
B) Let a = — 1, b = + 1, W(x) = xw(x). It suffices to determine the maxi-
maximum and minimum of
{p(x)} 2xw(x)dx :J
{p(x)\ 2w(x)dx,
if p(x) is an arbitrary irm, not identically zero with real coefficients. Having
done this, we must replace w(x) by A - x2)w{x), A ± x)w(x), respectively; see
below. Let x0, Xi , ¦ ¦ ¦ , xm be the zeros of the orthogonal polynomial pm+i(x)
associated with w(x); according to C.4.1), we find for the ratio G.72.6), the
representation
G.72.7) 53 ^IpWI^-Z
where X, denote the Christoffel numbers. Therefore, the maximum and mini-
minimum in question coincide with the greatest and least zero of pm+\{x), respec-

188
INEQUALITIES
VII
tively. (Cf. §3.4 C).) If p denotes the greatest zero of p(x), it is seen from
G.72.3) that the maximum of G.72.2) is, in this special case,
G.72.8)
if n = 2m,
if n = 2m + 1.
max (pm+] , qm)
max (fm+] , Sm+])
The result for the minimum is similar.
C) Here the general discussion of Tchebichef ends (cf. 7, p. 395). We can
prove, however, that the expressions G.72.8) are pm^i and fm+] , respectively,
so that the following theorem holds:
Theorem 7.72.1. Let w(x) be a weight function on the interval [— 1, + 1].
Let f(x) be an arbitrary irn , not identically zero, and non-negative in [— 1, + 1].
Then the maximum of
G.72.9)
/
f(x)xw(x) dx : / f(x)w(x) dx
J
is the greatest zero of pm+\(x) if n = 2m, and the greatest zero of pm+2(— \)pn+i{x) —
Pm+\{ — l)Pm+2(x) if n = 2m + 1. Here \pn{x)\ is the set of the orthonormal
polynomials associated with w(x) in the interval [—1, +1].
According to Theorem 2.5,
G.72.10)
A — x2)qm(x) — const.
A + x)rm(x) = const.
A — x)sm(x) = const.
pm{\)
First, let ?o > ?i > • • • > ?m be the zeros of pm+i(x) in decreasing order. We
show that the first determinant in the right-hand member of G.72.10) is non-
nonzero 0 if ?o < x < 1. Indeed, according to C.2.1)
Pm(x)
=z -'T-m+2
Pm(l) Pm-fl(l) Pm+l(l)
h(x) 1 x
h(-l) 1 -1
h(l) 1 1
where h(x) = pm(x)/pm+i(x). Now, by using C.3.9), we see that the last
determinant is positive in ?o < x < 1, since

FURTHER RESULTS 189
_ 2A - x2) _
(i
On the other hand, h(x) decreases from + w to -- w between {, and f0, and
from + °° to h(l) between ?„ and 1 (cf. the proof of Theorem 3.3.4). Further-
Furthermore, h(- 1) < 0, h(l) > 0. Thus the greatest zero of rm{x), or h(x) ~ h(~ 1),
is greater than the greatest zero of sm(x), or h(x) - h(l), x < 1.
D) Tchebichef discusses in detail the case
G.72.11) a = - 1, b = + 1, w(x) = 1, W(z) = x.
(Problem of the "centroid," see loc. cit., p. 399; cf. also Szego 13, pp. 627-629.)
Save for constant factors, we now have
VmAx) = Pm+iCz); gm(x) = PLhi)(x) = const. P'm+1(x),
G.72.12) rm+i(x) = P^(x) = const. {Pm+]C;) + Pm+t(x)\(l + a;);
s»+i(a;) = P?+°Hx) = const. {Pm+]C;) - Pm+2(x)}(l - x)~\
This yields the following theorem:
Theorem 7.72.2. Let f(x) be an arbitrary irn, not identically zero, and non-
negative for -1 gi| 1, Then the maximum of
+i r+i
/
f+i r+
G.72.13) / xf(x)dx : /
is the greatest zero of Pm+\{x) if n = 2m; if n = 2m + 1, it is the greatest zero of
Pm+l{X) -f- Pm+2\.X).
The distance from the maximum value to 1 is ~ n~2 as n —» °o. The minimum
is obviously the corresponding negative value.
For other refinements of the mean-value theorems, the reader is referred to
Tchakaloff 1. Cf. Problem 43. Concerning other extremum problems for poly-
polynomials and connected inequalities, see Geronimus 2, 3, 4, and Shohat 2.
7.8. Further results
A) From Theorem 1.82.5 we conclude the following refinement of Sonin's
Theorem 7.31.1. Let y = y(x) satisfy the differential equation G.31.1) and let
y(x) have an infinite set \xm\ of zeros ordered in the increasing way. xt < x2 <
xz < • • • . The function <p(x) should be positive, continuous, and decreasing.
Let p be a fixed positive number. Then the integrals
/ \y{x)\"dx
are increasing.
A similar statement holds if 4>{x) is increasing.
For v _^ o this reduces to the assertion of Theorem 1.82.2. Taking the pih root
of the integral, for p —> * we obtain the assertion of Sonin's theorem. Hence,

190 INEQUALITIES [ VII ]
the assertion above is a common generalization of both theorems (Makai 2).
In Makai 2 one finds various remarkable special cases of this useful theorem.
B) Let /x,.,n be the successive relative maxima of \Pn(xy when x decreases
from +1 to — 1. We have (Theorem 7.3.1):
I > Mi.» > M2.7, > • ¦ • > Hh.r, where h = [n/2].
Now for a fixed r, we have. (Szego 23)
G.8.1 ) yu,.r, > /*r.H+i, n > r + 1.
These inequalities also hold for the relative maxima of \Pna"p)(x)\/Ptti"'p)(l)
when a ==/3 > - i Szasz 2, and from this for a > /3 = - i by use of D.1.5).
The same result for a> $> - \ is probable, but still open. Somewhat
surprisingly these inequalities fail for
and graphical evidence suggests that the inequalities G.8.1) are reversed
for this function. See Askey-Gasper 4 for a problem which would be solved
by these inequalities.
For the orthonormal Hermite functions defined by
H (x)e~x'/2
Szasz 3 proved inequalities similar to G.8.1) and used them to prove
\K(X)\ g
Stronger inequalities in which the right-hand side goes to zero in n are
known. See Askey-Wainger 1 and Muckenhoupt 3. They also obtain refine-
refinements similar to that given in Problem 40 and similar inequalities for Laguerre
polynomials.
C) Many inequalities, involving in particular the classical orthogonal
polynomials, have been investigated by Karlin-Szego 1. Problem 70 (Turan's
inequality) is a special case. See Gasper 5 for a Turan type inequality for
Jacobi polynomials.
D) Theorems 7.31.2, 7.32.3, and 7.6.5 have been sharpened for a > - \ by
Lorch 3.
E) The inequalities G.71.12) have been refined in Schoenberg-Szego 1.
F) Turan 3 considers the problem of maximizing the Markoff-type functional
over polynomials irn of degree g n. He shows that the exact maximum is
1
2sinir/Drc+2) "

CHAPTER VIII
ASYMPTOTIC PROPERTIES OF THE CLASSICAL POLYNOMIALS
The consideration of the asymptotic properties of the orthogonal polynomials
\pn(x)}, n —* oo, leads to two fundamental problems: the asymptotic behavior
of the polynomials in question outside the orthogonality interval, especially
in the non-real domain, and the asymptotic behavior on the orthogonality
interval itself. In general, the second problem is deeper and more difficult than
the first one. In our treatment we start with a discussion of Legendre poly-
polynomials, obtaining various important asymptotic formulas for them. We
intend not only to give a survey of results, but also to point out the various
methods used. The extension to ultraspherical and general Jacobi polynomials
will also be indicated. The asymptotic investigation of Laguerre and Hermite
polynomials, in general, requires new considerations, although essentially the
same methods as before can also be applied to these cases.
The simplest special case,
Tn{x) = \{zn + O, x = \{z + z'1),
the case of Tchebichef polynomials of the first kind, furnishes a good illustration
of the characteristic features of our results. If x is located outside the interval
[— 1, + 1], we can take \z\ > 1, and we then see that
Tn(x)g*zn/2, n -> «.
On the interval [— 1, + 1], we write z = e\ Tn(x) = cos nd. Here the poly-
polynomials have an oscillatory behavior.
These results need only a slight modification for Legendre, and even for Jacobi,
polynomials as long as x ?± =fcl. A new difficulty will, however, arise in the
vicinity of the end-points ± 1, which are in some respects exceptional. This
is ultimately due to the fact that the coefficient of dd in
A - x)a(l + xfdz = -A - cos 6)a{\ + cos df sin Odd
vanishes, in general, or becomes infinite at 6 = 0 and 6 = ir. When this occurs,
functions of the type cos nd are not suitable for the approximation of the poly-
polynomials in question in the neighborhood of x = ±1. For this purpose we shall
use certain Bessel functions.
Usually, the problems and results for Laguerre and Hermite polynomials are
similar. But it is rather curious that, in the corresponding asymptotic expres-
expressions, the quantity ft* appears instead of ft. In the general Laguerre case,
Bessel functions are needed near x = 0, whereas for the Hermite polynomials
the origin x — 0 does not play an exceptional role. In both cases new difficulties
arise due to the fact that the interval of integration is infinite. For the expan-
191

192 ASYMPTOTIC PROPERTIES OF CLASSICAL POLYNOMIALS [ VIII ]
sion problem it is of great importance to have asymptotic formulas that hold in
intervals which become infinite as n —> ».
8.1. The formulas of Mehler-Heine type
These important formulas are elementary in character, and we shall discuss
them briefly before entering into the general considerations of §§8.21-8.23.
A) Theorem 8.1.1. Let a and C be arbitrary real numbers. Then
(8.1.1) lim n-aP(na^ (cos -) = lim n'aPia'e) (\-^\ = (z/2)-aJa(z),
where Ja(z) has the same meaning as in A.71.1). This formula holds uniformly
in every bounded region of the complex z-plane.
For Legendre polynomials, a = /3 = 0, formula (8.1.1) is due to Mehler C,
p. 140) and Heine C, vol. 1, p. 184). Concerning further literature we refer
to Watson 3, p. 155. In case a = /3 = 0, a very simple proof follows from the
first integral of Laplace [D.8.10), A.71.6)]. For a = =fc ? the function in the
right-hand member of (8.1.1) is a constant multiple of z~l sin z and cos z, re-
respectively (cf. ,A-71.2)). Formula (8.1.1) is trivial for the elementary cases
D.1.7) and D.1.8).
The proof can be based on D.21.2). In fact, we have for the (v + l)st term
of D.21.2) if x = cos(z/n), z and v fixed and n —> oo ; the following asymptotic ex-
expression :
v\(n-v)\ T(n + a + 0 + 1) T(v + a + 1)
(-#•
Here we exclude the case of a negative integer a. Passing to the limit under the
summation sign is valid because of the existence of a dominant for the total sum
which is readily derived. Indeed, we have, if n is large enough,
n~a Y{n + a + j8 + v + 1) T(n + a + 1)
(n - v)l T(n + a + j3 + 1) 2'n2'
S tVBn + a + ^(" + " + 1) _ 0A)
n! 2"n2"
uniformly in v, 0 ^ v ^ n. The argument needs only a slight modification if a
is a negative integer.
Formula (8.1.1) gives a complete characterization of the function Pi" i&) (cos 0)
for d = O(n~l). As an important consequence we note the following:
Theorem 8.1.2. Let zln > x2n > ¦ ¦ ¦ be the zeros of P^'^ix) in [— 1, + 1]
in decreasing order (a, j3 real but not necessarily greater than — 1). If we write

[8.1] FORMULAS OF MEHLER-HEINE TYPE 193
x,n = cos dyn , 0 < 0,n < 7T, then for a fixed v,
(8.1.3) lim ndm = jn
n —»oo
where jy is the vth positive zero of Ja(z).
B) Relation (8.1.1) enables us to derive some properties of Bessel functions
from the corresponding properties of Jacobi or Legendre polynomials. We use
the symbol (a) —> (b) to indicate that in writing x = cos (z/n), a certain formula
(a) is transformed into another formula (b) by the limiting process n —> oo.
Then we have the following relations:
D.1.7) and D.1.8) -> A.71.2),
D.2.1) -> A.71.3),
D.22.2) -> J_;(z) = (- l)lJi(z), I an integer,
D.24.2) -> A.8.9),
each of D.8.6), D.8.10), D.9.3) -> A.71.6) in the special case a = 0, and
D.9.19) -> A.71.6) in the general case.
See also Problem 44.
From Theorem 1.91.3 (Hurwitz's theorem) the reality of the zeros of z~aJa(z)
follows for a > - 1. From F.6.5) and F.6.3) (or F.6.2)) we obtain for the
positive zeros j, of Ja(z), A = a + \,
(8.1.4) (v + \a - \)ir fg ],-? vtt, v = 1, 2, 3, • • • , - \ ? a fg + i
The upper bound can be replaced by (v + o)tt if — ^ ^ a ^ 0 (cf. F.6.6)).
Furthermore (cf. Watson 3, p. 49, A); cf. G.31.5)),
(8.1.5) G.33.1) -> T(a + l)(z/2ra | Ja(z) | ^ 1, 2 > 0, a ^ - i,
(8.1.6) G.33.5) -> z1/2 I JaB) I ^ B/ttI/2, . z>0, -^afg+i
The expansion D.9.17), (8.21.5), and the inequality (8.21.6) of the remainder
term furnish the important formula (Stieltjes 8, p. 242):
-Z^Vv-1 {1-3---B,
~ \7z) h 2-4---
< /2
= W
2,
2V{l-3---Bp-l)}2
W
2-4 ...2p
Thus the error ep(z) is numerically less than the first neglected term (replacing
cos by 1). For p = 1 we obtain the special case a = 0 of A.71.7) (with a
numerical constant in the bound of the remainder).
C) Theorem 8.1.3. Let a be arbitrary and real. Then for an arbitrary com-
complex z
(8.1.8) lim n~aL(na\z/n) =z-al2JaBzi),
n-»oo
uniformly if z is bounded.

194 ASYMPTOTIC PROPERTIES OF CLASSICAL POLYNOMIALS [ VIII ]
This formula is of a type similar to (8.1.1) and yields similar results. The
proof can be given along the same lines as there. In the special cases a = =fc \,
we again obtain trigonometric functions. From this case an analogous formula
for Hermite polynomials can be derived. See Problem 45.
Both formulas (8.1.1) and (8.1.8) can be extended to an asymptotic expan-
expansion. Concerning the case of Laguerre polynomials, a = 0, see Moecklin 1, p. 28.
8.21. Asymptotic formulas for Legendre and Jacobi polynomials
From the point of view of the asymptotic problem, the Legendre polynomials
Pn(x) represent the simplest non-trivial case. We start with an enumeration
of some classical results concerning the behavior of Pn{x) as n —> ». The
proofs, based on various methods, are given in subsequent sections. In what
follows, e denotes a fixed number with 0 < e < tt/2, so that the interval [e, it — e]
lies wholly in the interior of [0, r]; p is a fixed positive integer.
A) Theorem 8.21.1 (Formula of Laplace-Heine; Heine 3, vol. 1, p. 174).
Let x be an arbitrary real or complex number which does not belong to the closed
segment [— 1, + 1], Then as n —> «=,
(8 21 1\ P (r\ ~ (iTrr)\~ll2(r2 — IT1'4 !r 4- (r2 — i\1/2!n+1/2
Here (x2 - 1)~1/4, (x2 - 1I/2, and \x + (x2 - iI/2}«+1/2 are real and positive if x
is real and greater than 1. This formula holds uniformly in the exterior of an arbi-
arbitrary closed curve which encloses the segment [— 1, + 1], in the sense that the ratio
tends uniformly to 1.
Theorem 8.21.2 (Formula of Laplace; Heine 3, vol. 1, p. 175).
P«(cos 6) = 2m(irn sin d)~m cos {(n + \)B - tt/4) + O(n~312),
(8.21.2)
0 < 6 < 7T.
The bound for the error term holds uniformly in the interval e ^ d 5ji tt — e.
Theorem 8.21.3 (Generalization of Laplace-Heine's formula). Let x be in
the complex plane cut along the segment [— 1, + 1]; let x = %(z + z~x), \z \ > 1.
Then
P(x) =a zn T 1-3 ••• B,-1) 2y( _ 2)-,-i
fo o1 o\ ^ Bn - l)Bn - 3) • • • Bn - 2j» + 1)
{0.Z1..6)
Here g, has the same meaning as in D.9.2), that is,
_ 1. _ 1-3... B?- 1) _-, o q
Formula (8.21.3) holds uniformly in the same sense as in Theorem 8.21.1.

[8.21] LEGENDRE AND JACOB I POLYNOMIALS 195
Theorem 8.21.4 (Darboux's generalization of Laplace's formula; Darboux 1,
p. 39).
D ( a\ o V* 1-3 ••• Bf - 1)
(8 214) = ^ 9' V^W^sr^fr -27+ i)
//ere gn has the same meaning as in Theorem 8.21.3. The bound for the error term
again holds uniformly in e ^ 6 ^ it — e.
Theorem 8.21.5 (Stieltjes' generalization of Laplace's formula; see Stieltjes
7,8).
Pn(cos0) = -
tt3-5- •¦ Bn+ 1)
V h cos{(n + y + ^)g- (y + ^)
(8.21.5) & *' BihT^ + ^W'
O<0<7T.
Here h, has the same meaning as in D.9.18), that is,
•¦¦B,-1) i|...(,-i)
=1- h -
2-4---2, (n + |)(n + S)---(n + ^ + i)'
, = 1, 2, 3,
-M
(8.21.6) ' pW tt3-5 ••¦ Bn+1) p(
M = max (| cos d p1, 2 sin d).
The factor M is between 1 and 2. 7'/iiis the error is numerically less than twice the
first neglected term (replacing cos by 1).
Theorem 8.21.6 (Formula of Hilb; Hilb 1).
(8.21.7) Pn(cos 6) = @/sin fl)*J0{(n + \N\ + O(n~l),
uniformly for 0 ^ d ^ tt — e. Afore precisely, for the error term we have the
bounds
(9JO(n""!) if en ^ (9 g tt - e,
(8.21.8)
02OA) i/ 0 < 0 ^ en,
u>/iere c is a fixed positive constant.
B) Some of these results can be extended to Jacobi polynomials. The
extension of (8.21.1) is due to Darboux A):

19G ASYMPTOTIC PROPERTIES OF CLASSICAL POLYNOMIALS [ VIII ]
Theorem 8.21.7. Let a and /3 be arbitrary real numbers. Then
Pia'"\x) S (x - iyan{x -t \ym{{x + 1)* + (x - l)*}a+/!
(8.21.9)
¦Bm)-*(x'- lr^z+tz2- II}4,
w/iere x is outside of the closed segment [ —1, +1]. This formula holds uniformly
in the same sense as in Theorem 8.21.1. The determination of the multivalued
functions occurring in this formula is obvious.
The extension of (8.21.2) is also due to Darboux A); this is the important
formula to which we referred in §7.32:
Theorem 8.21.8. Let a and j3 be arbitrary real numbers. Then
Pi"'^(cos 6) = n~hk{6) cos (NO + 7) + 0{n~*),
('cos 0
(8.21.10) k(e) = tTj ('sin 0 ('
7 = - (a + -Dtt/2, 0 < 6 < 7T.
TTie bound for the error term holds uniformly in the interval [e, t — e].
The extension of Theorems 8.21.3 and 8.21.4 to Jacobi polynomials is readily
achieved. However, the law of the coefficients is, in this case, rather com-
complicated.
Theorem 8.21.9. Let a and j3 be arbitrary real numbers. There exists a se-
sequence of analytic functions 4>,{z) = 4>,{a, j3; z) which are real for real z and regular
for I z I > 1 and \z\ = 1, z 9^ dbl, such that
(8.21.11) z-nP\a'&\x) = ?*,B)rr-J + O(n-p-h); x = \{z + z~l), \z\>\,
uniformly for \ z \ ^ R, R > 1.
Furthermore,
(8.21.12) Pl^Ccos 6) = 2dlieine E ^(e* V~4 + O(rf P~J), 0 < 6 < *-,
I J
uniformly for e ^ d ^ r — e.
These extensions attain, in the ultraspherical case, the following more pre-
precise form:
Theorem 8.21.10. Let x = |(z + z~~l), \ z \ > 1, and X > 0 orX < 0, X ^
-1,-2,-3, •••.
(8.21.13) Pnix)=an ^o ^n + A-mn + A
•z-2'

[8.211 LEGENDRE AND JACOB I POLYNOMIALS 197
Furthermore,
pw, as 9 v A ~ X)B - X) ¦ • • (v - X)
Pn (COS 6) = 2an 2 <*
f^o "> + X-l)(n + X-2) ••• (n + X-«0
(8.21.14)
B^ey^ + 0(nX~"~lj' ° <° < *¦
Here av has the same meaning as in D.9.21). Regarding the uniformity, the same
remark holds as in the previous theorem.
The special case of an integral value of X is discussed in §8.4 E).
An extension of Theorem 8.21.5 to ultraspherical polynomials is the fol-
following:
Theorem 8.21.11. Let 0 < X < 1. We have
9-1 n/ 1 \ \ti/,. \
V — A
r(x)
(8.21.15)
where
/ooiiflN \ d (a\ ^(n/\- ^ r(n + 2x) r(p + x)r(p - x + i) m
(8.21.1b) \ KAu) < B/7T)Sin XtT t-t . ; ; ; -r- -rz:—-
F(X) p!F(n + P + X + l) B si
Here M has the same meaning as in Theorem 8.21.5.
C) Finally, we mention the following formula of "Hilb's type" (cf. Szego
17, p. 77; Rau 2, pp. 691-692).
Theorem 8.21.12. Let a > — 1, and let /3 be arbitrary and real. Then we
have
( sin -) ( cos - ) P(na J) (cos 0) = N~a - -^ F/sin d^JdNd)
(8-21J7) f dh0{n-*) if cn~l ^ d ^ tt - e,
+20(na) i/ 0 < 6 ^ cn~\
where N has the same meaning as in (8.21.10); c and e are fixed positive numbers.
Obviously, the remainder term is always d^OirT*). If we use D.1.3), a
similar formula can be obtained in the intervals e ^ d ^ tt — cn~l and
7T - en S 6 < 7T provided j3 > — 1. In view of A.71.7) this leads to the
following important result:
Theorem 8.21.13. Let a > — 1, /3 > — 1. We have, with the same notation
as in (8.21.10),

198 ASYMPTOTIC PROPERTIES OF CLASSICAL POLYNOMIALS [ VIII ]
Pi"'p\cos 6) = rT*A;@) {cos {NO + 7) + (n sin 6)~lO{\)\,
(8.21.18)
en ^ d ^ ir — en.
Here c is a fixed positive number.
This formula (Szego 17, p. 77) is more precise than (8.21.10); for a = j3 =
X - \, 0 < X < 1, it follows from Theorem 8.21.11, p = 1. The restriction
a > — 1, /3 > —1 is not essential (cf. Szego, loc. cit.). See also Obrechkoff 2.
D) Analogous formulas hold for Jacobi's functions of the second kind Q(na'?>(x)
and, particularly, for Legendre's functions of the second kind Qn(x), if x is
in the cut plane, as well as for Qn(cos 0) if 0 < d < r. Here we point out
only the analogue of Laplace's formula (8.21.2):
Theorem 8.21.14. For 0 < 6 < tt
(8.21.19) Qn(cos 6) = 7rJBn sin 0)~* cos \{n + 1H + tt/4! + O(n~l).
This holds uniformly in the interval [e, it — e].
8.22. Asymptotic formulas for Laguerre and Hermite polynomials
Similar, but slightly more complicated, formulas hold for Laguerre and
Hermite polynomials. In what follows n —* =o ; we denote by e and u fixed
positive numbers, e < w, by p a positive integer.
A) Theorem 8.22.1 (Fejer's formula; Fejer 3). Let a be an arbitrary real
number; we have
L(na)(x) = tTWa/2~V/2~} cos {2(nxf - air/2 - tt/4)
(8.22.1)
+ O(na/2-3), x > 0.
The bound for the remainder holds uniformly in [e, w].
Theorem 8.22.2 (Perron's generalization of FejeYs formula; Perron 2, p. 78,
D9)). Let a be an arbitrary real number; we have for x > 0
a7r/2 - tt/4)
(8.22.2)
+ r e x 'n sin {2{nx) — air/2 — tt/4}
where Av{x) and Bv(x) are certain functions of x independent of n and regular for
x > 0. The hound for the remainder holds uniformly in [e, w].
We notice that AQ(x) = 1 and BQ(x) = 0.

[8.22] LAGUKRRK AND HHIIMITK POLYNOMIALS 199
Theorem 8.22.3 (Perron's formula in the complex domain; loc. cit.). Let a
be an arbitrary real number. Then
Lia(x) = |7r-|e'/2(-x)-a/2-1na/2"
(8.22.3)
where Cv{x) is again independent of n; it is regular in the complex plane cut along
the positive part of the real axis. Formula (8.22.3) holds if x is in the cut plane
mentioned; ( — x)~al2~' and ( — x) must be taken real ami positive if x < 0. The
bound for the remainder holds uniformly in every closed domain with no points in
common with x ^ 0.
Here we have Cq(x) = 1.
Theorem 8.22.4 (Asymptotic formula of Hilb's type). For a > —1 we have
e-xl2xal2lJna)(x) = ira/l-rln-±^-±i? Ja[2(Nx)l\ + O(na/2),
(8.22.4) n\
N = n + (a + l)/2, x > 0,
the bound holding uniformly in 0 < x ^ u>. More precisely, the following bounds
are valid:
zw*0(na/2-f) if cn~l ^ x ^ «,
(8 22 5)
xalw0{na) if 0 < x < cn~\
In case a = 0 the last bound is to be replaced by x~ log (aTV); in (8.22.5)
c is a fixed positive number.
Evidently, the remainder term (8.22.5) is equal to x&H0(na'2"i) throughout
0 < x ^ co.
As a consequence of (8.22.4) we obtain the following analogue of (8.21.18),
which is more precise than (8.22.1).
Theorem 8.22.5. Let a > —] and cn~l ^ x ^ o>; then.
Lia\x)
(8.22.6) .,,„!,.
= 7i Y'-x "'¦ 'n"n l{oos[2(rw-)' - aw/2 - tt/4J + (nx) '0(l)\.
Here c and co are fixed positive constants.
We observe that iV* — n" = O(n~'2) where AT has the samo meaning as in
(8.22.4).
B) Substituting a = ±.\ in (8.22.4), we find, by using E.6.1) and A.71.2),
a formula of Hilb's type for Hermrte polynomials. This is contained in the
more general theorem:
Theorem 8.22.6 (Asymptotic expansion for Hermite polynomials). For a
real x

200 ASYMPTOTIC PROPERTIES OF CLASSICAL POLYNOMIALS [ VIII
\-nle-x*t2Hn{x) = cos (Nhx - mr/2) ? uAx)N~v
sin {N*x - n7r/2) 2, v,(x)N~' + 0(n'p),
N = 2n + 1,
X = T(n + 1) T(n + 2) ,
" r(n/2 + l) ° r(n/2 ^
where
according as n is even or odd. The coefficients u,{x) and vv(x) are polynomials
depending on v; they contain only even and odd powers of x, respectively. The
bound for the error term holds uniformly in every finite real interval, whether it
contains the origin or not.
For arbitrary n, we have Xn = (T(n + l)/T(n/2 + 1)){1 + 0{n~l)\, and
uo(x) = 1, vo{x) = x3/6, so that
y + 1} e~*V2
e~*V2Hn(x) = cos (N'x - mr/2)
T(n + 1)
(8.22.8)
+ % N~l sin (N*x - n7r/2) + OU).
0
Theorem 8.22.7 (Asymptotic expansion for Hermite polynomials in the
complex domain). The expansion (8.22.7) holds in the complex x-plane if we
replace the remainder term by exp {N j $(x) \\O(n~p). This is true uniformly
for | x | ^ R where R is an arbitrary fixed positive number.
C) Finally we deal with another type of asymptotic formulas requiring a
more elaborate consideration.
Theorem 8.22.8 (Formulas of Plancherel-Rotach type for Laguerre poly-
polynomials). Let a be arbitrary and real, e and u fixed positive numbers. We have
(a) for x = Dn + 2a + 2) cos2 4>, e ^ <t> < tt/2 - m~{,
e xl Li (x) = (-1) Grsin0) *x *n /2 '
(8.22.9)
{sin [(n + (a + l)/2) (sin 20 - 20) + 3tt/4] + (nx)"§O(l)J ;
(b) /or x = Dn + 2a + 2) cosh2 0, e ^ 0 ^ w,
e~r 2Ln'
(8.22.10)
• exp {(n + (a + D/2) B0 - sinh 20)} {1 + OU)};
(c) for x = \n + 2a + 2 — 2Bn/3)^, « complex and bounded,

[ 8.23 ] REMARKS ON PRECEDING RESULTS 201
(8.22.11) e~xl2L(na)(x) = (-D"^-"-^*
where A{t) is Airy's function defined in §1.81.
In all of these formulas the O-terms hold uniformly.
The corresponding formulas for Hermite polynomials (Plancherel-Rotach 1)
are given by the following:
Theorem 8.22.9. Let e and u be fixed positive numbers. We have
(a) for x = Bn + 1)* cos <j>, e ^ 0 ^ ir — e,
e~xV2Hn(x) = 2n/2+i(n!)iGrnri(sin </>)"*
(8 22 12)
• {sin [(n/2 + i)(sin 20 - 20) + 3tt/4] + 0{n~x)};
(b) for x = Bn + 1)* cosh <j>, e ^ 0 ^ w,
e~xV2Hn(x) = 2n/2-i(n^(rn)ii
(8.22.13)
• exp [(n/2 + i)B0 - sinh 20)] {1 + 0{n~l)} ;
(c) for x = Bn + 1)* — 2" 3~}n~^, t complex and bounded,
(8.22.14) e-xV2Hn(x) = Z^-^'^Hn^rr^Ad) + Oin'1)}.
In all these formulas the O-terms hold uniformly.
Note that (8.22.12) holds uniformly in the vicinity of x = 0.
8.23. Remarks on the preceding results
A) Of all the formulas enumerated in §8.21, formula (8.21.1) has the simplest
character. It can be proved by various methods. An extension to an asymp-
asymptotic series is given by (8.21.3). Corresponding formulas hold for Jacobi poly-
polynomials (§8.21 B)). The following simple consequence of (8.21.9) is important
for various purposes:
(8.23.1) | Pia^(x) \lln ^ | x + (x2 - if |, n -> » ;
here x is in the cut plane. The right-hand member is > 1 and represents the
sum of the semi-axes of the ellipse with foci at ±1 and passing through x.
We compare (8.23.1) with the following formula for Jacobi's functions of the
second kind:
(8.23.2) \Q(na^(y)\lln^\y- (y2 - DM-
Here y is again in the cut plane and the right-hand member is <1. (Cf.
(8.71.19).)
We shall give also several proofs for the classical formula (8.21.2) of Laplace.
It can be similarly generalized in various directions.
Darboux's formula (8.21.4) is the most important illustration of the method
due to him A). This method furnishes asymptotic formulas for the coefficients

202 ASYMPTOTIC PROPERTIES OF CLASSICAL POLYNOMIALS [ VIII ]
of power series whose singularities on the circle of convergence have a certain
simple character. The same method yields (8.21.3), as well as similar expan-
expansions of Jacobi polynomials. (Cf. §8.4.)
The significance of the formula of Stieltjes is due to its unrestricted validity
in the interval 0 < 6 < it (although the bound for the remainder given by
(8.21.6) becomes infinite if 6 —> 0 or 6 —> it) . In view of this fact, it can be used
in various cases not only in a fixed interval in the interior of [0, w], but even in
the vicinity of the end-points where the formulas of Laplace and Darboux
fail in general. For p = 0 it holds in the sense that Pn(cos 6) = RQF). Then
(8.21.6) furnishes inequality G.3.8) but with a larger factor on the right side
(with 2B/tt)* instead of B/tt)*). From the formula of Stieltjes an arbitrary
number of terms in (8.21.4) can readily be derived. However, it seems diffi-
difficult to obtain the general law of the coefficients of (8.21.4) in this manner.
The importance of Hilb's formula also lies in its unrestricted validity in the
neighborhood of 6 = 0, with the additional advantage that the remainder term
tends to 0 uniformly in this neighborhood. It furnishes the Mehler-Heine
formula (8.1.1) immediately and yields Laplace's formula (8.21.2) by means
of A.71.7). The bounds (8.21.8) are a slight improvement over Hilb's result.
Our proof (§8.62) is essentially the same as that of Hilb. Szego A5) gives
an asymptotic expansion in terms of Bessel functions of increasing order and
generalizes Hilb's result. Analogous formulas hold for Legendre's function of
the second kind. Szego also obtains A5, p. 450) an analogue of (8.21.1) which
holds in the cut plane, arbitrarily near its boundary. This formula involves
Bessel functions with imaginary arguments.
Another formula of a type similar to (8.21.7) has been given by Watson B)
with a numerical estimate of the remainder. It involves Jq{z) and Y0(z).
Theorem 8.21.12 is the extension of Hilb's formula to Jacobi polynomials.
The proof given in §8.63 follows the same line of argument as that in §8.62.
The proofs of Theorems 8.21.1-8.21.14 are based on the following methods:
(a) Explicit series or integral representations;
(b) Darboux's method;
(c) method of Liouville-Stekloff (method of the integro-differential equation);
(d) method of steepest descent.
A short survey of these methods will be given at the proper places.
B) Fejer's proof of (8.22.1) is based on the generating function E.1.9),
which in this case has the essential singularity w = 1 on the circle of con-
convergence | w | = 1. This argument is of a character similar to Darboux's
method. The more complicated type of singularity in this case naturally
requires a more careful discussion; it is carried out by Fejer by an elementary
method similar to the second mean-value theorem of the integral calculus.
In Perron's first proof of (8.22.3) (in the special case p = 1, see 1), complex
integration is used. He obtains the complete expansions (8.22.2) and (8.22.3)
by using certain general asymptotic results concerning confluent hypergeometric
functions.

f 8.23 ] REMARKS ON PRECEDING RESULTS 203
Further proofs of FejeVs formula (partly giving more exact bounds for the
remainder and holding on certain segments the end-points of which tend to +0
and +«) have been given by Rotach A), Szego A0), and Kogbetliantz A4).
They use either the method of steepest descent or similar arguments. FejeVs
formula is contained in (8.22.4); the latter result follows from certain general
asymptotic theorems of Hilb's type due to Wright A, p. 261, however only
for fixed x). We give a proof for (8.22.4) by using the Liouville-Stekloff
method (§8.64).
The special cases a = ±| are equivalent to Hermite polynomials (see
(8.22.7)); in these cases, FejeVs theorem was known previously by Adamoff A).
Adamoff obtains the remainder term with certain numerical bounds.
We derive (8.22.7) by using the method of Liouville-Stekloff. Uspensky's
formula E.6.5) then leads immediately to a corresponding asymptotic expan-
expansion for Laguerre polynomials involving Bessel functions. We indicate this in
§8.66. Concerning this expansion see Wright, loc. cit.; its first term is (8.22.4).
From this expansion Perron's formulas (8.22.2) and (8.22.3) follow readily. A
second proof of these formulas can be based on the method of steepest
descent (§8.72).
The first term of the asymptotic expansion mentioned for Hermite poly-
polynomials furnishes (8.22.8). Adamoffs formula is less precise. (On the other
hand it contains numerical constants.) Comparison of (8.22.8) with Us-
Uspensky's formula A, p. 597, F)) indicates that it is convenient to work with
N = 2n + 1 instead of N = 2n.
A very detailed asymptotic investigation of Hermite polynomials is due to
Watson A, second paper).
We mention the following simple consequence of Theorems 8.22.3 and 8.22.7:
Let x be in the complex plane cut along the non-negative real axis. Then
(8.23.3) n~Mog | L{na\x) | -> 2dl{(-x)i}, n -> «.
Here { — xf is taken real and positive if x < 0. However, if x is non-real,
(8.23.4) BnrMog|r^2+iNl) | Hn(x) \\ -> \3(x) |, n->«.
Theorems 8.22.8 and 8.22.9 are closely related to the important results of
Plancherel-Rotach A). These authors deal exclusively with Hermite poly-
polynomials and use the method of steepest descent; they obtain a complete asymp-
asymptotic expansion in all three cases of Theorem 8.22.9. Their argument has been
applied to Ln(x) by Moecklin A). We shall derive (§§8.73-8.75) only the
principal terms of these expansions, however, for general Laguerre polynomials
L{f\x), by using the method of steepest descent. Our argument is based on
the generating function E.1.16) and on the asymptotic expansion of Bessel
functions jn the complex domain.

204 ASYMPTOTIC PROPERTIES OF CLASSICAL POLYNOMIALS [ VIII ]
The formulas (8.22.9)-(8.22.11) describe Laguerre polynomials in the
"oscillating region," in the region beyond this, and in a certain vicinity of the
largest zero, respectively. The same is true for (8.22.12)-(8.22.14).
Van Veen A, 2) derives the asymptotic series corresponding to (8.22.12)
with numerical estimates. Schwid A) applies the method of LiouviHe-Stekloff
(in the more precise form due to Langer A)) to the asymptotic investigation of
Hermite polynomials.
8.3. "Elementary" proof.of the formulas of Laplace-Heine and Laplace
A) We start from the representation D.9.4) and first prove (8.21.1). Let
x — \i.z + •z), I z | > 1; then
71 71
(8.3.1) Pn{x) = ]C gn,gn-m^~'m = 0»z" H "—" S'mZ""""'.
7/1=0 m=0 gn
We next show that
(8.3.2) lim ? (?"-'" - l) gmz~^ = 0,
m=0
uniformly for | z | 2s R, R > 1. Indeed, the expression in the brackets tends to
zero if m is fixed and n —¦* <». On the other hand, it is easy to find a dominant;
we have, for instance,
0 < yJL=!" - 1
~ \ Qn
Now (n + l)Vn is bounded from zero and from infinity, and
(" + »' si, 0 S m i n.
(n - m + l)*(m + II
B) We can prove (8.3.2) in another way, by use of certain very elementary
properties of the sequence {gn}- Let 8 > 0 be arbitrary, and let M be a positive
integer, such that
V p-2m x
2L, K < o.
The numbers gn-m/gn — 1 are positive and increase with m; we therefore have,
if n > M,
M / \ / \ M
V / gn-m _ I 1 7?~2m < I "n~M _ 1 | Y^ n 7?"~2m -
/ \ oo
/, gmt\>
The last expression tends to 0 as n —> °o since gn/gn-i —> 1. On the other hand,
71

[ 8.3 ] FORMULAS OP LAPLACE-HEINE AND LAPLACE 205
Now gmjgm-\ is increasing, so that
, _?±i=??_ ^ 1 or ^ 1
according as m ^ (n + l)/2 or m ^ (n + l)/2. Consequently, grn_mgrm attains
its maximum, as m varies from 0 to nr for to = 0 or m = n; hence gn-mgm ^ ^n ,
so that
Qn-
m-M+l \ gn
This establishes (8.3.2).
Therefore the expression
(8.3.3) (g
R
~2m
m=0
tends to 0 as n —> °o, uniformly for | z \ ^> R, R > 1. The last sum tends to
A — z~2)~*; whence (8.21.1) is readily derived.
C) For the proof of (8.21.2) we use D.9.3). We have for 0 < 0 < it
. . . i«Z?i
Pn(cos 6) = 2gn
(8.3.4)
m-0
im
where E' has the same meaning as in G.4.5). The sequence {gm} tends
monotonically to 0, so that for 0 < 6 < ir
E
This series converges uniformly for e ^ 6 ^ it — e.
Now, if 5 is an arbitrary positive number, we determine the positive integer M
so that
(8.3.5)
e2imB
M" > M' > M.
The numbers grn_m/grn — 1 being positive and increasing with m if n > 2M, we
have, according to Abel's inequality,
(8.3.6)
-— — 1) max
u
limi
where K is a fixed constant. On the other hand,

20G ASYMPTOTIC PROPERTIES OF CLASSICAL POLYNOMIALS [ VIII 1
if! _ 1
(8.3.7)
<7n
— 1 1 max
e2imS
<
gn
Since gn/g2n is bounded, we see that the first 23' in the right-hand member of
(8.3.4) tends to zero. Hence we have
(8.3.8) B^n)-1Pn(cos 6) = dl{e-inS.ei'"i-ei2\2 sin 0)""*} + 6n ,
where <5n —> 0 uniformly in f g Sg ir - e. This is the formula of Laplace with
a remainder term otn'*).
D) Although they do not lead to the remainder term 0(n~!) stated in (8.21.2),
these elementary arguments are important since they use only some very
simple properties of the sequence {gn\. At the same time they yield certain
asymptotic formulas for FejeVs polynomials Fn(x), introduced in §6.5, valid
in the cut plane and in— 1 < a; < +1, respectively, provided certain condi-
conditions regarding the sequence {an} are satisfied. We have the following:
Theorem 8.3. Let {am} be a positive sequence, am —> 0, and let am/am_i t 1.
Then the following asymptotic formula holds:
00
-2 m
(8.3.9) Fn(x) ^ anzn 2^ amz~2m, n
m—0
where x is in the cut plane, x = \{z -\- z x), \ z\ > 1. If-in addition am/a2v
remains bounded, we have
(8.3.10) Fn(cos0) = 2an9Ke 2. o^1 + o(an), n
m-0
where 0 < 6 < ir. Both formulas are valid uniformly in the same sense as (8.21.1)
and (8.21.2), respectively.
The generalization (8.3.9) of the Laplace-Heine formula is new; concerning
the generalization (8.3.10) of Laplace's formula, see Szego 11, pp. 186-187.
We observe that the series in (8.3.9) is convergent, and the series in (8.3.10)
is uniformly convergent in e ^ 6 ^ w — e. With regard to the latter fact we
note that am/am_i ^ 1, so that am is decreasing. As an application of Theorem
8.3, we obtain the analogue of (8.21.1) and (8.21.2) (with a less precise estimate
of the remainder in the second case) for the ultraspherical polynomials P(n\x)
provided X > 0.
8.4. Darboux's formula proved by Darboux's method
A) We shall prove formula (8.21.4), as well as others, by means of an impor-
important method due to Darboux A), and we begin with an illustration of this
method. Supposing 0 < 6 < ir, let us consider the generating function of

[ 8.4 ] DARBOUX'S FORMULA 207
Legendre polynomials (cf. D.7.23))
(8.4.1) h(w) = A - we~i6Y\l - weiB)~h
in the neighborhood of elB:
( 2:8 ^ -J
= a - "» o»{ J !
A similar representation holds in the vicinity of e~'°. Denoting the Lth partial
sums of these expansions (stopping at the terms v = L) by s^\w) and si2)(u/),
respectively, let us consider the difference
(8.4.3) H(w) = h{w) - 8?\w) - s[2\w).
We see immediately that the Lth derivative H{L\w) possesses continuous
boundary values in | w \ S 1. Thus if we expand H(w) in a power series about
w = 0, the coefficients of H{L'(w) tend to 0. This simple remark shows that
the coefficients dn of H(w) satisfy the condition
(8.4.4) lim nLdn = 0.
n —*oo
Each of the terms of the finite sums s(l\w) and s^iw) has only one singu-
singularity on the unit circle. The vth term of s(l\w) contributes to the coeffi-
coefficient of wn in h(w) (and therefore to Pn(cos 6)) an expression of the form
n
The sum s{l\w) contributes the conjugate of (8.4.5). Both terms are 0(n~ ).
For a fixed value of p the coefficient of wn in H{w) is of higher order than n~ ,
provided L is sufficiently large.
By use of the same argument, the following general theorem can be obtained:
Theorem 8.4. Let h(w) be regular for \w\ < 1, and let it have a finite number of
singularities
(8.4.6) e'*1, e'*\ ¦•¦ , e{*\ e{*" * e'*', a * /3,
on the unit circle \ w \ = 1. Let
00
(8.4.7) h{w) = 2-i c« A — we ) > * — 1) ^> ¦¦¦)'»
m i/ie vicinitxj of e"t'k, where bk > 0. Then the expression
(8.4.8)

208 ASYMPTOTIC PROPERTIES OF CLASSICAL POLYNOMIALS [ VIII ]
furnishes an asymptotic expansion for the coefficient of wn in h{w) in the following
sense: if Q is an arbitrary positive number, and if a sufficiently large number of
terms is taken in the sum X/"=o zn (8-4.8), we obtain an expression which approxi-
approximates the coefficient in question with an error equal to O(n~Q).
A simple discussion shows that it suffices to stop at the term v = p — 1 of
the sum, where p is a positive integer such that
(8.4.9) p ^ max bZx{Q - $l(ak) - 11.
Proper logarithmic singularities of h{w) can also be admitted.
B) In the case of Pn(cos 6), this asymptotic expansion becomes
(8.4.10) 2»{g ,.A - .")%?-,)'(' ; »)(-.-)•
The general term is 0(n " ); thus, if we stop at v = p — 1, the error is 0{n p *).
This agrees with formula (8.21.4). It is also clear that the bound for the
remainder holds uniformly in e r§ 6 ^ -k — e.
The same method can be used for the proof of the expansion (8.21.3), which
corresponds to the Laplace-Heine formula (8.21.1). (Actually, this case is
simpler than the preceding one.) Indeed, let | z \ > 1; then
h(w) = A - zw)~Kl - z~lw)~h
(8-4.11) .. _._1,-
whence (8,21.3) readily follows.
C) The infinite series which corresponds to Darboux's formula (8.21.4) is
convergent in the ordinary sense and represents Pn(cos 6) provided 2 sin 6 > 1,
that is, tt/6 < 6 < 5tt/6. In fact, the representation (8.4.2) holds uniformly
near w = 0 if
(8.4.12) |A - we-a)(eM - I)1 < 1.
See 8.92 D).
D) Darboux's method also applies to the general Jacobi, and in particular
to the ultraspherical, polynomials and leads to the Theorems 8.21.9 and 8.21.10.
Another method of deriving the expansions of Theorem 8.21.9 will be indi-
indicated in §8,71 D) and E).
E) Finally we observe that the expansion (8.21.14) stops at v = X — 1 if X
is a positive integer. Then we obtain the exact representation
V A - X)B -X) ¦¦¦ („ -X)
Pn (cos 6) = 2an
(8.4.13)
,=0 (n + \ — \){n + \ — 2) ¦ ¦ ¦ {n + \ — v)
cos {(n - v + \N - {v + X)tt/2)
B sin 0)"+x
= 0, 1,2, ¦••; X = 1,2,3, •••; an =
Th

[ 8.5 ] PROOF OF FORMULA OF STIELTJES 209
In fact, the difference (8.4.3) (with L = X — 1) is in this case a rational func-
function which has no singularities for any w (including w = «) and which van-
vanishes for w —> oo. Hence it must be identically zero.
The analogous representation of PlX) (x) for | z \ > 1, X a positive integer, is
slightly more complicated; it can be readily derived from (8.4.13).
8.5. Proof of the formula of Stieltjes
A) Stieltjes' formula D.9.17), D.9.18) has been derived in §4.9 C) from the
integral representation D.8.17). This argument furnishes for the remainder
Rp(d) of (8.21.5) the representation
(8.5.1) RpF) = ?3 je'<"-»>V(W4-fl/2)B sin 0)~* f f A - t)^Pp{t) dtl.
Here {gv has the same meaning as in Theorem 8.21.3)
p-l >(fl-7r/2)
(8.5.2) Pp{t) = A - z)-> - ? g,z'; z = A - I) e—— .
v-o ? sin 0
According to Stieltjes we have
—i / . %v . j.
gv = t I sin <p aq>,
(8.5.3) p_x \
A - z)~* - Ylgvz" = iTl I Z Sm. d<t>.
v*=o Jo 1 — z sm2 0
The last formula is obvious first under the assumption that \ z \ < 1; it then
can be extended to the whole strip 0 5j 9?(z) ^ \ without restriction. Now,
writing A — t) sin2 0 = r, we find
11 - z sin2 0 |2 = 11 - r/2 + (zr/2) cot 0 |2 = [sin 0 - r/B sin 0)]2 + cos2 0.
The minimum of this expression is cos2 0 or B sin 0)~2, according as 2 sin2 0^1
or 2 sin2 0 ^ 1, so that
(8.5.4) | pp(t) | ^ gp{\ - t)p{2 sin 6)~PM,
where M has the same meaning as in (8.21.6). Thus
2 if1 i
|flP@)| ^ - Bsin0)~* / T(l - 0A - OP^PBsin0)~pM^
T Jo
(8.5.5)
_ 2 r(n + l)r(p + 1) M
tt
which is equivalent to (8.21.6).
The analogous formula of Theorem 8.21.11 for the ultraspherical polynomials
Pix)(cos 0), 0 < X < 1, results from D.82.3) (cf. Szego 17, pp. 57-60). It is
the expansion D.9.25) completed with an estimate of the error if we stop at the

210 ASYMPTOTIC PROPERTIES OF CLASSICAL POLYNOMIALS [ VIII ]
term v = p — 1. The error is again less than twice the first neglected term in
which cos is replaced by 1. The proof is the same as before; we use for the
quantities ay of D.9.21) the representation (cf. F.5.9))
-i f' x-
(8.5.6) a, = ir sin Xtt / | tan 0 j2 l sin2' 0 d^.
Jo
The same remark as in §8.4 C) applies to the infinite series corresponding to
Stieltjes' formula and to its generalization. (Cf. D.9.17), D.9.25).)
8.61. Method of Liouville-Stekloff; formula of Laplace
We shall now derive the formula of Laplace from the differential equation
G.3.5). The essential idea is the transformation of this equation into an
integral equation of Volterra's type which permits a successive improvement of
the asymptotic formula in question. The idea is very old, and appears in the
investigations of Liouville on differential equations of the Sturm-Liouville type.
Stekloff A) applied this method to the asymptotic discussion of certain classical
polynomials.
Recently, Langer A, 2, 3) employed this method systematically and improved
its efficiency considerably. He generally considers "singular" cases like D.24.2),
or any of the equations E.1.2), in the neighborhood of 6 = 0 and x = 0, respec-
respectively, and obtains general asymptotic formulas of "Hilb's type." He takes up
also applications of this method in the complex domain.
A) We write the differential equation G.3.5) in the form
^ j {(sin0)*P»(cos0)} + (n + IJ (sin 6)h Pn(cos 6)
(8.61.1) '
= _(sin fl)*P»(cos 0)
~ 4 sin2 d '
Interpreting this relation as a non-homogeneous equation for (sin 0)\Pn(cos 0),
we can apply A.8.12); the corresponding homogeneous equation has the funda-
fundamental system {cos (n + \N, sin (n + §N], so that with certain constants
(sin 0)*Pn(cos 0) = a cos (n + \N + d sin (n + \)d
(8-61-2) 1 [' sin {(n + h)(fi - t)} , . ^ D f A Jt
— r- / ————-r— — (sm t) Pn(cos t) at.
n + k Je0 4 sm21
If we assume 60 = tt/2, 0 < 0 < tt, the last integral and its derivative vanish
for 0 = 7r/2. This remark enables us to determine Ci and c2 . We find
(sin 0)iPn(cos 0) = Xn cos \{n + fH - tt/4J
(8-61-3) l H sin {(n + \)($ - t)} , . a
n + | Jw/2 4 sin21

[8.61] FORMULA OF LAPLACE 211
where
(8.61.4) \n =
/2 if n is even,
+ 1
0
if n is odd.
n +
This is the Volterra equation mentioned above.
If 6 is confined to the interval [e, tt — e] and Mn denotes the maximum of
the absolute value of the left-hand member of (8.61.3), we have
JIT
(8.61.5) Mn ^ Xn + s-^rf r%- ¦
2n + 1 4 sin2 e
Therefore, if n is sufficiently large,
Mn < 2Xn = 0(n-i),
or
(8.61.6) (sin d)kPn (cos 6) = Xn cos {(n + JH - tt/4) + O(ri~h),
which readily furnishes Laplace's formula.
B) Successive application of (8.61.3) leads to an expansion of Darboux
type, namely, to the formula
fp-i
(sin0)*Pn(cos d) = cos (n + \H
f
where v4,,@) and jBv@) are certain functions analytic in 0 < 6 < w and inde-
independent of n and p. The explicit determination of these functions, that is, the
identification of (8.61.7) with the formulas of Darboux or Stieltjes, seems,
however, to be rather difficult.
For the proof we use mathematical induction. Assuming (8.61.7), we obtain
from (8.61.3),
(sin 0)*Pn(cos 0)
= Xn cos {(n + \)B - tt/4) - -Lj f —
W + 2 Jt/2
•<cos (n + ^)i E vlX^n"
+ sin (n + \)t Z~> B,(t)n " >dt +
4 sin2
Now2sin {(n + ^)(e-0) cos (n + h)t = sin (n + \)9 + sin {(n + ?)(# - 20),
and integration by parts yields

212 ASYMPTOTIC PROPERTIES OF CLASSICAL POLYNOMIALS [ VIIIJ
f
Jr/2
4 sin21
= cos (n + h) dl 2J a,(^)n-"| + sin (n + J) ^ ? &,(e)n""j + O(rCp),
where a^O) and b^ifi) are certain functions of the same type as Av{6) and BfF).
The integrals involving B,{t) can be dealt with in the same way. This leads
by use of Stirling's series (cf. P61ya-Szego 1, vol. 1, pp. 29, 193, problem 155),
to a representation of the form
(sin 0)}Pn(cos 6) = cos (n + §) 0
+ sin (n + h)e(pQ Bi1\e)rr'-i\ + OCrr*"*).
Comparing this with (8.61.7), we see that
A?\e) = A,(e), B?\e) = B,(e), v = o, 1,2, • •., v - 1.
This establishes the statement.
We have
(8.61.8) AoF) = B/tt)* cos (tt/4) = tt, BoF) = B/tt)* sin (tt/4) = iT\
The same method can be used for the asymptotic evaluation of Qn(cos 6),
(see Problem 18), as well as of the ultraspherical polynomials. In the first
case we obtain Theorem 8.21.14. The application of the method to general
Jacobi polynomials is more difficult because no explicit values for these poly-
polynomials are known at the point 8 = x/2 (or at any other fixed point in 0 <
e < x).48
8.62. Method of Liouville-Stekloff; formula of Hilb
A) We again deduce an integral equation for Pn(cos 8), different from
(8.61.3) and involving Bessel functions. Writing G.3.5) in the form
(J-J {(sin 0)*Pn(cos 6) }+il + (n + §)j (sin fl)*Pfl(cos 6)
(8.62.1) ^ ^ ^
(sin0)*PB(cos0),
1 1
402 4 sin2 6
we apply A.8.12). The corresponding homogeneous equation is A.8.9) (a = 0,
k = n -f- |) with the solutions
(8.62.2) 6kJQ{(n+\N} and 0* Y0{(n +
48 Cf., however, Korous 3.

[ 8.62 ] FORMULA OF HILB 213
Consequently, with certain constants do, ci, c2, we have
(sin 0)*Pn(cos 0) = Ci0*Jo{(n + hN] + c2dhYo[(ri + §H}
0* [' }/o{(n + hN}Yo{(n + $)«} - F0{(n + §H}J0{(n + §
[
(8.62.3) +n + §V >o{(n + })«JTo{(n +J)«} - Fo'{(n +
(sin ^)}^»(c°s t)dt.
\2 4 sin
According to A.8.14)
(8.62.4) J'0{{n+\)t}Y0{{n+h)t} - Fo{(n
Furthermore, t~2 — (sin t)~2 is analytic at ^ = 0; that is, we can take 60 = 0,
and we then obtain
(sin 0)*PB(cos 0) = ciflV0{(n + ?Hj + csfl*F0{(n + |Hj
°H - F0{(n +
\it2 4 sin2
If this equation is divided through by d\ and 8 approaches 0, the last term
tends to 0, whereas the left-hand member tends to 1. Hence (cf. A.71.10),
A.71.1)) c2 = 0, ci = 1, and we find for 0 < 0 < t
¦0) = Jo{(n + hN} + ~ [J0{(n + ^H}Fo{(n
\ G /
(8.62.5)
- Y0{(n
This is the integral equation required.
B) First assume 0 < nd ^ 1. Then according to A.71.1) and A.71.4)
Jo{(n + ^H}Fo{(n + m - Y0{(n + \N)JQ{(ri + \)t)
= Jo{(n + \N) I"? log {(n + h)t}Jo{(n + \)t) + 0AI
(8.62.6) To 1
- Joi(n + J)«} yilog {(n + ^H}Jo{(n + \N) + 0A)J
J{( +
g^ + 0A) = 0A) log^ + 0A).
T It
Therefore, the integral of the right-hand member of (8.62.5) is (by G.21.1))
0A) [' t log j dt + 0A) f'tdt = O(lH2 f\ log \dt + O@2) = O@2).
Jo t Jo Jo t

214 ASYMPTOTIC PROPERTIES' OF CLASSICAL POLYNOMIALS [ VIII ]
C) On the other hand, let nd ^ 1, 0 ^ x - e. Then, according to
A.71.10) and A.71.11), the contribution of the interval 0 < t ^ n~l to the inte-
integral previously considered, is
t\YQ(nt)\dt + 0{(n6)~h\ I t\J0(nt) dt
Jo
Finally, we find for the contribution of the interval n~l g t g 6, according
to G.3.8),
>dt = 0@* tT!).
8.63. Method of Liouville-Stekloff; extension of Hilb's formula to
Jacobi polynomials
By use of complex integration, Szego A7) has extended Hilb's formula and
the corresponding asymptotic expansion mentioned in §8.23 A), to ultraspher-
ical, and even to general Jacobi, polynomials. Following Rau B), we deduce
the principal term of this general expansion by means of the Liouville-
Stekloff method and obtain formula (8.21.17). The bounds for the remain-
remainder are better than those in Szego 17, p. 77, D7), and in Rau 2, pp. 691-692,
B9), C0).
A) Let a > -1. We write D.24.2) in the form
d2u , [}-
+ JV>= ^ i. + G-aV-i i-\ u,
d02 I 02 ' 1. O0 ' ^ ' 02 .9
^ J |4cos2(r \ 4sn2?
(8.63.1) I 2 \ 2/
Again applying A.8.12), we obtain, because of A.8.9),
(a\«+i / o\P+i
sin-1 (coso) ^n"lft(cos0) = a&JJiNO) +c20}/-«(^0)
0} fe _t Ja(Ne)J^(Nt) - J-a(N6)Ja(Nt)
+ Njeot J'a(Nt)JUNt) - f-a(Nt)Ja(Nt)
(8.63.2)
(na^ (cos t)dt.
Here and in what follows J-a(z) must be replaced by Ya(z) if a is an integer.
We refer again to the identity A.8.14), by which

[8.63] EXTENSION OF HILB'S FORMULA 215
(8.63.3) Ja(Nt)J-a(Nt) - j'-a(Nt)Ja(Nt) = 2 *"* "*" .
irJSt
(If a is an integer, sin air is to be replaced by — 1.) Therefore,
sin IJ (cos |J Pia'»(cos 0) = c.JMe) +
cos ~
{Ja(Ne)J-a(Nt) ~ J-a(N0)Ja(Nt) }«*/(«) (sin M (c
¦ Pia^ (cos t)dt,
where/(?) is regular in 0 ^ t < w, and independent of n.
The last integral is convergent for 0O = 0; as 6 —> +0 it becomes (to fixed)
0A) f @-r" + e-T)tHa+i dt- ~ o(ea) f tdt + o(e~a) f ^2o+1 d^ = o(ea+2).
Jo Jo Jo
This is true whether a is an integer or not [A.71.10)], except for a = 0. Then
we obtain
0A) j* (log I + log I) H dt = O (V log fj.
Dividing (8.63.4) by 6% 6 -+ +0, we find [A.71.1)] a relation of the form
+ O(d2).
(The last term must be modified for a = 0.) Hence if a j^ 0,
c2 = 0 and cx = 2~hN~a T(n + a + l)(n!)"x.
The same result holds when — 1 < a < 0 if we take into consideration the
fact that the "principal term" of d~aJ_a(N6) is 6~2a. Thus, for 0 < 6 < *•,
( 0( 0 6) =
(8.63.5) + ?
in 0"^ (cos
B) Now let w —> oo. We find bounds for the last integral in a manner
similar to that in §8.62. First, let 0 < nd ^ 1. Then for a ^ 0, the integral
is (cf. the second bound in G.32.5))
0A) f \{nd)tt{nt)~a + (ne)~a(nt)a}tHa+inadt = O(na6a+2).
Jo
In case a = 0, we reason as in §8.62 B) and obtain the same bound, that is,

216 ASYMPTOTIC PROPERTIES OF CLASSICAL POLYNOMIALS [ VIII ]
O@2). On the other hand, let n~l ^ 0 ^ w - e. The contribution of the
interval 0 < t ^ n~l is (cf. the second bound in G.32.5))
/•n-i
0A) / (n0)-*{ | J-a(Nt) | + | Ja(Nt) | (A^n" dt
i2 }S tf tf
(Here /_« must be replaced by Yo if a = 0.) For ? ^ to, we use the first
bound in G.32.5) and obtain
0A)
f (n
8.64. Method of Liouville-Stekloff; asymptotic formula of Hilb's
type for Laguerre polynomials
By use of the fourth equation in E.1.2) this method readily leads to (8.22.4).
The third equation could likewise be used, but the calculation would then be
slightly more complicated. Concerning an extension of (8.22.4) to an asymp-
asymptotic expansion (at least for a > — -|) see §8.66.
A) Let a >> —1. Writing the equation in question as
(8.64.D ^
N = n+(a+
we can apply A.8.12) and A.8.9). Hence with certain constants x0, cx, c2,
- J-aBNix)JaBNit) -(*/2.«+2w«w.2
faBNh)J-.aBNh) - j'-aBNH)JaBNh)e " ^
Once again we use (8.63.3) and obtain
—I—
2 sin ax
If
Jxa
dt.
— UJLJ.X 14./I ./ XQ
a is an integer, J-a(z) must be replaced by Ya(z) and sin ax by —1.
Let x0 = 0. For a fixed to, as a; —* + 0, the last term is
0A) / (x t + x t )t dt = 0{x ).
Jo
(If a = 0, this bound must be multiplied by log (I/a;).) Therefore, as in §8.63,
c2 = 0 and L(na)@) = CliVa/2{r(a + I)}; whence
(8.64.2) a = N"
a'2 T(n + a

[ 8.64 1 FORMULA FOR LAGUERRE POLYNOMIALS 217
Consequently,
n\
2 sin .cw Jo
{H -t2l2ta+3Lia)(t2) dt.
B) As to —* oo, the remainder term can be estimated in the same way as
in the previous cases. However, we avoid here the use of bounds of the type
G.32.5). (In §7.6 C) we derived such bounds as a consequence of (8.1.8)
and of Fejer's formula (8.22.1); this is almost (8.22.4), which is just what we
must prove now.) In the following proof we apply only the elementary
formula (8.1.8) of the Mehler-Heine type, in particular only the second bound
in G.6.8).
First, let 0 < x ^ n~K Then we have Ln"\t2) = 0(na) for 0 ^ t ^ x.
It then follows that the integral term in (8.64.3) is
(8.64.4) 0A) P (nal2xan-al2ra + 7ra/2aTV/2O«a4V dt = 0(z"+V);
Jo
for a = 0 this bound must be multiplied log (x~lri~*).
Now let n~* ^ x ^ w}, where « is a fixed positive number. Let Mn be the
maximum of e~x l2x" \ L<na) (x2) in this interval. Then the contribution of the
part 0 ^ t ^ n of the integral term is, a 9^ 0,
(8.64.5) 0A) (rTix-in-a"ra + n^x-^'hy^n" dt = O(afV/2-9/4).
Jo
The same result holds if we have a = 0. The contribution of the other part
n~} ^ t ^ x is
(8.64.6) 0A) / rTkx~hrrH~h3Mndt = 0(l)n~lxzMn = Mn-o(l).
Jo
Taking account of (8.64.3), and using the same argument as in §8.61, we find
that
(8.64.7) Mn = 0(n~"/2H(n"H(n"}a;~}) = 0(x~ina'"~i).
(This is, of course, identical with the first bound in G.6.8).) Therefore, in
view of (8.64.5), (8.64.6), and (8.64.7), we obtain for the remainder term, if
n~h ^ x ^ «*,

218 ASYMPTOTIC PROPERTIES OF CLASSICAL POLYNOMIALS [ VIII ]
8.65. Method of Liouville-Stekloff; Hermite polynomials
A) The integral equation (8.64.3) assumes a particularly simple form in case
a = =b|, that is, for the Hermite polynomials. Using E.6.1) and A.71.2),
and writing n = 2m, a = ~ ?, and to = 2m + 1, a = + §, respectively, we obtain
e~x*l2Hn(x) = Xncos (N*x - tott/2)
(8.65.1)
+ N~* / sin {N\x - f)}te-tS'2Hn(t) dt,
Jo
where
(8.65.2) Xn = | HM |, or | #n@) I N~\
according as to is even or odd, and N = 2n + 1. However, it is more con-
convenient to deduce this directly from the second equation in E.5.2).
We prove (8.22.7) by mathematical induction. The statement is true for
p = 0, replacing both sums ^2'-o by 0. In fact, if Mn denotes the maximum
of e~*2/21 Hn(x) | in a fixed real interval, we find from (8.65.1) that
(8.65.3) Mn ? Xn + 0(rCh)-Mn .
Then Mn = X»OA).
B) Now assuming (8.22.7) for an arbitrary p, we obtain from (8.65.1)
e~xV2Hn(x) = Xn cos (N*x - rwr/2)
sin ^x ~ $ \ (cos (^« - W7r/2) S f
+ iV~} sin (N*t - rnr/2) 2 ? vf(t)N-*\ dt + XnO(n~p-}).
,-o J
The second term of the right-hand member contains expressions of the fol-
following type:
XnA?~'~} /"'sin [N\x - t)} cos (N*t - mr/2)tk dt
Jo
= Bk + 2)-1XniV""-}a;i+1 sin (iV}a: - nw/2)
+ |XniV"""} f t" sin {iV}(a; - 2«) + nir/2) dt,
(8.65.5)
XniV"^1 / sin \Nh(x - t)} sin (rft - mr/2)tl dt
Jo
= -B1 + 2y1XnN~"~1xl+1 cos (N*x - n7r/2)
Jo
«' cos {iV}(a; - 2«) + nw/2} dt,
where k is even and Z is odd (k ^2,1 ^3). Integration by parts furnishes

[ 8.66 ] LAGUERRE POLYNOMIALS 219
/ tk sin {N*(x - 2t) + tit/2} dt = %N~ixk cos (N*x - tit/2)
Jo
- JfciNT* f t1 cos {iV}(a; - 20 + tit/2} dt,
(8.65.6)
/ tl cos {iV}(a; - 20 + tit/2} dt = %N~hxl sin (Nhx - tit/2)
Jo
+ JZAT* / «' sin {JV*(a; - 20 + tit/2} dt;
Jo
the second formula holds also for 1 = 1, and
f* TO
(8.65.7) / sin [Nh(x - 20 + tit/2} dt = < '
jo l^A' cos (A'1 a; — tit/2),
according as h is even or odd.
This consideration leads to a formula of the type
-x*l2Hn(x) = Xnjcos (N*x - tit/2) ? u?(.x)N-'
+ JV~* sin (iV*a; - n7r/2) 2 ©^(^iNT* + O(n~p~})|.
By repeated application of the same argument
e-xV2Hn(x) = Xn (cos Q^x - tit/2) J2 ul2\x)N
{
sin (N*x - tit/2) ? »i2)(a;)iV-' + O(jTir-1)\.
v-0 J
We readily see that u[l)(x), ul2)(x); v^ix), vB)(x) are polynomials of the same
type as uf(x); vf(x), respectively, and uf(x) = ull)(x) = ul2\x), vv(x) = vll)(x) =
v?\x), v g, p - 1; whence (8.22.7) follows.
The proof of Theorem 8.22.7 can be given along these same lines.
8.66. Application to Laguerre polynomials
A) The asymptotic expansion (8.22.7), combined with the formula E.6.5)
of Uspensky, readily furnishes an asymptotic expansion of Hilb's type for the
Laguerre polynomials L(na)(x), at least for a > — ?. We shall give only an
outline of the proof.
Substituting (8.22.7) in E.6.5), we obtain an asymptotic expansion of which
the general term is, apart from trivial constant factors depending on n,
f A - t2y-hext2'2 {cos [Dn + l)hxH]uf{xh)
(8.66.1) J-i
+ Dn + 1)"} sin [Dn + 1)* a;*«] 0,C*0} dt.

220 ASYMPTOTIC PROPERTIES OF CLASSICAL POLYNOMIALS [ VIII ]
Here the uf(x) and the vv(x) are even and odd polynomials, respectively, and
both are independent of n. The expression (8.66.1) is a linear combination of
terms of the form
cos {Dn + lyxH}dt,
(8.66.2)
D?i + 1)"* / A - *2r~V2/V*)' sin {Dn + l)ixit}dt,
r
where k and Z are non-negative integers, k even and I odd. If we expand
e/2Tk'2 in a power series about t = x, we obtain for the first integral terms of
the type
A - f)a+«-} cos {Dn + l)*a;}*} eft,
with q integral, and q =g 0. This can be expressed in terms of Bessel functions
[A.71.6)]. A similar method furnishes for the second integral (8.66.2) terms
of the type
/ A — t) q t sin {(An -f- 1) x t\ dt,
q again being integral, and q =? 0. These can also be expressed in terms of
Bessel functions (combine the second formula A.71.5) with A.71.6)).
If we stop the expansion of the first expression (8.66.2) at a certain term, the
remainder appears in the form
(8.66.3) I ' A - t2)a-hj[x{\ - t2)] cos {Dn + ifxH] dt,
where f(t) = cmTm + cOT+iTm+1 + • • • is an integral function with a zero of order
m at t = 0; here m is an arbitrary integer. The remainder is of similar form
for the second integral in (8.66.2). Writing
g(t) = A - O"~*/[*U - t2)],
we see that the functions g(t), g'{t), g"{t), ¦ • ¦ , 0<m~1)(?) vanish at t = ± 1;
they are all xm0(l), where 0A) is uniformly bounded in —1 ^ t ^ +1, and in
a fixed finite interval a tg a; ^ & containing the origin or not. Integrating by
parts, we find for (8.66.3) a bound of the form O(ri~K), where K is arbitrarily
large with m, uniformly in x, a ^ x % b.
Similar remarks hold for the second remainder.
B) The first term of this expansion furnishes (8.22.4). We also obtain
readily an extension of (8.22.4) to the complex domain.
Now assume 0 < e ^ x ^ «. Then applying A.71.8), we find Perron's ex-
expansion (8.22.2). (Cf. Uspensky 1, pp. 608-610.) There is no difficulty in
deriving the complex formula (8.22.3) of Perron in the same way.

[ 8.71 ] LEGENDRE POLYNOMIALS AND RELATED FUNCTIONS 221
In all these considerations we assumed a > -•§. An extension of Perron's
formulas to arbitrary real values of a is possible by means of the second formula
in E.1.13). Concerning a second proof of Perron's formulas (by use of the
method of steepest descent), see §8.72.
8.71. The method of steepest descent; Legendre polynomials and
related functions
A) This method can be used for approximating integrals of the form
(8.71.1) j \F(t)}ng(t)dt = I enmg(t)dt,
extended over a certain arc or closed curve, where F(t) = e/u) and g(t) are
given analytic functions regular in a certain part of the complex 2-plane, and
n —» oo. According to Cauchy's theorem, a deformation of the contour is
possible. In many cases it is convenient to make it pass through some of the
points U> at which f'(t0) = 0 (saddle point); in addition, the direction of the
contour at to (critical direction) must be determined according to the condition
that (assuming f"(t0) 9^ 0) the expression
(8.71.2) nf"(to)(t - toJ/2
is real and negative if t is sufficiently near to. Then
[F(t)}n = {F(to)}nexp {nf"(to)(t - t0f/2 + nf'"(to)(t - to)*/6 + • • • }.
Hence under proper conditions concerning the behavior of f(t) on the comple-
complementary part of the path of integration, the neighborhood
(8.71.3) t - to = O(na~*), 0 < 5 < i
of the saddle point furnishes the "principal" part of the integral as n —* <».
Tts contribution is of the form
(8.71.4) enfMg{to)n-h f+" exp {-aP2)dp ^ enfUa)g(.to)(—) ; a > 0, n-» «.
(Additional difficulties arise if f"(to) = 0; concerning a case of this type, cf.
§8.75.)
The term "method of steepest descent" arises from the following consider-
considerations. Let t = u + iv. Then representing u, v, 9?[/(?)] as cartesian coordi-
coordinates in the ordinary euclidean space, we obtain a surface with a saddle point
at t = t0, and the curve with the critical direction is the "steepest" curve on
the surface through this point.
There is, of course, considerable freedom in the choice of the contour; only
its direction through the saddle point is restricted. However, the exact calcu-
calculation of the saddle points to from f'(t0) = 0, and particularly that of the corre-
corresponding critical directions, might well be a complicated task in certain cases.

222 ASYMPTOTIC PROPERTIES OF CLASSICAL POLYNOMIALS [ VIII ]
For this reason, the following observation may greatly simplify matters. In-
Instead of the critical direction itself, any other direction through the saddle point
can be taken for the path of integration, provided it forms an angle of less than
7r/4 with the critical direction. Then the constant a in (8.71.4) becomes com-
complex with a positive real part. Geometrically, this condition means that in
passing through the saddle point on the surface, no higher level is reached in
the neighborhood of this point than at the point itself.
For our purposes, we shall prefer a contour satisfying the last condition, and
along which 1R[f(t)] varies monotonically. Then the discussion of the integrand
exterior to the neighborhood (8.71.3) becomes comparatively simple.
Concerning the history, further details, and important applications of this
method, the reader is referred to Watson 3, pp. 235-236.
B) As a first illustration let us consider the Legendre function of the second
kind, that is, the special case a =. j8 = 0 of D.61.1). Let x = cos 0 — iO,
0 < 0 < x. Then
dt
(8.71.5) QiM) (cos0 - iO) = Qn (cos0 - iO) = f J Q j^r~d/ cos 0 - f
The original path of integration, — 1 ^ t ^ +1, can be deformed into the upper
half of the unit circle described in the negative sense. The saddle point con-
condition is
/« »¦. *n d /l t2 - 1 \ 1 t2 - 2t cos 0 + 1 A , . ±,9
(8.71.6) -r. ( ^ . ) = x —T. ~- = 0, whence t = <T ,
dt \2 t — cos 0/ 2 (t — cos 0J
and we see that the path of integration passes through the saddle point t = e*.
If t = etS we have
rs 7i 7) fYf M = 1_
^" ; \dtj \2 t - cos 0/ i sin 0'
so that near this point
(8.71.8) \ ±JZ±. = e<° + %=A± + • • • •
2 ^ — cos 0 2t sin 0
Along the critical direction, e~t{8+rl2\t — el6J must be real and negative; that is,
(8.71.9) arg (t - eie) = 0/2 + 3x/4 or 0/2 - tt/4.
Now, the angle between this line and the tangent to the unit circle at the point
(8.71.10) arg {«,••<•'«+»"«: «,'<•+"»} = x/4 - 0/2,
and | x/4 - 0/2 | ^ tt/4 - e/2 if e ^ 0 ^ v - e. We can therefore use the
circle | t \ = 1 as the path of integration.49
49 The critical direction is given by that of the bisector of the acute angle between the
tangent at the point eie and the horizontal direction.

[ 8.71 ] LEGENDRE POLYNOMIALS AND RELATED FUNCTIONS 223
Substituting t = el4>, 0 S <t> ^ *", we find that
12—1 _ sin <j>
2 t - cos 0 ~ f(cos <f> - cos 0J + sin2<?}*
(8.71.11)
j/ - cos 0Y p
0Y
/ J
is an increasing function of <f> for 0 ^ ^ ^ 0, and a decreasing function for
0 fS <? ^ 7T. We next consider the contribution of the arc
(8.71.12) <j> = 0 + n^p, -na ^ p ^ + n* ,
where 5 is a properly chosen positive number. On this arc we have
e " - e = e (exp [m 'p] - 1) = e < in 'p + —^ + • • • f,
and, in view of (8.71.8),
5 f^
provided 8 < \. Here cx, c2, • • • are certain functions of 0 independent of n, for
which cm — 0(Am) holds uniformly in e ^ 0 ^ x — e, and in m; 4 = -A(e).
Now
\2 t-cosd
where
2\ w
I sin 0
Let 8 < |; then if ilf is an arbitrarily large positive integer, W and ew can be
reduced to a finite sum plus a remainder which is O(ri~M). This yields the re-
relation
- * ~1 ) =eln8exp<^ e—o2\
2 t-cosdj { 2isin0 J
•{1 + Wi(p,0)n -\-uz(p,d)n + uz(p,d)n + •••}.
The series in the braces is an asymptotic expansion. If only m terms of this
are taken, the error is less than an arbitrarily large power of n provided m is
sufficiently large; here w,(p, 0) is a polynomial in p and a function of 0, analytic in
e ^ 0 ^ x — e. In addition uv( — p, &) = ( — Vf uv{p, 0).

224 ASYMPTOTIC PROPERTIES OF CLASSICAL POLYNOMIALS [ VIII
C) Now multiply (8.71.13) by
1 dt = 1 ie^d<f> = 1 uF^'n+dp
2 cos0 — t 2 cos 0 — e'-* 2 cos0 — eitein~^
where {cmj is a sequence similar to {cm}; this does not change the essential
character of equation (8.71.13). We obtain, as the contribution of the arc
(8.71.12),
(8.71.14)
( * l ! + •¦¦jdp,
where {^(p, 5)j is a sequence of polynomials similar to [uv{p, d)\. The last
series is again an asymptotic expansion of the same type as (8.71.13).
At the end-points of the arc in question the modulus of the integrand of
(8.71.14) is O(e~cpi), that is, O(e~cni ), c > 0; the same is true of the contribution
of the complementary arc becaase of the monotonic character of the function
(8.71.11). Thus,
Qn(cos 6 - tO)
t(n+lH
2 sin 6
n
+ O(e~cn2S).
The terms corresponding to the odd powers of n * vanish after integration.
Upon completing the interval of integration, we obtain
(8.71,15)
\ 2 0 J
2 sin (9 \ 2 sin 0 J
il(n + 1M + tt/4]}
sin
the bound for the remainder holding uniformly for e ^ 6 ^ x — e.
This method leads to a complete asymptotic expansion of Qn(cos 0 — iO) of
the type (8.61.7) for t ^ 6 ^ x — e, but seems to be too difficult for use in
finding the general law of the coefficients.
From (8.71.15) we obtain the corresponding asymptotic expansion for
Qn(cos 6 + iO) by merely replacing % by —i. From this, and D.62.8), we
easily derive Laplace's formula. Formula (8.61.7) can also be derived in this

[ 8.72 ] PERRON'S FORMULAS FOR LAGUERRE POLYNOMIALS 225
way. At the same time we obtain a similar asymptotic expansion for QJ,°'o) (cos 6)
= Qn(cos 0) with the principal term (8.21.19).
D) The analogous considerations for the Jacobi polynomials Pna'P)(cos 6)
are immediate. Here a and 0 are arbitrary and real, n —> °o, and 6 again
satisfies the condition t^Hi-e. According to D.4.6) we have
(8.71.16)
/ i-t Y( l
\1 - cos (9/ \1
+ cos (9/ cos 0 - </'
where the contour is the same as that used in B).60 The additional factor
If '
7TZ \1 -
(8.71.17) Vf
Z \1 COS 0/ \1 + COS
does not cause any new difficulty and furnishes for t = elS
(8.71.18) ^-(sin|) (cos|
Therefore, we find (8.21.10) and an expansion of the type (8.21.12).
E) The same method can readily be applied to the functions Qn(x), Pn(x),
or more generally to Ql"l/3) (x) and Pi"l/3) (x), where x is arbitrary real or complex
but not on the segment [ — 1, +1]. This leads to (8.21.9) and also to the
expansion (8.21.11). In the case of Qia'^(x), we start from D.61.1). It.is
convenient to replace the half-circle used above by the circular arc through
±1 and 2 = x — (xl — l)\ \ z\ < 1, (x is in the cut plane), which was intro-
ducedin§4.81 A). The resulting integral is the same as D.82.4). Applyingthe
method of steepest descent, we obtain a formula of the type
(8.71.19) (x - l)a(x + lYQia'"\x) ^ n*{x - (x - l)h\n+1<j>(x),
where j x — (x1 — 1)' | < 1, and 4>(x) is independent of n, and regular and non-
nonzero in the cut plane.
8.72. Method of steepest descent; Perron's formulas for Laguerre polynomials
As a further application of this method we again prove the expansions (8.22.2)
and (8.22.3). The present proof is based on the integral representation E.4.1),
which is valid for an arbitrary real a, provided n is sufficiently large.
A) Let x be arbitrary but nonzero. We start from the asymptotic expan-
expansion A.71.8) of the Bessel function Ja(z), z complex. The contribution of the
segment 0 ^ t ^ 1 in E.4.1) is (w!)~'O(l). We can therefore confine our at-
attention to values of t ^ 1, so that A.71.8) may be applied. Substitution of
this expansion into E.4.1) leads to integrals of the type
50 We must avoid the points t = ±1 by moans of small semi-circles.

226 ASYMPTOTIC PROPERTIES OF CLASSICAL POLYNOMIALS [ VIII ]
1 f e_(r+a/2(to)_m/2_1/4 cog {2(te)i _ ^/2 _ ^/4j ^ ^ ^^^
(8.72.D y« ^
n\ Jo X Sm X
m = 0, 1, 2,
(Here the range of integration has again been completed to 0 ^ t < °o.) The
remainder term will be of the form
(8.72.2) i I e-(f+*>2-< exp [2th | 3?(-x)J |) (ft,
with g a fixed positive number which can be chosen arbitrarily large; the deter-
determination of (—x)J is the same as in Theorem 8.22.3. If x is a fixed positive
number, this is obviously O(nal2~"). If x is complex, the discussion of this
remainder term requires greater care. Subsequent considerations furnish a
bound in this case also.
B) For the sake of convenience we shall first discuss the integral
T f
(8.72.3) -. f <T'f exp [«*{]# = ?^lT f (e1"'*)" exp [nU^]dt,
m Jo n\ Ju
(? 7± 0, arbitrary complex) to which the integrals (8.72.1) and (8.72.2) can be
reduced; here n —> + =o, but n is not necessarily an integer. In the last integral
the saddle point is "essentially" t = 1, and the positive real axis corresponds
to the critical direction.51 If we write, as in (8.71.12),
(8.72.4) t = 1 + n~lp, -n* ? p ? +ns,
we obtain, for 0 < 5 < 1/6,
(e'-'O" = exp {n(l - t) + n log [1 + (< - 1)]!
(8.72.5) = <T'2/2 exp (n ? (-
where the w/p) are polynomials in p independent of n. This is an asymptotic
expansion similar in character to that in (8.71.13). Furthermore,
exp [r?h] = exp {n}? + n*$([l + « - 1)]J - 1)!
(8.72.6)
= exp j
where the cr are certain numerical constants. This furnishes exp (n ? + p?/2)
61 This case is not a direct application of the method indicated in §8.71 A) since the
integrand has the form {F(t)}n{G(t)}nl/\

[ 8.73 ] LAGUERRE POLYNOMIALS 227
multiplied by an expression similar to that in the braces in (8.72.5). The
coefficients corresponding to the uv(p) are in this case polynomials in p and ?.
We also see that the contribution of the range of integration complementary to
(8.72.4) isO(exp [—cn2a]), c > 0. Therefore, we have in the same sense as in
(8.71.14)
[+nl
= n * exp [n*|] / exp [ -P2/2 + p?/2] {1 + Vi(p, ^)n~J + v2(p, On'1 + • • • Up
J-n*
/+oo
exp [—p2/2 + p?/2] {1 + vi(p, i^n"* + t>2(p, ^n + • • •} dp,
00
where the v,{pK ?) are polynomials in p and ?. Now if q is a non-negative integer,
we have, in general,
7—
"exp [-P2/2 + p?/2] p«dp =
and the last integral is a x9 in ?. (For g = 0 we get Bx)Je{2/8.) According
to Stirling's formula an expansion of the type
exp
results for (8.72.3), in which the tv(?) are polynomials.
By applying this result to the expressions in (8.72.1), (replacing n by
n + a/2 — m/2 — 1/4), we obtain the required expansions. The special case
? > 0 yields the required bound for the remainder term (8.72.2).
8.73. Method of steepest descent; Laguerre polynomials for
1 ^ x ^ D — ri)n
Here and in the next two sections we derive formulas (8.22.9), (8.22.10), and
(8.22.11) by the method of steepest descent. We notice that in the first case
the condition x = Dn + 2a + 2) cos2 <j>, e ^ <f> ^ x/2 — en~l, means that x
satisfies the inequality xo ^ x ^ D — rj)n, where e, Xo, and rj are fixed positive
numbers, e < x/2, r? < 4, n large. The parameter a is arbitrary and real.
A) We start with formula E.1.16) (cf. the remark at the end of §5.2), in
which we replace x by ?2 and w by — w2/4; thus
oc
(8.73.1) I
^=o r(n + a + 1)
Consequently,
(8.73.2) (-*)" r/i(^.T, = 2^C / e-'^w-^—'e^J^e-^tw) dw.
T(n + a + 1) 2xt J

228
ASYMPTOTIC PROPERTIES OF CLASSICAL POLYNOMIALS [ VIII ]
The integration is extended over a contour enclosing the origin. We choose
it as a circle with center at the origin and radius to be determined later. The
function (8.73.1) is real if w is real.
Since ? = x',
(8.73.3)
= h cos <j>, ln = Dn + 2a + 2)*,
Then ? is bounded from zero. According to A.71.9),' as
uniformly in 0 ^ arg w ^ w,
> «), we have
w
eja(e
(8.73.4)
+
Hence from (8.73.2), for
(8.73.5) (-i)n -
where
exp [-?w + (a
ln cos ^>, tt' = lnz,
w
JL.
2tti
= /
exp { - | ?z2 + Z^z cos <j> - \ l\ log z} dz,
(8.73.6) ff = e
la+i)"
exp { - J Z2n22 - l\z cos 0 - ^ Z2n log 2)
i^ = OCr'O I I exp { - i Z2 22 ± Z2n2 cos 0 - IZ2 log 2} I I d2 |.
Here the integration is extended over the upper half-circle | 2 | = 1, and log 2
is zero if 2 is 1. In the last integral we choose the plus or minus of the ambigu-
ambiguous sign according to which gives the integral the larger value. If now we set
(8.73.7)
/(z) = - *22 + 2 cos 4> - \ log 2,
we find the saddle points of the first integral from the equation f'(z) = —z/2
+ cos<j>- {2z)~l = 0, that is, 2 = e±1>. Since/"(e1*) = sin<? e~i(*+W2), we have
in the neighborhood of e'*,
(8.73.8) /(z) = /(e*) + | sin 0 e-'(*+"J)(z - e1*J + • • • .
Therefore, the critical direction can be found in a manner similar to that used
in §8.71 B).
B) On the circle 2 = e*
^S.73.9) . W(e*)l = - i cos 2^ + cosi/' cos<^
is increasing if 0 ^ ^ ^ ^>, and decreasing if <j> ^ \p ^ x. Consequently, it
suffices to consider the contribution of the arc
iS.73.10)
In p,

[ 8.73 J LAGUERRE POLYNOMIALS 229
where 8 is a fixed positive number, 5 < 1/6. Now we have
(8.73.11) dz = e'*iC |l + iCp + (i^- + • • • j dp,
Whence from (8.73.8)
(8.73.12) f(z) =f(e«) - * sin 0ei(*-'/2)Z;V{ 1 + ClCp + c2(CpJ + ••• },
where the c, are functions of <f>, independent of n and p, and cm = 0(^1m) uni-
uniformly in e ^ <? ^ 7T — e, and uniformly as to m; A = A(«). (We note that
this condition for <f> is more general than that in (8.22.9).) Hence, we have in
the same sense as in (8.71.14), the following asymptotic expansion:
G = <r1>/2 l^
n
[ exp { - J sin <*> e'(*-)p2} {1 + Cicip + c'lp3)
(8.73.13) J—b
+ In (C2 p + C2 p + C2 p ) + • • • I dp,
where c[ , c[ , c2 , c2 , c2 , • ¦ • are constants. The principal term furnishes
G = '^ 2^ i*1l*'w/2/
(8:73.14)
•exp {\l\co^<j> + \fn + \il\ sintf>costf>!{l + 0(l~2)\,
since the integrals with odd powers of p vanish. The bound for the error holds
uniformly for « ^ <f> = * ~ «•
The integral in H is obtained by replacing <j> by t — <j>, so that
H = I -— ) exp {(a + $¦)«' — iBn + a + l)(x — </>) + 3xi/4j
(8.73.15) \sin0/
• Z^1 exp {-^ Z2n cos2 <f> + \fn — \itn sin <j> cos </>) {1 + O(Z^2)).
Taking the absolute values of the integrands in G and H, we obtain, « 5g <f> ^
x/2 - «»""J,
(8.73.16) iC = (Kf'OC exp{| Z2n cos2 <^ + ill}.
We easily see that, except for terms of higher order, H = —G. Therefore,

230 ASYMPTOTIC PROPERTIES OF CLASSICAL POLYNOMIALS [ VIII ]
(8.73.17) I27rt Z7rt j \7rsin
• {sin [| fn sin <f> cos <? — Bn + a + l)<j> + 3x/4] +
since l^2 = 0(?~1Z^1). Introducing this into (8.73.5), we obtain
7 (a
r_ l)n tUL.
Z;2n—5 exp {§ ft cos2 * + 1 ft
(n + « + f) X(sin ^)r
• {sin [i fn sin 4>cos<t>- Bw + a + 1) 0 + 3x/4]
Now
L = 2nJ exp f^-1) U + OivT2)},
4"r(n + a + 1) = ^22B+in"+a+le""{l + OirT1)}.
And since n = O(^~1C1), we have
•exp {-n - (a + l)/2) exp {i^2 + n + (a + l)/2]
• {sin [* Z2n sin 0 cos 0 - Bn + a + 1) 0 + 3tt/4] + OifX1)},
(8.73.18)
• {sin [(n + (a + l)/2) sin 2<j> ~ Bn + a + l)<j> + 3tt/4] + 0(^01 ¦
This is identical with (8.22.9).
We observe that in the application of this result to Hermite polynomials
certain simplifications are possible. For a = i| the O-terms in A.71.9) and
(8.73.4) vanish identically, so that K in (8.73.5) can be cancelled, and ? can be
arbitrarily near zero. Therefore, (8.22.12) follows readily by use of E.6.1).
8.74. Method of steepest descent; Laguerre polynomials for
D + ri)n ^ x ^ An
We start again from (8.73.2) and integrate along a proper circle about the
origin. Using a notation analogous to that in the previous section, we assume
that
(8.74.1) ? = ln cosh <t>, e ^ <f> S w, w = lnz.
Then we have
Lna)(?) a -a -| 2»-o-J
* T(n + a + 1) "
(8.74.2)

[ 8.74 ] LAGUERRE POLYNOMIALS 231
where
G[ = z~* exp { -\fnz2 + tnz cosh <f> - \fn log z\ dz,
(8.74.3) H[ = e("+i)" f z~* exp { -\tj - l\z cosh <f> - ill log z\ dz,
K' = Oir'C) j I exp { -\l\i ± ?z cosh tf> - *? log z\ \ \ dz \.
In each of Gi and Hi, we integrate along both arcs
- 7r/4 ^ arg 2 ^ + 7r/4, 3x/4 ^ arg z ^ 5x/4,
of the circle | z | == e'*, while in G2 and H'2 (which have the same integrands as
G'i and H[, respectively) we take x/4 ^ arg 2 ^ 3x/4; in K' we take the arc
0 ^ arg 2 ^ x of the preceding section. Then (8.73.4) can again be used
[cf. A.71.9)].
In this case we discuss
(8.74.4) f(z) = -iz2 + z cosh <f> - | log 2.
The condition/'B) = 0 furnishes 2 = e**, and we have/"(«"*) = (e2* - l)/2
= e* sinh ^>. Therefore, the circle | 2 | = e~* passes through the saddle point
with the smaller modulus and has the critical direction at that point. For the
second integral we obtain the saddle points — e**, from which it follows that the
circle | 2 | = eT* can be used here again. Obviously, for 2 = e~*+1^,
(8.74.5) W(z)} = ~h~U cos 2^ + e~* cosh 0 cos * + 0/2
decreases as ^ increases from 0 to tr.
On writing ^ = Cp, -nl ^ p ^ + n\ 0 < 8 < 1/6, we obtain
2 =
(8.74.6) dz =
whence
(8.74.7) /(z) = f(e~*) - ?sinh <f> e~*Z~2p2{ 1 + dl~lp + c2{l~x pf + ¦¦ ¦ }.
Here the coefficients Ci, c2, ¦ • • (and also c[ , c" , c2 , c2 , c2", ¦ ¦ ¦ below) are
analogous to those in §8.73. Hence, in the same sense as before, we have the
asymptotic expansion

232 ASYMPTOTIC PROPERTIES OF CLASSICAL POLYNOMIALS [ VIII ]
f+n*
G[ = e+12 exp{l2nf(e-+)\e-+iC I exp { -^sinh <f>e~%2\
J-ni
• { 1 + inVlP + C['p3) + 1-V2P2 + cJV + C'2"p°) + • • • } dp
= e*12 exp {Z2n/(e-*)|e-^7;1B7r)i(sinh^»)-ie*/2{l + 0@}
4> - \tne2*){l
The "principal parts" of G[ and H[ are given by the arcs
-tt/4 ^ arg z ^ +tt/4, 3tt/4 ^ arg z ^ 5tt/4,
respectively. If we replace z by e"z in #1, we see immediately that these
principal parts are identical. Consequently, we have G[ = H[ except for terms
which are of higher order than the remainder term in (8.74.8); furthermore, G'2
and Hz are of higher order than the same remainder term. Therefore, because
of ?~V = 0(l~2), we have
\-Kf
'Z-kx ~ 'Z-kx VZin / vzin I
(8.74.9)
= {-4^- ) eBn+o+1) V exp {& cosh2 <f> - \l\e2*} {1 + 0(l~2) \,
so that
(8.74.10) r(n "*
•exp{?2 — il2ne2<t> + Bn
or
(8.74.11)
•exp {[n + (a + l)/2]B* - sinh 2^)j {1 + OijT1)).
Returning to the variable x, we obtain (8.22.10). From this result (8.22.13)
follows immediately.
8.75. Method of steepest descent; Laguerre polynomials for
x = 4n + O(n )
A) First, let t be real and bounded. We write as before
X = ?2, ? = In - (Qln)~% W = lnZ,
(8.75.1) r(<
(_ ) = 2TB^)C2ft .(? + .
T(n + a + 1) (^2x1 2x1
where

[8.75] LAGUERRE POLYNOMIALS 233
G" = J z^exp {llMz) - Q-Hltz} dz,
(8.75.2)
with
(8.75.3)
H"
K"
6 J
= o(i-2) 11
/i(z) =
h(z) =
z * exp {I
exp {fnf,{
~\z2 +
— \ z2 —
i/,W + 8"*
z) =F 6~*Z!Bfe
0 - 5 log Z,
z - \ log z.
dz,
dz I, k = 1, 2,
The integrals in (8.75.2) are extended over the upper half of a proper curve
symmetric to the real axis for which | z \ and | z I are bounded.
The "saddle point condition" f[{z) = -z/2 + 1 - Bz)~1 = 0 furnishes
0 = 1, and we notice that/('(I) = 0 and f["(l) = — 1. Therefore, this saddle
point is of a different character from the preceding ones.
We first integrate along the segment
(8.75.4) Z=l + e^pe21, 0 ^ p = n" ,
where 5 is a fixed positive number, 5 < 1/6, then along the segment symmetric
to this one with respect to the imaginary axis, and finally along the arc of a
circle with center at z = 0 which connects the ends of the segments mentioned
(see Fig. 9). For certain constants ct, c5, ¦ ¦ ¦ , we have
(8.75.5)
Furthermore, if r is the radius of the arc of the circle, then
(8.75.6) Wi(relV)} = - |r2 cos 2^ + r cos ^ - \ log r
is decreasing for 0 S ^P ^ ir, since r cos \j/ < 1. We now obtain the following
expansion

234 ASYMPTOTIC PROPERTIES OF CLASSICAL POLYNOMIALS [ VIII ]
G" = exp {\i2n - e-^j-eW'"*
(8.75.7) • f exp { - p3 - Pe2"'3t} {1 + (c'lP + c"P*) I?
Jo
+ (cf2P2 + ciV + c;V)tf + • • • Up.
This is an asymptotic expansion in the usual sense. At the end-point p = n
we have the bound O(exp [—cnu]), c > 0, for the integrand, which isatthesame
time a bound for the contribution of the remainder of the contour. (The slight
modification of the circle around z = — 1 is immaterial.) Thus
G" = exp
|^ exp (- p3 - Pe230 dp + O(tf)j.
In the same way we see that the principal part of H" is due to the small segment
(8.75.9) z = -1 + 6*Z;V73, 0 ^ p ^ n*.
We find
/2(z) = /2(- 1) + ~ (z + 1)^"(- 1) + • • •
(8.75.10) 3!
= 3/4 - iV/2 - z;V + • • •.
Therefore,
(8-75.11)
• ^ie'm{J exp (- p + pe3o rfp + °(rM >
and
Now we can readily show that H" = — G" except for terms of higher order.
Consequently,
(8.75.13) ivAt-t-a-t-u
exp (-p -
it -
The imaginary part in the braces is Airy's function A(t) (Problem 2). Sub-
Substituting the approximate values of 4"r(n + a. + 1), l~2n~a~7/6, and if"*"*
= C"}! + O(C/3)}, and observing that
(8.75.14) ?2/2 = fn/2 - 6~HU +

[ 8.8 ] DIFFERENTIATION OF ASYMPTOTIC FORMULAS 235
we obtain
which is (8.22.9) with the less precise remainder term OirT*).
B) Direct calculation of a further term in (8.75.5) leads to
(8.75.15) c4 = I/<4)AN4/Y*t73 = |64/Y*t7\
Furthermore, from (8.75.4)
(8.75.16) *T* = 1 - h$e2wil%tfP + O(C/3P2).
Consequently, the expression in the braces in (8.75.8) can be written as follows
f exp (-p3 - pe2Hnt)dP[l +(-*6Y3p + |64/Y3p4)C} + O(C4/3)].
Jo
The corresponding more precise form of (8.75.13) can be obtained by replac-
replacing the expression in the braces by
exp(-p3 - pe2ril3t)
- Wrnls[eiH'3 ^ exp (-p3 - pe2*il3t)pdp~\
+ i64/3CJs[e10't73 J^exp (-p3 - pe^th'dp'j + O(C13)
= A(t) +
= A(t) +
Here the differential equation A.81.2) has been used. Now (8.75.14) can be
also written in the more precise form
so that
e-^2 = exp [-11/2 + 6*^]{1 - |6~W + O(rnil3)}.
From this, (8.22.9) readily follows with the remainder term O(J^13) = O(n~l).
C) The case of a general complex t can be settled by a slight modifica-
modification of the argument in A).
The corresponding asymptotic formula for Hermite polynomials follows
immediately by means of E.6.1).
8.8. Differentiation of certain asymptotic formulas
Differentiation of an asymptotic formula with respect to a parameter occurring
in that formula is in general not permitted. In some of our previous formulae,

236 ASYMPTOTIC PROPERTIES OF CLASSICAL POLYNOMIALS [ VIII ]
however, the permissibility of differentiation is readily established. We must,
of course, modify the remainder in a proper way. We shall discuss the formulas
(8.21.18) and (8.22.6), involving Jacobi and Laguerre polynomials, from this
point of view, using the important identities D.21.7) and E.1.14).
A) First let us consider (8.21.18) which contains. Darboux's formula (8.21.10).
We shall prove that
I- \PiaJ)(cos 0)\ = n'kF)\ - sin (Nd + 7) + (n sin ey'OH)},
(8.8.1) d0
'a > — 1, /3 > -1, crT1 ^ 0 ^ 7T - cn~\
with the same notation as in (8.21.10). We note that k'F) = /c@)(sin 0)~1OA).
In the proof, we write for the left-hand member of (8.8.1),
-| sin 0(n + a + 0 + l)P^ll/s+1)(cos 0)
= -?sin0(n + a + 0 + l)(n - l)~'^@)
—l
sin v- cos ^ ] {cos (Nd + 7 — 7r/2) + (to sin 0)'
2 2/
which establishes the statement. Evidently, the remainder term of (8.8.1)
can be replaced by d~a~iO(n~*) if cn~x ^ 8 ^ ir — e, and by 0(n~*) if
€ ^ 0 ^ 7T — €.
From (8.8.1) we derive the following important formulas:
02)
01 — 02
cos (N6i + 7) ~ cos (iV02 + 7)
= n
+ dr^Oin'1') + (tt - 02)~^5O(n~i),
(8.8.2) Pia^(cos0Q - Pia^(cos02)
COS 0i — COS 02
- -'Me ) cos (Ndl + 7^ ~ cos ^g2 + ^
COS 0i — COS 02
a > — 1, /3 > —1, cn~l S 0i < 02 ^ 7r — en.
Similar formulas hold with &@2) instead of kFi) in the right-hand member.
For the proof we apply the mean-value theorem to
0@) = p(na^ (cos 0) - n~'kF) cos (NO + 7)-
Thus

8.9 ] APPLICATIONS; ASYMPTOTIC PROPERTIES OF ZEROS 237
di — d2 cos 0i - cos d2 sin t2' 0i < t2 < 02-
But, we have ^'(n) = n*fc(Ti)(n sin ^"^(l), which is fll""^^1) or
(it — 02)~^~§ 0(n~*), according as tx S 7r/2 or n ^ 7r/2. Moreover,
cos (iV0i + 7) - A;(fe) cos (N62 + 7)
01 — 02
+ 7)
cos
"l —  v\ — 02
The latter ratio is k'ir) = 0Ta''iO(l) + (tt - 02)^~§OA), 0x < r < 02. A
similar argument leads to the second formula in (8.8.2).
If both 0i and 02 are confined to an interior interval e ^ 0 ^ 7r — e of [0, tt],
the remainders in the formulas (8.8.2) are simply O(ri~').
B) Next we consider (8.22.6), which contains Fejer's formula (8.22.1).
We write (8.22.6) in the form
Lla)(x) = k(x)nal2~{ {cos [2(nxY' + 7] + (rw)H0(l)},
(8.8.3) lfc(aO = 7r~*el/2afa/2-*, 7 = - (a + |)tt/2,
a > — 1, cn~l ^. x S u,
where c and u are fixed positive numbers. By means of E.1.14) we now obtain
(8.8.4) —^ {L[a\x)\ =2k(x)nal2+i{ - sin [2(rw)» + 7] + (nx)-l'0(l)},
noting that d{k(x)\/d(xi) = k{x)O(x~'), and that (n - 1)* - n* = O(n"').
The mean-value theorem furnishes, if both xx and a;2 belong to the interval
- Lia)(x2)
en 1 s x s w,
x\ - x\
(8.8.5) _ u s a/2-i cos [2(rwi)* + 7] - cos [2(na%)* + 7]
— K\Xijn j j
> >
l2-1) + i^Odi'^), a > -1.
The proof is similar to that used in the case of Jacobi polynomials.
8.9. Applications; asymptotic properties of the zeros of Jacobi and
Laguerre polynomials
A) We shall first point out some consequences of the formulas (8.21.18)
and (8.8.1), which will be important for the discussion of interpolation with
Jacobi abscissas (Chapter XIV). Setting c = 1 in (8.21.18), and using the
same notation as before, we have the result:

238 ASYMPTOTIC PROPERTIES OF CLASSICAL POLYNOMIALS [ VIII ]
Theorem 8.9.1. Let a > -1, /S > -1, and let 0 < dx < 02 < • • • < 6n < w
be the zeros of P(na'e) (cos 0). Then
(8.9.1) e, = rr1{vir + Oa)\,
with 0A) being uniformly bounded for all values ofv = 1,2, • • • n; n = 1, 2,3, • • • .
Furthermore,
(8.9.2) | Pia'Ph (cos 0,) | ~ Jra~ina+2, 0 < 0, ^ tt/2,
in ?/ie sense that the ratio of these expressions remains between certain positive
bounds depending only on a and /3.
Let p be a fixed number, 0 < p < 7r/2. Substitute in (8.21.18)
(8.9.3) N6 = (v - §> - 7 ± p, v > 0, ? an integer, 0 < 0 ^ tt/2.
Then the first term on the right, that is, ±( — l)"n~^k(d) sin p, is the principal
term provided the remainder d~a~iO(n~i) is less than n^k{6) sin p. This
is so if v and n are sufficiently large, v ^ M = M(a, /3, p). The same is true
for the formula (8.8.1). We have also 8 > 0 if M is properly chosen. Further-
Furthermore, let (v — |Or — 7 + p ^ iV7r/2, so that 0 ^ ir/2. Then for the values
(8.9.3)
(8.9.4) sgnPia^(cos0) = (-1)' and (-1)'+1,
respectively. Hence P(na'^(cos d) has at least one zero between the bounds
given by the values (8.9.3) and, since (8.8.1) shows that it is monotonic in this
interval, it has only one zero there. Also, we see that in the same interval
(8.9.5) I- {P^tcos 6)} - nhk{e) - n*^,
ad
and in the "complementary intervals" of [0, 7r/2],
(8.9.6) | P(naS\cos 6) | ~ <n* ^^
The statements (8.9.5) and (8.9.6) hold in the sense that the ratio of the ex-
expressions in question remains bounded from zero and infinity, uniformly in 0
if 6 lies in the intervals mentioned; here n is sufficiently large, v 2: M(a, /3, p),
and (v — \)ir — 7 + p ^ Ntt/2.
Thus Pla'^(cos 6) has no zeros in the "complementary intervals."
In the interval 0^0^ N^HM — \)ir — y — p} we have a bounded number
of zeros which have the form n~l{jk + «n), where jic denote the positive zeros of
Ja(z) and en -> 0 (cf. (8.1.1) and (8.1.3)). This furnishes (8.9.1) for 0 <
0, ^ 7r/2. Taking into consideration the formula D.1.3), the full statement
(8.9.1) follows.
In addition, we find, by use of (8.9.5), that
(8.9.7) | P^ ii *

[ 8.9 ] APPLICATIONS; ASYMPTOTIC PROPERTIES OF ZEROS 239
provided the conditions mentioned are satisfied. The extension to the total
interval follows again by means of (8.1.1). Combining this result with (8.9.1),
we obtain (8.9.2).
For the zeros dv from a fixed interval [a, b] in the interior of [0, w], that is,
0 < a < b < 7r, (8.9.1) can be written in the more precise form
(8.9.8) 6, = N-'{(v - a)tt - 7
where K is a fixed integer (depending only on a, (j, a, b), and «„ —> 0. In this
case, (8.9.2) attains the simpler form
(8.9.9) |Pj,ai/S)/(cos0,)|~n*.
B) The analogous statements for Laguerre polynomials follow from (8.22.6;,
(8.1.8), and (8.8.4). We use here the same notation as before.
Theorem 8.9.2. Let a > — 1, and let Xi < x2 < ¦ ¦ ¦ <xn be the zeros of L(n\x);
then we have for the zeros xv from a fixed interval 0 < x ^ w
(8.9.10) 2x\ = n~i{vTr
Moreover,
(8.9.11) | Lla)'(xv) | - xrl2-l
Let p be a fixed positive number. Introducing in (8.22.6)
(8.9.12) 2{nxf = (v - \)v - y ± P, v > 0, v an integer, 0 < x ^ «,
we obtain values of opposite signs provided v 2; M = M(a, p), and n is suffi-
sufficiently large. Then we see from (8.8.4) that Lia\x) is monotonic between
the corresponding values of x, and hence L^a)(:r) has precisely one zero in the
corresponding interval. Taking (8.1.8) into account, we find (8.9.10) as in A).
Again, from (8.8.4) and (8.1.8), 0 < xr S u,
For the zeros xv from a fixed positive interval e S xv S w, e > 0, we obtain
(8.9.13) | Llah(x>) | ~ nal2+i.
C) Let a be arbitrary and real. The preceding results (especially formula
(8.22.11)) enable us to discuss the largest zeros of L^'Or,. Let u < it <
i3 < ¦ ¦ ¦ be the zeros of Airy's function .4@; then the only zeros of L(n\x) in
the interval
(8.9.14) x = An + 2a + 2 - 2Bn/3)*«, t real and bounded,
are those corresponding to the values t = iv. (Near t = iv we have precisely one
zero if n is sufficiently large; this follows from the uniform validity of (8.22.11)
for bounded complex t, by use of Theorem 1.91.3 (Theorem of Hurwitz).)
On the other hand, the upper bound F.31.7) of the zeros belongs to this interval.

240 ASYMPTOTIC PROPERTIES OF CLASSICAL POLYNOMIALS [ VIII ]
Consequently, if we arrange the zeros xi > x2 > z3 > • • • of L^ix) in decreasing
order, we have
(8.9.15) xv = x>n = in + 2a + 2 - 2Bn/3)*(i, + «»), Km e» = 0, v fixed.
This is identical with F.32.9). An analogous result holds for Hermite poly-
polynomials, namely F.32.5). It follows immediately from (8.22.14). This
formula was proved in §6.32 by use of Sturm's method.
8.91. Applications; asymptotic properties of the maxima of Laguerre and
Hermite polynomials
Another application is the discussion of the magnitude of the Laguerre and
Hermite polynomials for non-negative and for real values of x, respectively.
A) Theorem 8.91.1. Let a. be arbitrary and real, a > 0, 0 < t\ < 4. We
have for n —> <*>
(8.91.1) max ,-», | L(x) | [
f:i;:
[n if x ^ a.
These maxima are taken, respectively, in the intervals pointed out in the right-
hand members.
These asymptotic formulas play an important r61e in the discussion of
Laguerre series (Chapter IX).52
A more exact characterization of these maxima when n —> oo is also possible.
We can replace ~ by ^ by introducing into the right-hand members the
constant factors
B/x)iD1? - 1)J, *~lQ-2)h max A{t);
(8.91.3) . ,
Tr-^-1)*, ir A8)* max A(t),
respectively. This follows readily from the subsequent proof. We note that
these factors are independent of a and a. The first maximum of A(t) is posi-
positive, and is actually the greatest value of | A (t) \ for all real values of t. (Cf.
Theorem 7.31.1 and A.81.2).)
B) Let t = ji be the first maximum point of A(t), 0 < jx < ix ¦ Now choose
two values t', t" such that 0 < t' < t" < jt and denote the corresponding
62 Kogbetliantz B2, p. 144;23, pp. 39, 51-53), states erroneously the appraisal L^a)(z) =
O(exl2x~al2~1lina/~~1/4), x^a. The error is made in 22, p. 154, where a certain bound valid for
Hn(x), if I x | < c-n}'2, is applied to an arbitrary Hm{x), m ? n. Unfortunately, the main
results of the paper 23 (and partly also those of 24) are based on this erroneous statement.

[8.91 ] ASYMPTOTIC PROPERTIES OF THE MAXIMA 241
values of x in (8. 9. 14) by x' and x", respectively, so that x' > x" > Xm if n
is sufficiently large. Then A(t') < A{t"), so that by (8.22.11)
(8.91.4) (-l)ne-''l2Lia\xr) < (-1)V"/2Z,iaV).
Therefore, x' and x" both cannot be on the left of the last extremum point of
e~xi2L{na\x), whence this extremum point lies in the interval (8.9.14).
The same consideration applies to e~xl2x^L(n\x), where \ is an arbitrary but
fixed real number. Indeed, zx = Dn)x{l + 0{n~1)}. Thus, a formula analo-
analogous to (8.22.11) holds for e'xl2xxL{na\x) with the additional factor Dn)x in
the right-hand member.
C) In the interval a ^ x ^ D — tj)n formula (8.22.9) can be applied.
Since
-V'-V1-* = x< (ain«)-*na/1-» ~ nal2 (cot «)*,
xal2+i (8in«)-»a:-a/>-*na/*-* = (sin </>)" V'2"*,
the maxima in question are ~na/2 and na/2-i, respectively.
In order to calculate the maximum in the interval x ^ o, we use Theorem
7.6.2. The sequence of the relative maxima of
(8.91.5) e-xla+1)l2\Lia)(x)\ and e~x/2xa/2+i \ Lia\x) | .
is increasing for x > xo, where xo is a certain non-negative number depending
on a, and n is sufficiently large. Therefore, on account of A), the absolute
maxima of the functions (8.91.5) for x ^ a are attained in the interval (8.9.14).
According to (8.22.11) these maxima are ~ nia+1)i2n~i and nal2+*ri~*, respec-
respectively. Between a and Xo, FejeYs formula (8.22.1) must be used. This
furnishes the complete proof.
D) We also readily prove the following more general theorem:
Theorem 8.91.2. Let a and X be arbitrary and real, a > 0, 0 < y\ < 4. Then
for n —> oo
(8.91.6) max e~x/2xx \ L^ix) \ - nQ,
where
(max (X - \,a/2 - \) if a ^ x ^ D - jfin,
(8.91.7) Q =
[max (X — -|, a/2 — ?) if x ^ a.
In the first case, that is to say, in the interval a ^ x ^ D — ??)n, we again
apply (8.22.9). Now
zx(sin ^)-^-a/2-intt/2-i - nx-*<cos </>JX~a- J(sin <t>y\
This expression attains its maximum in the interval e ^ <j> ^ x/2 — en-i
for <f> = e or <f> = tt/2 — en~* according as X ^ a/2 + \ or X < a/2 + i-
The maxima are ~nx~* and ^^(n"*J""" = n"'2, respectively. This

242 ASYMPTOTIC PROPERTIES OF CLASSICAL POLYNOMIALS [ VIII ]
establishes the first part of the statement. We also see that the maximum is
attained near x = D — 17O1 if X ^ a/2 + j, and near x = a if \ < a/2 + \-
In the interval D — 17O1 ^ x ^ 4n + O(n*), we have
max g'/y I LLa)(z) | - n*<»+»'8 max g-*/y1 L() |
Because of (8.91.1), this furnishes the second part of the statement. The
maximum is attained near x = 4n if X — J > a/2 ~ \, and near x = a if
X - | < a/2 - f
E) By use of E.6.1), or directly from (8.22.12) and (8.22.14), we obtain the
corresponding results for Hermite polynomials.
Theorem 8.91.3. Let X be arbitrary and real, a > 0, 0 < 17 < 2. Then
(8.91.8) max e~l2/V | Hn{x) \ ~ B"n!)V,
where
(max (X/2 - 1/4, -1/4) if a <, \ x \ g [B - r,)n]\
[max (X/2 - 1/12, -1/4) if \ x \ ^ a; x real.
We have, for instance (Theorem 7.6.3),
(8.91.10) max e"l2/21 Hn{x) \ 9* B"n!)i2i3}7r~in/12 max A(t).
Here x and t range over all real values. (Cf. Hille 1, p. 436, C0).)
8.92. Further results
A) Gatteschi A, 2) investigated various asymptotic formulas involving the
classical polynomials and their zeros, and replaced the (usually unspecified)
constants occurring in the 0-terms by numerical values. As an illustration we
mention the following remarkable refinements of Theorem 8.21.6.
We have
Pn (cos 6) = @/sin 6)iJo{ (n + 1H} + cr
where
\a\ < 0.09 02 if 0 < 6 ^ x/Bn),
a\ < 0.63 6*n-i if 7r/Bn) < 6 ^ x/2.
The following formula is even more informative:
~~J Pn(cos6) =
where
<x'\ < 0.03 04 if 0 < 6 ^ x/Bn),
< 0.25 e^n-s- if w/Bn) < 0 ^ x/2.
These formulas can be used for the asymptotic evaluation of the "first" zeros
with specified constants in the error terms. See also Tricomi 4.

[8.92] FURTHER RESULTS 243
B) Further information concerning the asymptotic behavior of the Jacobi
and Laguerre polynomials and their zeros can be found in Bateman Manuscript
Project, vol. 2, Chapter 10, pp. 196-202; also Tricomi 5, pp. 219-224, Thorne 1,
and Erd^lyi-Swanson 1. See also the literature quoted at these places.
Recently, Erdelyi 3 has obtained two new asymptotic formulas for the
Laguerre polynomials L^ix) valid for x ;S aDn + 2a + 2) and for x ^
frDn + 2a + 2) where a and b are fixed, 0 < b < a < 1. These formulas
involve Bessel functions and Airy's function, respectively, and hold on the
above ranges which overlap and cover the entire real axis. This is an important
result.
Erdelyi only proved these asymptotic formulas for a ^ 0. They were ex-
extended to a > —1 by recurrence relations by Muckenhoupt 1.
C) The formula (8.21.1) of Laplace-Heine has been used in the estimation
of the smallest eigenvalue of the truncated Hilbert matrix A/0'¦-\-j + l)M-=o-
See Szego 27 and Widom-Wilf 1. The size of this eigenvalue gives a measure
of the degree of difficulty in finding the inverse of this matrix.
D) The remark in §8.4 C) is not quite correct. The infinite series corre-
corresponding to Darboux's formula (8.21.4) is convergent for tt/6<0 < 5tt/6,
but it converges to 2Pn(cos0) rather than Pn(cos0). See Olver 1. While
asymptotic series in the sense of Poincare must converge to the "right" function
if they converge this is not true of more general asymptotic expansions; so
great care must be exercised when using them.

CHAPTER IX
EXPANSION PROBLEMS ASSOCIATED WITH THE
CLASSICAL POLYNOMIALS
The formal expansion of a function in terms of general orthogonal poly-
polynomials having been defined in §3.1 (cf. C.1.3)), we shall now turn our atten-
attention to expansions in terms of the classical polynomials. In this connection we
shall deal with the following problems:
Expansion of an analytic function in series of Jacobi, of Laguerre, and of
Hermite polynomials; discussion of the domain of the convergence.
Expansion of an "arbitrary" function in series of Jacobi, of Laguerre, and
of Hermite polynomials; discussion of equiconvergence and summability
theorems.
In the main, our principal concern will be with the second problem, and we
agree that "arbitrary" function shall mean a function restricted only by certain
conditions of integrability, or continuity, and by conditions involving the
existence of certain integrals.
Two series ^™=o un and 2»=o vn are called equiconvergent if the series
2Zn=o (m« — vn) or, more generally, if
00
(ttn — AVn), A 5* 0,
n=0
is convergent. We shall try to find simple trigonometric (Fourier) expansions
equiconvergent with a given type of polynomial expansion. Such a procedure
will enable us to reduce the discussion of the polynomial expansion to the dis-
discussion of trigonometric series under very general conditions.
Of the various methods of summability, we shall be primarily interested in
that of Cesaro. A series 2Z"_o un is called (C, &)-summable, k > — 1, with
the sum s if
„(*)
where
n=0 n=0
A _ r)-*-1 = S C?> r" = E (n + k) rn.
n-0 n=0 \ n /
Obviously, k = 0 corresponds to convergence in the ordinary sense. If k' > k,
it is readily shown that (C, /c)-summability involves (C, A/)-summability with
the same sum.
244

[ 9.1 ] RESULTS 245
A necessary condition for (C, /c)-summability, k ;> 0, is un = O(nk).
Cesaro summability of any order k implies Abel summability; that is, the
existence of the
00
lim XI un rn
with the same sum.
9.1. Results
A) The formal expansion of a function fix) in a Jacobi series is (compare
D.3.3))
(9.1.D
hlnaJ>)an = J^ A - x)a(l + xff{x)P^\x)dx.
The expansion of f{x) in a Laguerre series is (cf. E.1.1))
/(*)~Zo.?-a>(*),
(9.1.2) , n~°
T(a + l)(n + a)an = I e-xxaf(x)L(na)(x)dx;
\ n / Jo
and in an Hermite series it is (cf. E.5.1))
f(x)
n=0
*r*2"n!a« = / e'xif{x)Hn{x)dx.
In all these cases f{x) is a measurable function, and the existence of all
integrals occurring is required.
In what follows we assume a > —1,/3> — 1 in the Jacobi case and a > — 1
in the Laguerre case.
B) In this connection the following results may be stated:
Theorem 9.1.1 (Expansion of an analytic function in a Jacobi series). Let
f(x) be analytic on the closed segment [ — 1, +1]- The expansion of f(x) in a
Jacobi series is convergent in'the interior of the greatest ellipse with foci at ±1,
in which f(x) is regular. The expansion is divergent in the exterior of this ellipse.
Using the notation (9.1.1), we have the following representation for the sum R of
the semi-axes of the ellipse of convergence
(9.1.4) R = liminf \an
-l/n

246 EXPANSION PROBLEMS FOR THE CLASSICAL POLYNOMIALS [ IX ]
Theorem 9.1.2 (Equiconvergence theorem for Jacobi series in th!e interior
of the interval — 1, +1). Let f{x) be Lebesgue-measurable in [ —1, +jl], and let
the integrals
xY\f{x)\dx,
(9.1.5)
' " A - aOa/8-*(l + a;/'2-* |/(x) | rfz
exist. If sn(x) denotes the nth partial sum of the expansion of f(x) in\ a Jacobi
series, and sn(cos 0) the nth partial sum of the Fourier (cosine) series of
(9.1.6) A - cos 0)a/2+i(l + cos 0)m+if(cos 0),
then for — 1 < x < +1,
(9.1.7) lim {sn{x) - A - x)-an~\\ + xy'^s^x)] = 0, |
uniformly in — 1 + c ^ x ^ 1 — e, where e is a fixed positive number, je < 1.
Theorem 9.1.3 (Summability theorem for Jacobi series at the ejid-points
x = dbl). Letf{x) be continuous on the closed segment [— 1, +1]- The expansion
of f(x) in a Jacobi series is (C, k)-summable at x = +1, provided k ^ a + \.
This is in general not true if k = a + \ . An analogous statement holds for x
= — 1, a being replaced by /3.
Theorem 9.1.4 (Generalized summability theorem for Jacobi series). Let
f(x) be Lebesgue-measurable in [—1, +1] and continuous at x = -\-l.\ Then if
we assume the existence of the integral
(9.1.8) jTV -x)a(l + xY\f{x)\dx,
the Jacobi series is (C, k)-summable, k > a -\- \, at x = -\-\, provided that in the
case
(9.1.9) jS > -i a + I < k < a + jS + 1,
the following additional "antipole condition" is satisfied: the integral
(9.1.10) jT(l +x?'*-i\f(x)\dx
exists. (For k ^ a -f- /3 -f- 1 no antipole condition is necessary.) For k ^ a + |
or for k > a + |, but without the antipole condition, the statement is not trtue.
i
i
Theorem 9.1.5 (Equiconvergence theorem for Laguerre series for a; > 0).
Letf(x) be Lebesgue-measurable in [0, + °°L an<^ ^ ^e integrals
(9.1.11) I" xa\ f(x) | dx, f xal2~{\ f(x) | dx
Jo Jo

[ 9.1 ] RESULTS 247
exist. If the condition
fC\ 1 1 O\ / ~XW~ a/2—13/12 I /./ \ l j / —iN
(9.1.12) ex \fix) \dx = o(n ), n—> °o,
y n
is satisfied, and if sn(x) denotes the nth partial sum of the Laguerre series of fix),
we have, for x > 0,
(9.1.13) lim <sn(x) - tT1 [' +*/(r2) ~n {2n!^ ~ r)i dr) = 0,
n-oo ^ Jxl~& X* — T J
where d is a fixed positive number, 8 < xK This holds uniformly for every fixed
positive interval e 5= x 5= a, d < e .
T/ie some equiconvergence theorem (9.1.13) is valid if the integrals (9.1.11) exist
and condition (9.1.12) is replaced by the following:
I e~x ixai~i | f(x) | dx is convergent;
(9.1.14)
n
Theokem 9.1.6 (Equiconvergence theorem for Hermite series, x arbitrary
and real). Let fix) be Lebesgue-measurable in [—°°, +00], and let the integral
(9.1.15) [ \f(x)\dx
exist for every a > 0. If the condition
(9.1.16) f" e~xV2x~H\f(x) | + \f(~x) \}dx = oin-1), n
zs satisfied, and if sn(x) denotes the nth partial sum of the Hermite series of f(x),
we have, for an arbitrary and real x,
(9.1.17) hm <sn(a;) - tt I f(t) ~—-~ — dt> = 0,
n-oo I, Jx-S X ~ t )
where 5 is a fixed positive number. Moreover, (9.1.17) holds uniformly in every
finite interval.
The same equiconvergence theorem (9.1.17) holds if the integral (9.1.15) exists
and we replace condition (9.1.16) by the following:
/ e~*"*x~1{\f(x) \ + \f(-x)\)dx is convergent,
(9-1.18)
/ e'x\~\ | /(*) |2 + I /(-*) I2} dx - o(n-3), n -> «,.
Jn
Theorem 9.1.7 (Summability theorem for taguerre series at x — 0). Let

248 EXPANSION PROBLEMS FOR THE CLASSICAL POLYNOMIALS [ IX ]
f(x) be Lebesgue-measurable in [0, + °° ] cmd continuous at x = 0. If we assume
the existence of the integral
(9.1.19) f' e-"axa-k-i\f(x)\dx,
the Laguerre series of f{x) is (C, k)-summable at x = 0 with the sumf@) provided
k > a + f. This statement is not true for k ^ a -f- §.
9.11. Remarks
A) Theorem 9.1.1 is well-known in the Tchebichef and Legendre cases.53
The determination (9.1.4) of the ellipse of convergence is analogous to the well-
known Cauchy-Hadamard formula.
B) The importance of Theorem 9.1.2 is due to the possibility of applying it
to convergence and summability problems for Jacobi series similar to those
which are found in the classical theory of ordinary Fourier series. This theorem
has been proved in the special case a = /3 = 0 (Legendre series) by Haar B)
and W. H. Young A) with conditions concerning f{x) somewhat different from
those arising from our general theorem in the special case a = /3 = 0. W. H.
Young considers also series proceeding in terms of Legendre polynomials, with-
without being Fourier expansions in the ordinary sense. The proof of Theorem
9.1.2 given in §9.3 is due to Szego A7, pp. 88-92). Recently, Obrechkoff B)
has treated the same problem, using as his main tool Darboux's formula in the
more precise form (8.21.18).
Summability theorems for interior points were investigated earlier (in the
ultraspherical case) by Adamoff B) and (in the ultraspherical as well as in the
general Jacobi case) by Kogbetliantz A, 2, 3, 7, 18, 19). Concerning properties
of these expansions analogous to those treated in Riemann's theory of trigo-
trigonometric series (in particular theorems of uniqueness) see Kogbetliantz F, 20)
and Zygmund A).
In setting up the expansion in Jacobi series, we must require the existence of
the first integral (9.1.5). Using an obvious notation, we readily see that
(9.11.1) f A - x)a \f(x) | dx ±+ f A - x)"-* \f(x)
Jo Jo
dx,
according as a ^ — | or — 1 < a ^ — |. That the condition of the existence
of the second integral cannot be improved upon for a > — | follows when we
consider the special function f{x) = A — x)" with /x = —a/2 — 3/4 (cf. §9.3
D)). The existence of both integrals (9.1.5) follows from that of
(9.11.2) jT A - x)a(l + xf{f{x)fdx.
63 The Legendre case is usually attributed to F. Neumann (cf. Whittaker-Watson 1,
pp. 322-323).

[9.11] REMARKS 249
Instead of (9.1.6) the function
(9.11.3) A - cos 0)a+J(l + cos 0)"+V(cos 6)
may also be considered (cf. Szego, loc. cit.). The change, necessary in (9.1.7)
for this function, is at once apparent. Both functions (9.1.6) and (9,11.3) are
integrable in — x g 6 g +x. The difference (9.1.7) may be written as
(9.11.4) J-i
- (i - x)-al2~K\ + x)-m-K\ - ty^-Ki + (f*-lK?'-*\x, t)\dt,
where the notation D.5.2) has been used. Comparison of sn{x) = sn(a, /3; x)
with sn(y, 5; x), where y and 8 are arbitrary, is also possible.
C) Theorems 9.1.3 and 9.1.4 have an extensive literature. Gronwall A, 2)
has investigated the special case a = /3 = 0 (Legendre series). His proofs have
been simplified to a considerable extent by Lukacs B), Hilb A), and Feje*r (8).
The ultraspherical case has been considered in great detail by Kogbetliantz
B, 4, 19; 21, pp. 70-73) (cf. also Obrechkoff 1). The method used in §§9.4-9.42
is new and comparatively simple. It is based on a peculiar relationship between
the Jacobi polynomial P(na+k+1'p\x) and the "kernel" of the kth Cesaro mean
of the Jacobi series which corresponds to the parameters a and /3. (In the case
of Laguerre series the corresponding relation is trivial (cf. §9.6).) The case
of a Legendre series (if we consider the series at the end-point x = +1) is equiva-
equivalent to a Laplace series; whence the term "antipole condition". At the end-
point x = — 1 a similar theorem holds; in this case the summability index k
must exceed /3 -f- \, and an "antipole condition" must be satisfied near x = -f-1.
Theorem 9.1.3 furnishes the convergence of the Jacobi series at the end-point
x = +1 provided fix) is continuous in [—1, +1] and —1 < a < — | (cf. also
Rau 1; Szego 17, §20; Lorch 1, 2).
In the case of Legendre series the "kernels" of the Cesaro means of the second
order are non-negative (Feje"r 4). This fact has, of course, important conse-
consequences. Concerning similar theorems in case of ultraspherical expansions,
cf. Kogbetliantz 19.
In the special case of Legendre series Kogbetliantz A6) gave important
refinements of Theorems 9.1.3 and 9.1.4. He studied, among others, the C-
summability of proper order at the end-points x = ±1 if f(x) becomes infinite
of a certain order at the antipole.
As regards certain older (very complicated) results on the convergence of ultra-
ultraspherical expansions at the end-points x = ±1, see Adamoff 2.
D) The equiconvergence problem for Laguerre and Hermite expansions
has been treated (the first only in the special case a = 0) by Rotach A), and
(in all cases) independently by Szego A0). For the special Laguerre case a = 0
the conditions of Szego are more restrictive than those of Rotach, and the same

250 EXPANSION PROBLEMS FOR THE CLASSICAL POLYNOMIALS [ IX ]
is true in the Hermite case. The conditions formulated in §9.1 are, however,
slightly more general than those of Rotach.54
That the condition of the existence of the second integral (9.1.11) cannot be
improved upon is shown by considering the expansion of the function f(x) = x",
m = —a/2 — 3/4, in a Laguerre series (§9.5 F)).
Neither of the two sets of sufficient conditions formulated in Theorem 9.1.5
contains the other. Indeed, the function/(a;) = exl2x~al2~h satisfies (9.1.14),
but not (9.1.12). On the other hand, if
0, otherwise, m = 1, 2; 3, • • • ,
then (9.1.12) is satisfied, but not the second part of condition (9.1.14).
Condition (9.1.12) implies, of course, that the integral on the left side exists.
From the condition (9.1.12) the first part of (9.1.14) follows. In fact, if we write
(9.11.6) u{x) = e-«*t2-131121 f(t) | dt,
Jx
we find from A.4.4) that
(9.11.7) / xhdu{x) = «*«(«) - «A) - | j x~iu(x)dx
is bounded as w —> °o, since u{x) = o{x ).
A sufficient condition for the validity of (9.1.13) is that
(9.11.8) f(x) = Oie^x"*2-*-5), 8 > 0, x -» +°°.
Then (9.1.14) is satisfied. On the other hand, for f(x) = exi2x~a/2+i conditions
(9.1.11) are satisfied, but not (9.1.12) and (9.1.14), and (as we are going to show
in §9.5 G)) the Laguerre series is divergent for x > 0.
A sufficient condition for the validity of (9.1.17) is that
(9.11.9) f(x) = O(elV2 | x T), S > 0, x -» «.
Then (9.1.18) holds. An analogous "Gegenbeispiel" here is f(x) = xex"/2.
From Theorems 9.1.5 and 9.1.6 there follow the usual theorems on the con-
convergence and the summability of Laguerre and Hermite expansions. Indeed,
the integrals occurring in (9.1.13) and (9.1.17) are essentially the partial sums
of order [nJ] of a Fourier series (cf. A.6.4)).
As early as 1907 Adamoff B) obtained a convergence theorem for Hermite
64 Rotach's second condition 62 on p. 8 makes the first part of 61 superfluous. His
first set of conditions is equivalent to (9.HI) plus (9.1.14) (for a = 0), whereas his second
set is more restrictive than (9.1.11) plus (9.1.12). In the theorem on p. 6, the second
condition 62 must be corrected to read "Jfe~*2/i |/(±z) | zz/2dz exists." (I owe this to a
written communication from Mr. Plancherel.) This, of course, implies 61. Here the
first set of conditions is more restrictive than (9.1.18), and the second set is more restrictive
than (9.1.16). (Rotach's notation differs from ours; we must write 2 = 2^x.)

[ 9.2 ] JACOBI, LAGUERRE, AND HERMITE SERIES 251
expansions. There is an extensive further literature on this subject; E. R.
Neumann A), Galbrun A), Wigert A), Hille A, 2, 3), Cramer A), Uspensky A),
Korous A, 2), Stone A), Miintz A), and Kowallik A) have all given direct
treatments of Laguerre and Hermite expansions of an "arbitrary" function.
(Concerning the expansion of an analytic function into Hermite series see Wat-
Watson 1 (first paper).) These authors have obtained convergence and summability
results but.no equiconvergence theorems, except Korous (loc. cit.). The con-
conditions which all these authors use at x = =c are more restrictive than those in
our theorems. For instance, in case of Laguerre expansions, the convergence of
(9.11.10)
is required by Korous; for Hermite expansions, convergence of the integrals
(9.11.11) I e-x2\f(±x)\2dx, ^ e~xix\f{±x)fdx
are required by Uspensky and Kowallik, respectively. In addition, we mention
the earlier treatment of Laguerre and Hermite expansions by means of the
theory of integral equations (Myller-Lebedeff 1, Weyl 1).
Concerning Kogbetliantz 22, 23, 24, see the footnote in §8.91. Concerning
Theorem 9.1.7, see Szego 10 and Kogbetliantz 10. The condition regarding
(9.1.19) is satisfied if
(9.11.12) f(x) = OVx"—*-*), 8 > 0, x -> +«.
On the other hand, the series is not (C, A:)-sumiriable for the function f(x) =
exl2xk~a, k > a + 1/2 (cf. §9.6 C)).
Gibbs' phenomenon in the case of Hermite expansions has been studied by
Jacob B).
9.2. Expansion of an analytic function in Jacobi, Laguerre, and Hermite series
A) Theorem 9.2.1. Assume a > -1, 0 > -1, and let P(na'®(x), Qia'0)(x),
and h(na'® have the same meaning as in Chapter TV (see §4.61 and D.3.3)); then
(9.2.1) ± i^i"^) ikjzJL^L+lLl
±
± {i}i\)Q^(y) i,
«=o I y — x
where x lies in the interior of, and y in the exterior of, an arbitrary ellipse with foci
at ±1. This expansion holds uniformly if x and y belong to closed sets which are,
respectively, in the interior and exterior of the ellipse mentioned.
The functions (y - l)a(y + lHQia'0\y) are single-valued and regular in the
complex ?/-plane cut along [—1, +1].
This important formula is well-known in the special case a = /3 = 0 (Heine 3,
p. 78). The general formula is obtained from the identity D.62.19) by letting
n approach «. We write x = *(« + z'1), y = i(f + f); K i 2I < I f I •

252 EXPANSION PROBLEMS FOR THE CLASSICAL POLYNOMIALS [ IX ]
use of (8.23.1) and (8.23.2) we find the "remainder term" of D.62.19) to be
2-"-f) T(n + 2)r(n + a + 0 + 2)
2ft + a + 0 + 2 T(n + a + l)T(n + 0 + 1)
x — y
as ft —* oo, with e > 0 arbitrarily small. This tends to 0 as n —> °° provided e is
sufficiently small.
B) Now let/(x) be regular if x is in the interior of the ellipse | z | = R > 1.
On multiplying (9.2.1) by (Tri)~\y - l)a(y + l)pf(y) and integrating over
the ellipse | f | = i2 — e, 0<e< i2/2, we obtain an expansion of f(x) in a Jacobi
series which is uniformly convergent for | z | < R — 2e. The usual term-by-
term integration over the segment [ —1, +1] identifies this expansion with the*
expansion (9.1.1), and for the coefficients an we obtain the representation
(9.2.3)
n = 0, 1, 2, • • • ,
where the integration is extended over the ellipse | f | = R — e in the positive
sense.
C) By means of (8.23.1) we can discuss formal series of the type
(9.2.4) aoP(oa*\x) + aiP{°"w(x) + a*Pi"'m(x) +¦¦•+ anP(na'P)(x) + ¦¦• ,
which are not necessarily Fourier expansions in the ordinary sense. Let R
have the same meaning as in (9.1.4), and assume R > 1. Then (9.2.4) has as its
domain of convergence the ellipse of Theorem 9.1.1. In the interior of this el-
ellipse it represents an analytic function.
D) Expansions in terms of Jacobi's functions of the second kind, which are
the analogues of Laurent expansions, can also be readily discussed.
Theorem 9.2.2. Assume a > — 1, /3 > -1, and let F(y) be regular at y =
oo with F(oo) = 0. Then
(y - l) ' jW iW iS\
(925)
This expansion is convergent in the exterior of the smallest ellipse with foci at ± 1,
in the exterior of which F(y) is regular. It is divergent in the interior of this ellipse.
The sum of the semi-axes of this ellipse is given by
(9.2.6) p = lim sup | bn
n-»w
Obviously, p ^ 1. In case p = 1 the statement needs a slight modification.
The following representation holds for the coefficients:

[9.3 ] PROOF OF THEOREM 9.1.2 253
(9.2.7) bn = Ivihl"-*)-1 J F(x)Pla^(x)dx, n = 0,1,2, ¦¦¦,
where the integration is extended over the ellipse | z | = p + e, x = \{z + z'1),
e > 0, in the positive sense. For the proof we again use (9.2.1).
Formal series proceeding in terms of Jacobi's functions of the second kind can
also be discussed.
E) The boundary of the convergence domain of a Laguerre series
(9.2.8) aMa)(x) + aiLia)(x) + a*tia\x) + ¦¦¦ + anL(na)(x) + ¦¦¦
can be characterized by the condition 9?{(—x)h) = const. Therefore, this
boundary is a parabola with its focus at the origin. The series is convergent in
the "interior" of this parabola and divergent in its "exterior". The analogue
of Cauchy-Hadamard's formula holds. The proof is based, on (8.23.3).
For an Hermite series
(9.2.9) a0H0(x) + cnH^x) + a2H2(x) + • • • + anHn(x) + • • •
the corresponding condition is | Q(x) \ = const., which defines a strip with the
real axis as axis of symmetry (cf. (8.23.4)). The series is convergent and di-
divergent in the "interior" and in the "exterior" of this strip, respectively. The
analogue of Cauchy-Hadamard's formula holds again.
Concerning the expansion in an Hermite series of a given analytic function
regular in the strip | Q(x) | ^ a, a > 0, see Watson 1 (first paper).
9.3. Proof of Theorem 9.1.2
A) First let us replace f(x) by a polynomial p(x). Then sn(x) = p(x) if n
exceeds the degree of p(x). Furthermore, A — x)'2^^ + x)~^/2~isB(a;) —* p(x)
as n —¦ oc, according to elementary tests for the convergence of Fourier series
(see for instance Zygmund 2, p. 25). Now the integral
' (l OM + if IM »I * \l - min ("' it ~
(9.3.D f(l - OM + if IM - p»I <*, \l it }?
7-i Xb = min (/3, 0/2 - 1/4),
can be made arbitrarily small by proper choice of p(x) (cf. Theorem 1.5.2). It is
therefore sufficient to show that the difference (9.11.4) admits an estimate
(9.3.2) 0A) jT A - t)a(l + t)b |/@ I dt + 0(l), n -* «,
where both bounds hold uniformly in i, - 1 + e S i ^ 1 - e, and 0A) is
independent of f{x).
B) In subsequent considerations we use Darboux's formula (8.21.10) for
P(na '^'(cos 6), as well as the second formula (8.8.2) for the difference ratio. In
the latter case we assume that both arguments 0i , 62 lie in an interval which is
entirely in the interior of [0, ¦*].
According to D.5.2), we have

254 EXPANSION PROBLEMS FOR THE CLASSICAL POLYNOMIALS [ IX ]
(9.3.3) ( p(«.«M p(<*,0)/ \ p(<*,0)/.\ r,(a,0)( \
= x. yn ix) r_ pn+1 (x) rzri
where
(9.3.4) Xn = 2-°-l)-1{n + 0A)}.
Writing x = cos d, t = cos <j>, and using the notation of (8.21.10), we find for
-1 < x < +1, -1 < t < +1,
*<"«(*, t) = 2-^'*(« Wcos
( cos <j> — cos d
- cos [{N + 1H + y] cos (N<j> + 7) - cos (N8 + 7) + 0{1)\
cos <? — cos 6 J
= 2—g-'fc^jfc^) (cos [Ar(^ + g) + «^ + 27] + cos [JV@ - 6)
(935)
cos <? — cos 5
cos [N(<j> + 6) + fl + 2y] + cos [#@ - fl) - 6)
cos 4> — cos fl
sin
sin
Now assume — l + e^z^l — e. Then
2°+l)+1 (si* | j (^cos * j /(cos 0)lfi-« (x, 0 d*,
where cos 17 = 1 — e/2. Next replace Kna'p)(x, t) by (9.3.5). By Riemann's
lemma (Zygmund 2, p. 18), the result will be
1 x-, sin Bn
/(cos <?) d4> + 0A)
(9.3.6) V*J* .6-6
as w —> °o.
The bound of the term 0A) is independent of f(x). When we replace a and /3
by -|, and/@ by A - 0"/2+i(l + QW8+*/@» repeated application of Riemann's
lemma yields

r9.3
PROOF OF THEOREM 9.1.2
255
ix, t) dt
/'
A -
sin Bn + 1)'
sin
-6
(9.3.7)
rr
/
Ji
+ 0A)
/(cos
/l-t/2
1/@
l+«/2
dt
. 4> — e
sin-2-
|
-l + t/2
Consequently, the part of (9.11.4) which corresponds to - 1 + e/2
1 - e/2 has the form (9.3.2).
C) We consider now the expressions
(9.3.8) (Kn)\Plnajn(x)\ f
Jl-t/2
n+1
+1
(9.3.9)
Jl-t/2
n+1 W — -Tn+1 (X)
t - x
and the corresponding integrals extended over — 1 ^ t ^ — 1 + e/2. The first
integral is
= 0{nH(n*) P
Jo
Pi"+f(x) |}A - t)a(l + 0" 1/@ I*
i+? (cos 4>) \ ^2a+1 |/(cos 4>) \ d$
r
From G.32.6) and G.32.7) it follows that the first term of the right-hand mem-
member is
0A)
1/(cos 4>) | d4> = 0A) jH A - 0 """*(! + 0"'2"* 1/@ I dt,
or is

25G EXPANSION PROBLEMS FOR THE CLASSICAL POLYNOMIALS [ IX ]
0A) f\2a+1 |/(cos 0) d* = 0A) (+l A - 0"(l + 0" 1/@ I dt,
Jo J-i
according as a ^ -§ or a ^ -§. In view of D.1.7) the expression (9.3.9) is
A - 0a/2~*(l + 0"' 1/@ I dt
Jl
l-t/2
= 0A) jT\l - t)al2-\l + jf2"' 1/@
The integrals corresponding to —1 ^ t ^ — 1 + e/2 are similarly treated.
D) Finally, we consider the Jacobi series of the function (see, in the special
case a = /3, Kogbetliantz 19, p. 184, F2))
(9.3.10) f(x) = A - xy,
and we show that for proper values of m the first integral in (9.1.5) exists, but
that the second does not. Moreover, the Jacobi series is divergent in — 1 < x <
+1. Here a > -|.
To this end we note that, according to D.3.1) and D.3.3),
l r\i -
t+l (l - 0n+a(i + t)n+\l - tY~ndt
—2u—a—1
~ n as n —> °o .
If we assume that m + a > — 1, fx ^ 0,1, 2, • • • , and take 0 < 6 < x, the general
term of the Jacobi series has the form n~2"~a~i cos (Nd + 7). The required
values of m are those which satisfy the condition
(9.3.12) _l_a<M^ -a/2- 3/4.
9.4. Proof of Theorem 9.1.3; preliminaries
A) We first derive the following important identity:
p(«+*+i,0)/_\ _ 9<*+0+ir/ 1 ^ r(w + ff + 1)
rn {x) ~ t Ha + UT(n + a+-p + k + 2)
(9.4.1)
*p T(n + v + a + 0 + k + 2) «(A) ,, (a,w . _i p(a,
y-o F(n + j/ + a + /3+2)
where /jI^ and C^ have the usual meaning (see D.3.3) and the introduction to

[ 9.4 ] PROOF OP1 THEOREM 9.1.3; PRELIMINARIES 257
Chapter IX). This is a generalization of D.5.3) (last identity), to which it re-
reduces when k = 0.
For the proof we calculate
(9.4.2) jf" Pi°+k+^ (s)p,<«.« (x) (i _ xy{1 + x)t> dx_
According to D.3.1), this last expression is equal to
(i - rt + x)*»)dx
J
or, by D.21.7) and D.3.1), it is equal to
~ ^ (n + a + 0 + ft + 2)(n + a + 0 + ft + 3) • • • (n + a + 0 + ft + * + 1)
- xY+a(l + x
(-!)""" T(n + a + $ + k + y + 2)
2n+'v\(n - v)\ T(n + a + 0 + k + 2)
"{(l - x)n+a+k+1(l
(ft + n - v) T{n + a + & + k + v + 2)
2n+'v\{n-v)\ T(n + a + 0 + ft + 2)
which, if A.7.5) is taken into account, furnishes the statement.
B) On substituting the explicit expressions for P?"lW(l) and h\"'m (see D.1.1)
and D.3.3)), we obtain
/3a = r(n + 0 + 1) ^ T(n + v + a + l3 + k + 2)
p3 = ^
K ' r(n + « + 0 + ft + 2) ,"=0 r(n + * + a + 0 + 2)
(9.4.3)
•C^B, + a + 0 + 1) r(;+^+^+I) P^C
Since this is an identity in a, /3, k, we may replace a by a + ft + 1 and ft by
— ft — 2. Whence the inversion formula
P<«'»f ^ T(n + /3 + 1) y r(n + v + a + 0 + 1)
n W r(n + a + 0 + 1) ^ r(n + ^ + a + 0 + ft + 3)
• c?^> B, + a + 0 + ft + 2) r0/r(r+^i)+2) p'"

258 EXPANSION PROBLEMS FOR THE CLASSICAL POLYNOMIALS [ IX
follows. Consequently,
m—0
(9.4.5) = f^+T) 5 GM k)Bv + « + P+k + 2)
with
(9.4.6) G,(n, k) = Z C(klm Ci"l*~2)Bw + a
The expression S(k\x) represents the numerator of the nth Cesaro kernel of
order k.
9.41. Continuation; the Lebesgue constants of order k
A) For the proof of Theorem 9.1.3, according to Theorem 1.6 (Helly's the-
theorem), it is suflBcient to show that the sequence of the "Lebesgue constants"
(9.41.1) Vnk) = {C^r1 I A - s)a(l + xY \S?\x) \dx,
is bounded if and only if k > a + 1/2. Since for k > a + \ (cf. G.34.1), first
and third case),
/:
-x)a{\ +xY\Pia+k+1^{x)\dx
(9.41.2) = 0(l) TA _ x)a | pia+wr^fc + 0A) rA _ xy j P
Jo Jo
= 0{nk—1) + O(n"J) = 0{nk-°-1),
we have from (9.4.5)
(9.41.3) Lik) = 0{rTk) E I G,(n, k) \ vu+\
The last factor v2k+1 must be replaced by 1 for v = 0.
B) First let k be a non-negative integer. Then in (9.4.6) we must consider
only the terms v^m^v + k + l,so that G,(n, k) can be written in the form
G(n - v - 1), where55
" If v + A: + 1 > n, certain additional terms not permitted by (9.4.6) occur in G(u).
However, these terms vanish if we substitute u = n — v — 1, except if v — n and m =
n + k + 1. Now the contribution of this term to G(u) = G( — 1) is 0(m)(m + v)~k~* =
OCn"*)- This multiplied by the factor eu+1 = nu+\ which occurs in (9.41.3), furnishes
a total contributionO(n-*)O(n*) = 0A).

9.41 ] CONTINUATION; THE LEBESGUE CONSTANTS OF ORDER k 259
G{u) = E (-Dm v(U + k + \ m + V)
(9.41.4)
" ' T(m + v + a + 0 + k + 3)-
This is a x* in w, containing v and A; as parameters. Newton's formula shows us
(cf. MarkofT5, p. 15; notation as in B.8.4)) that
(9.41.5) G{u) = E(
and from (9.41.4) it follows that
— p )\m ~ V)
T(m + v + a + 0 + 1)
¦ Bm + a + 0 + 1)
+ * + « + 0 + & + 3)
-(k-Py.J0 ?i (m-y)i(p + i-m + v)V u ;
lf v+p+1 .m+v+a+P/-. t\k+l
;_1)m-,-i
(9.41.6) (A; - p)! ;0 \ mfn-iV ' (m-y- 1)!(P + 1 - m + v)l
y+f+1
+ Bv + a + -0 + 1) E (
1 / I 9/ 1
(k - p)!(p + l)! Jo 1
? L
(m — v)!
t U — t)
Bv + a + 0
The last integral formula also shows that AkG@) = 0, v ^ 1; in particular,
G(u) = 0 for k = 0, v ^ 1. Both facts hold also for v = 0, as analytic con-
continuation with respect to a + /3 shows. This settles the case k = 0 in view of the
remark in the last footnote. For k > 0,
(9.41.7) Gv{n, k) = E (» - ^'O^^2),
p-0
to which (according to the same remark) the term O(n~k~l) must be added if
v — n. In this case the factor (n — v)", in the case v = 0 the factor »> " ,
must be replaced by 1. For k > a + \, this provides, as desired,
fc-l n-l A-l
p-0 »=1 p=0
(9.41.8) ,
+ 0@ E n-^"-2n2i+1 + n-^n2^1 = 0A).
L-o J

2G0 EXPANSION PROBLEMS FOR THE CLASSICAL POLYNOMIALS [ IX ]
C) We now consider the case k > a + \, k not an integer. Then, according
to the previous result,
(9.41.9) Lik'} ^ A, .n=0,l,2,...,
where k' = [k] + 1, and A is a proper constant independent of n.
Let a > k'. If we write un(x) = {h(n"-0)}~1 P(na^(l)P(na^(x), the definition
(9.4.5) furnishes
gSf'OrK = (] - r)-*'-xi: un(x)rn.
Thus,
00 00 00
so that on \X) == ? jmnft G n m &m \x) and
n n
l-'n = (^ n i /L-d *-" n—m ^m i^m =s "¦ [^ n I jL-i ^ n—m ^m
Consequently,
(9.41.10) Lia) ? A, <r = &', n = 0, 1, 2, • • • .
This holds in particular ifa = /t + l,/t + 2, •¦•.
D) Now we prove the identity
T(n + v + a + 13 + k + 2) (A) = ?Bn_+ a + 0 + 2fc + 3) A , y
T(n + v + a + 0 + 2) "~" TBn +~a + 0 + Jfc + 3) ??a
(9.41.11) /
A + p\cu+P)__ k(k - 1) ¦ • • (k - p + 1)
2n + a + j8 + fc+3) ••• Bn + a+ 0 +H-p + 2),
0 ^ j/ ^ n.
The series on the right converges absolutely if n Si 1. (For p = 0, the last
fraction is to be replaced by 1.) For the proof we use the following trans-
transformation of the left-hand member:
(n - v)\T(k + 1) Jo
w + q + /3 + 2ft + 3) 1_ _
T(n + v + a + 0 + 2) (n ->0!r(& +1)
tn-v+k+"{\ -ty+v+a+f)+1dt.
Calculation of the last integral completes the proof. Term-by-term integra-
integration is permitted, since the terms are, save for a finite number of exceptions,
all of the same sign.

[ 9.41 ] CONTINUATION; THE LEBESGUE CONSTANTS OF ORDER k 201
Consequently, from (9.4.1) we have the representation
(9.41.12) *r(n + a + 0 + k + 2)rBn + a + /3 + fc + 3)
k(k - 1) • • • (k - P + 1)
Bn + a + 0 + k + 3) • • • Bn + a + 0 + k + p + 2) "
or
r(n + 0 + l)rBn + a + 0 + 2fc + 3) \"x d(«+*+i.«
T(n -f a + 0 + & + 2)rBn + a + 0 + & + ;
'(*)
(9.41.13)
- 1) • • • {k - P + 1)
' Bn + a + 0 + k + 3) • • • Bn + a + 0 + Jfc + p + 2)
On account of (9.41.2), this gives
(9.41.14) Lf = O(n-k)O(na+1)O(nk-a-1) + O(rCk)An = 0A) + 0(n~*LB,
where
(9.41.15)
Now, according to (9.41.10),
-x)a(l+xY\S<nk+")(x)\dx.
- 1) • • • (k - p +1)
Bn + a + 0 + A + 3) • • • Bn + a + 0 + & + p + 2) "
(n + A + 1) • • • (n + k + p)
Bn + a + 0 + A + 3) • • • Bn + a + 0 + k + p + 2)
= O(nk)
+ 2) • • • (k + p + 1)
= 0{nk),
since (n + A; + l)/Bn + a + /3 + fc + Z + 2) decreases as n increases provided
j>Q,_j_0_?-j_2. This completes the proof of the first part of the statement
in Theorem 9.1.3.

262 EXPANSION PROBLEMS FOR THE CLASSICAL POLYNOMIALS [ IX
E) Finally, let k = a + ^. Using the previous notation, we have
w , In^-ip,. , ^ rOj + 0+i)rB» + « + /3 + 2fc + 3) w
7u-)
r(n + a + /3 + Ar + 2)rBn
'(I - z)a(l + *)"
from (9.41.13). The first term on the right is ~ na+1n *n * log n = log n (ac-
(according to the second part of G.34.1H6); the second term is bounded, according
to the result of D). That is,
L[k) > A log n,
(9.41.17)
A > 0, k = a + \f
so that the expansion of a continuous function in Jacobi series is, in general, not
(C, k = a + |)-summable.
9.42. Proof of Theorem 9.1.4
A) Let f(x) be continuous at x = 1, let /(I) = 0, and assume k > a + |.
First we discuss the integral
(9.42.1)
r+
L
as n —+ oc. Denoting by e an arbitrary positive number, e < w/2, we decompose
the interval 0 ^ 6 ^ 7r, a; = cos 6, into
(9.42.2) 0 ^ 0 ^ e,
« ^ C ^ 7T - 6,
7T — € ^ V ^ 7T.
The corresponding integrals I, II, III, can be estimated as follows (cf. (9.41.2),
G.32.6), and G.32.7)):
I = max j /(cos 6) \
- *ra
= max
III =
|/(cos 0)
= O(nh a x) / I /(cos 6») I (?r - 6Kff+1 dd,
I/(cos 0) I Or - 0J/3+1Or - ^-^OOT*)^ = O(n-i) = o{nk~a-1),
66 If we use only G.34.2), we obtain L(nk) ^»asn-+»; this is sufficient for our purpose.

t 9.42 ] PROOF OF THEOREM 9.1.4 263
according as -1 < P ? -$, -$ < p ? k - a - 1, or p > k - a - 1. In
the last case we use the antipole condition of Theorem 9.1.4. Since e is arbi-
arbitrarily small, it appears that in all cases the integral (9.42.1) is o{nk~<x~1).
B) Next we introduce the constants
(9423) ^ = ^
n = 0, 1,2, ... ,
where Sf}(x) has the same meaning as in (9.4.5). We obtain the following
analogue of (9.41.3):
M(nk) = 0{n~k) E I G,(n, k) \ o(v2
The last factor o(v2k+1) must be replaced by 1 for v = 0. Therefore, if k is any
positive integer, we have as in (9.41.8),
M™ = O(n'k) E n" + O(n~k) E E (n - vYv'"'"-2o(v2k+1)
The same holds if k = 0, for then G(u) s 0 (see the remark in the footnote of
§9.41 B)).
C) For non-integral k we use (9.41.13) again. Let e be an arbitrary positive
number, and let n0 be so chosen that, for k' = [k] -\- 1,
(9.42.5) M(k>) ^ e if n ^ n0 .
Then for n ^ no, a > k' we find, as in §9.41 C),
m=0
We decompose the latter sum into the sums E'™^1 and E»'="o • ^n the sec-
second part (9.42.5) can be applied so that
no—1 n
1YJL n ^~ [ O 7i j / f O 7i —in O m -itJ. m ~\~ ^ \ ^ n j / j *-* n —m ^ wi
The second term of the right-hand member is less than €, Therefore,
(9.42.6) M[a) < e + AiC^T'Ci'-"'-",
where A is a positive constant depending on e and independent of <x. This
holds in particular if a = k + 1, k + 2, ¦ ¦ ¦ . We must replace Cn""*'5 by
CnT^+15 if a < k' + 1, which occurs when <r = k + 1, but not when <x ^ & + 2.
Consequently, we obtain, as in (9.41.14),

2G4 EXPANSION PROBLEMS FOR THE CLASSICAL POLYNOMIALS [ IX ]
O(jTk)O{na+l)o{nk
O(n~k)
-K V
[Hk -_^)^-^(^z p±J2L
Bn + « + 0 + k + 3)~- •"• Bn -fa~+ 0 +"ft+'p~+~2)
+
(9.42.7)
(For p = 1 we must replace &k+p k' u by C^^+i'.) The first term of the
right-hand member is o(l). The second term can be decomposed into two parts.
The first part has the form eO(l), according to §9.41 D). To estimate the
second part we use the fact that
F{n + k + /
«- ;n" t c t l) t.
r(n + ft + p +"i)r(ft~+ p -~ft') " ~ Vn + ft + P,
where B > 0 depends only on ft. For the second part we therefore obtain the
bound (cf. (9.41.16))
0A) ^
00
(n + ft + 1) • • • (n + k^±_p)__ _
Bn + a + 0 + ft + 3) ¦ ¦ • Bn + a + 0 + k + P + 2)
A'+l
ft + P
n + ft + p
Decomposing this sum into the sums X)p=i and X)"=p+i > where P is an arbitrary
positive integer, we obtain the bound
*'+i
(ft + 2) ¦ • • (ft + p + 1)
The first term tends to zero asn-> «. The second term is arbitrarily small if
P is sufficiently large. Therefore M^ = o(l) asn-> «.
Concerning the case ft = a + \, compare §9.41 E).
D) Remark. The continuity at x = +1 can be replaced by the more general
condition
(9.42.8)
I |/(cos 6) - /(I) | d0 = o(8),
./o
+0.
Only the estimation of the integral I (cf. (9.42.2)) must be slightly modified.
We have, by use of G.32.5), if/(I) = 0,
I = O(ri
a+k+l\
e2a+1\f(cose)\de
d°-k-\ |^cos ^1 ^
In both integrals we integrate by parts (cf. Fejer 8, p. 280). Let

9.42 ] PROOF OF THEOREM 9.1.4 265
/ \ f(cos t)\dt = FF).
Jo
Then we find
I = O(na+h+1)n-2a~1F(n-1) + 0(na+k+1) f" $2aF(d) dd
Jo
+ O(rri){ea-k-iF(e) + ^""^(n)} + 0(n-*) /"' ea~k^F{e)d6
Jn 1
max
+ O^4-0-1) max
E) Finally, we show that the assertion of Theorem 9.1.4 does not hold in
general if the "antipole condition" is not satisfied. We consider the function
(cf. §9.3 D))
(9.42.9) /(*) = A + x)".
Its expansion at the point x ~ 1 is
(9.42.10)
According to (9.3.11), up to a fixed constant nonzero factor, the principal part
of the general term of (9.42.10) is
(9.42.11) (-lJV-*-*1-1, or (-lyci"-^2"-^.
But (9.42.10) cannot be (C, fc)-summable if k ^ a - p - 2m -1=X. Indeed,
(9.42.12) A - r)-*-1 ? (~l)nC™rn = A - ^-^(l + r)""
0
n-o \ n //_i
-0
Darboux's method (§8.4) yields for the coefficient of rn in the power series
expansion of this function, the principal term
(9.42.13) ACik) + B(-l)nC™,
where A and B are fixed constants, different from zero.
Now let (9.1.9) be satisfied. If we take
(9.42.14) -13 - 1 < M ^ i(« ~ P ~ k - 1),
the Jacobi series exists, the integral (9.1.10) is divergent, (since we have
j(a _ ? _ k - 1) < -0/2 - 3/4), and the series (9.42.10) is not (C, fc)-sum-
mable.

266 EXPANSION PROBLEMS FOR THE CLASSICAL POLYNOMIALS [ IX ]
9.5. Proof of Theorems 9.1.5 and 9.1.6
A) We start from the representation
(9.5.1) «„(*)= [ e" taf(t)K{°\x, t)dt
Jo
of the nth partial sum of the Laguerre series of the function/(a:), where the
"kernel" K(na)(x, t) has the meaning in E.1.11). We write the formula given
there in the more convenient form
n
(a—1) /.\ t (a
Jn + l \t) - L\^
rC ~~" V
(9.5.2)
(nta)
x - t
L(a) /,\ t (a) / \
. ,- .,, . n + lKl) — Ln + l{X)
Ln+l W -—t
This follows from E.1.11) if for L(na)(x) we substitute Z/na+i(z) - L{n+i\x),
and similarly for L(na\t) (cf. E.1.13)).
First we assume that the integrals (9.1.11) exist and that the condition (9.1.12)
is satisfied. Then the first integral (9.1.14) exists (cf. the remark in §9.11 D)).
Let fix) be a polynomial p(x); then the statement (9.1.13) is true. Therefore,
according to the closure property pointed out in Theorem 5.7.3, it suffices to
show that the difference in (9.1.13) admits an estimate of the form
(9.5.3) 0A) f f | f(t) | dt + 0A) f e-? V1-* | fit) | dt + o(l), n — oo,
where a = min (a, a/2 — 1/4), and the bounds 0A) and o(l) hold uniformly in
x, e ^ x ^ a). Furthermore, both factors 0A) are independent of fix).
B) Let us consider the contribution to sn(z) of the interval O^i^ e/2. In
accordance with the first formula (9.5.2) and G.6.9), for a ^ \ this is
Oinl~a)
(9.5.4) = 0A) f r/2+i | fit) | dt + 0A) f r/2-J | /(o | dt
Jo Jo
= 0A)
If _ i ^ a < I we use G.6.9) and G.6.10); if a < - |, we use G.6.10). In the
first case the result (9.5.4) remains valid, while in the second case we obtain
0A) jUa\fit)\dt.

[ 9.5 ] PROOF OF THEOREMS 9.1.5 AND 9.1.6 267
C) We next consider the contribution of the interval e/2 ^ ( ^ 2a>, and we
apply (8.8.3) and (8.8.5) to the second formula (9.5.2). Here the variables are
confined to a fixed positive interval; so the remainders in (8.8.3) and (8.8.5)
depend only on n. We find (notation as in (8.8.3))
Kia)(x,t) = ni-n-»-*n<-"/»-i*
(g 5 5) •{<» cos [2H' + 7] sinj2(nO^ + ^-s
- *» sin [2(n*)» + y] C°S [2(nt) -±-7j " C°S [2M +-l! + 0AI
t1 — a:1 J
The mean-value theorem allows us to replace n' = n + 1 by n. Suppose,
for instance, that 0(m, t) = cos Bm} t + 7). Then
[<t>(n'} t) - <t>(n, t)] - [0(w;, x) - <t>(n, x)] = ^0
taken at a proper place ?n, ?, where m is between ?i and n', and J is between x
and <. This readily furnishes
K{:\x, t) =
(9.5.6) - sin [2(nz)> + 7] gggjg^+_?] ~ ^os [2(n«)* + 7]
^ — a:'
Thus, according to Riemann's lemma, if e < 1 < w,
e/2 Ji/2
+0A) r\f(o\di
(9.5.7) Jt/2
x'i-6 T — Xi
j 10< + 0A).
D) In the interval 2w ^ < ^ 3» (n large), we have, according to the first
statement (8.91.2),
(9.5.8) e-tl2tal2+i I Lia)(t) I = 0(na/*-»).
Consequently, from the first formula (9.5.2) it follows that

268 EXPANSION PROBLEMS FOR THE CLASSICAL POLYNOMIALS [ IX ]
J2u
(9.5.9)
fZn
i /-)/ l-a\ (a—l)/2-i I —t.a—1 I f/.\ I I t (
+ U{n )n let \j{t) \ \ Ljn
= O(n1-a)na'2~in(a~v'2~i / e~1"'2^ \ f(t) \ dt
J2u
\t) \ at
which is equal to
(9.5.10) 0A) jfV'V'*-* \f(t)\dt.
In the interval 3n g t < +°°, Theorem 8.91.2 (z ^ a, X = a/2 + 1/12),
tells us ?hat
(9.5.11) e t" | Z/J/* (t) | = O(n° ).
Therefore, from E.1.11), on account of (9.1.12), we have
re~ttaf(t)Kl"\x,t)dt
(9.5.12) = 0(n1->a'*-i (" e~l f~l \ f(i) \ { | Lia)(t) | + | L(n%(t) \) dt
\f(t)\dt =o(l).
E) If the condition (9.1.12) is replaced by the requirements of (9.1.14), a
treatment of the interval 2a> g t S 3n, the same as that given in D), applies.
In 3n ^ t < + °° we use Schwarz's inequality:
(9.5.13)
The last integral is 0(n°) (see E.1.1)), and this establishes the statement.
F) Let
(9.5.14) f(x) = x"
(Blumenthal 1, pp. 32-33). We shall show that for proper values of m the
first integral in (9.1.11) exists, but that the second does not; furthermore the
Laguerre series is divergent for x > 0. Here a > — f.
The coefficient on of the corresponding Laguerre series is given by
r(« + 1) (n + °)an = f e-xxa+>Ll°\x) dx
(9.5.15) V n J J°

t 9.5 ] PROOF OF THEOREMS 9.1.5 AND 9.1.6 209
(see E.1.5)). On integrating by parts, we have
T(jt + a + 1) Tin - n)
-- f(-g-r(»+ . + !)•
Here we assume n + a > -l,/i ^ 0, 1, 2, ••• . Then on ~ ri~"~a~1, and in
view of Fej6r's formula (8.22.1), the principal term of the Laguerre series
behaves like
(9.5.17) n-*—/*-*/« cos {2(na.)» _ a7r/2 _ T/4}
for x > 0. This series is therefore divergent (cf. Problem 47) when and only
when m + «/2 + | ^ i If a > -f and
(9.5.18) -a - 1 < M ^ -a/2 - 3/4,
the first integral (9.1.11) exists, but the second does not, and the series is
divergent for a; > 0.
G) We consider also the function
(9.5.19) f(x) = ex/V,
and we show that for a proper value of m (particularly for n = —a/2 + 1/4) the
integrals (9.1.11) exist, but the conditions (9.1.12) and (9.1.14) are not satis-
satisfied; in addition, the Laguerre series is divergent for x > 0. Here a > — 1,
a + n > -1.
The integral
(9.5.20) r(« + 1) (n + ") on = jT e~xl2xa+"Lia\x) dx
can be calculated by means of the generating function (see E.1.9))
{ ) ( ) anrn = A -
(95-21) =r(a +
- r)*(l
Thus, Darboux's method (§8.4) gives for on the principal term
(9.5.22)
where ^L and fi are fixed constants, A ?± 0, B ?? 0. We have A = 0 if
M = 0, 1, 2, ....
If z > 0, this shows, on account of Fejer's formula (8.22.1), that aJL" (x)
does not tend to zero if m = —a/2 + 1/4.
(8) Finally, we discuss the Hermite series
(9.5.23) f{x) ~ aoHoix) + 0^1@;) + • • • + anHnix) + ¦ ¦ • .
Writing y = x2, we readily see that (see E.6.1))
Z) (hmHzmix) and
m~0

270 EXPANSION PROBLEMS FOR THE CLASSICAL POLYNOMIALS [ IX J
are the expansions of
{/(*) +f(-x)\/2 = {/(*,*) +/(_.y*)}/2 and
x~l\f(x) -f(-x)}/2 = jTW) -/(-y*)}/2
into a Laguerre series with the parameters a = -\ and a = +§, respectively.
Applying Theorem 9.1.5 to these functions with a = — § and a = +§, respec-
respectively, we obtain the conditions (9.1.11), (9.1.12), and (9.1.14) in the fol-
following form:
|T* l/(±z/) I dy = 2 / |/(±jc) | dx exists;
±y*)\dy = o(n-i), or [" e^x* \f{±x) \dx = o(rrl):
f
e"y V11 /(±2/J) \dy = 2C e~x2'2x~11 /(±a;) Irfa: exists;
or
= o(n).
This establishes (9.1.17), provided x belongs to an interval not containing
the origin.
In order to accomplish the proof, if x lies in an interval of the form [ — e,
+ e], we have only to show that
(9.5.24) e~l E Byv\iri)~1Hy(x)HM) - * U K — = 0A)
t^O X ~ t
uniformly if both x and t belong to [ —«, +«]. The first member of this dif-
difference is, according to E.5.9),
( II (A II (~\ II fA u f~\
fazo^ ron+1*,i h-^-^Jw , sHn\t) — tin{x) u ( ^ /Jn+iw ~ nn+i\x)
yv.o.Zo) \z niTT) e <,rtn+i(.x) — — tin\x) —
^ x — t x ~ t
Now, by using the the notation of Theorem 8.22.6, we obtain
x;Vx2/2#n(z) = cos (Nhx - nir/2) + O(n~h),
_i e~x2f2Hn(x) - e~l2/2Hn(t) _ cos (Nkx - niv/2) - cos (N*t - mr/2)
Xn ___________ .... x_t i
The second formula follows by an argument similar to that used in §8.8 (cf.
the first formula in E.5.10)). Its left member can also be written in the form
x - t
Replacing Nh by (N + c)*, c a fixed constant, we find the error committed in
the right-hand member to be 0A).
These asymptotic formulas furnish for (9.5.25), by use of Stirling's formula,

[9.6] PROOF OF THEOREM 9.1.7 27
Bn+1n! w^KK+ie*^" jsin (rfx - nvr/2)
cos (NH - mr/2) - cos {Nhx ~ rnv/2)
x ~ t
- cos (N>x ~ mr/2) *nJ
X - t
C?.n\h(r. _ t)\
h
X ~ t
9.6. Proof of Theorem 9.1.7
A) The Cesaro means of the Laguerre series at x = 0 can be represented in a
particularly simple form. For, from E.1.7),
oo oo /"oo
(9.6.1) ?anL(na)@) = }r(a+ I)} E / e~ltaf(t)Lia)(t) dt.
n-0 n-0 7°
Hence, by applying E.1.9), the Cesaro means of order k are found to be
(9.6.2) {Cik)r(a + I)} r e-'taf(t)Lia+k+1)(t) dt. ,
Jo
Assume k > a + |, and subdivide the interval [0, +°o] into [0, e], [e, a>],
and [a>, + »]. Then we find for (9.6.2)
0(rfk) max | f(t) \ [ ta \ L(na+k+1)(t) \ dt
(9.6.3) + 0(n~k) j" e-'ta | f(t) | | Lla+k+1\t)
0(n"*) r e~'r |/@ | | Llna+k+1\t)
By use of G.6.8) the first integral becomes
(9.6.4) 0A) fn tana+k+1dt
The second integral is 0(n(o+*+1)/2~i) = o{n). Now, using Theorem 8.91.2,
with a replaced by a + k + 1, and X = & + ^, we see that X — ^ =
k > (a + & + l)/2 — i; whence the third integral is
(9.6.5) 0A) f e-ta\f(t) | t~k'\kdt = 0(n*) /"" e~//2r-*-s |/@ I dt.
Thus (9.6.2) can be represented in the form
(9.6.6) 0A) max |/@ | + 0A) [" e'^t"-^ \f(t) | dt + o(l).
o y
Here the bounds of the terms 0A) are independent of e and a>. If it is assumed
that /(O) = 0, the statement is established.

272 EXPANSION PROBLEMS FOR THE CLASSICAL POLYNOMIALS [ IX ]
B) Let k = a + %, and apply Theorem 1.6 (Helly's theorem) to the linear
operations
Un(/) = {C(nk)r(a+ I)} f e-'iaf(t)Lla+k+1)(l)dt,
(9.6.7) Jo
U(/) = /@).
It suffices to show that the second condition in A.6.10) is not satisfied. As a
matter of fact, if this condition is not satisfied, then a continuous function
S(x), 0 ^ x ^ 1, exists for which
lim U.(/) =
does not hold. Extending the definition of this function to x > 1 by means
of the condition /(z) = 0, we obtain the "Gegenbeispiel" required. (By use
of the remark made in connection with Theorem 1.6, a continuous function
Six) can be constructed for which (9.6.2) is unbounded when n —» °o.)
Now if Q is a positive constant independent of n,
{Cik)Tia + I)} f e-f | Lia+k+1\t) | dt
> An* f "V | Lia+k+1)it) | dt,
Jo
where A is positive and independent of S2 and n. The last expression is, ac-
according to (8.1.8),
This integral becomes arbitrarily large with Q.
C) Integral (9.1.19) exists if
(9.6.9) f{x) = O(exl2xk-a-i~i), 8 > 0, x -» + °o.
On the other hand, there is no difficulty in proving that the Laguerre series
Z)n-o anLia)(x) of Six) = exk~a is not (C, &)-summable at x = 0. Here the
condition k > a + | is satisfied. Indeed, from (9.5.21) we obtain, n = k - a,
(9.6.10) A - TO"* ? anLia\0)rn = ?^4\ 0 H
n-0 T(a +1)
Darboux's method furnishes for the coefficient of r" in the expansion of this
function the number
(9.6.11) C(-1)V + oin), C> 0, n -> oo.
This establishes the statement.
We also notice that the Laguerre series is (C, /b)-summable at x = 0 with
an arbitrary k > a + 1/2, provided/(x) is continuous at a; = 0, and
(9.6.12) Six) =Oie*2x-m), rc-»+oo.
Again, the Laguerre series of the special function Six) = exl2xi is not (C, k)-
summable at x = 0 with any k > a + 1/2.

[9.7] FURTHER RESULTS 273
9.7. Further results
A) Using a generalized translation operator which can be defined by means
of D.10.21) it is possible to use Theorem 9.1.3 to show that the Jacobi series
of a continuous function is uniformly (C,k) summable for k > max(a + ?,/5 + f)
when a,/3^ - \, or a+/3^ — 1, and a,0> — 1. See Askey-Wainger 4 and
Gasper 3, 4.
B) Fora =0 ^ - \ Kogbetliantz 19 proved the positivity of the (C,2a +2)
means of the formal reproducing kernel. A simple proof was given in Askey-
Pollard 1. For the general Jacobi case it is likely that the (C,a +0+2) means
are positive when a,fi^ — |. This has been proven for max(/3 — 1, — 0) g
a g, p + 1 and max(^ - 2, 3 - 0) g a g 0 +2. See Askey 8.
C) Theorem 9.1.2 can be used with the Carleson-Hunt theorem (Carleson
1, Hunt 1) to obtain almost everywhere convergence of Jacobi series when
the function satisfies the conditions of both theorems.
D) Slightly stronger equiconvergence theorems for Laguerre and Hermite
series were obtained by Muckenhoupt 4. Again almost everywhere convergence
theorems follow from these results. These improvements and the mean con-
convergence results to be given below use Erdelyi's relatively recent asymptotic
formulas in an essential way. These estimates are described in 8.92 B).
E) If f\\f{x)\ "dx < oo, if f(x) is expanded in a series of Legendre poly-
polynomials, and if sn(x) denotes the partial sums of this series, then Pollard 1
proved that
lim f \f(x) -sn(x)\»dx=Q
n— a> J —1
for 4/3 <p <4 and these bounds are best possible (Newman-Rudin 1). This
theorem has been extended to Jacobi series (Pollard 2, 3, Muckenhoupt 5),
and Laguerre and Hermite series (Askey-Wainger 1, Muckenhoupt 2, 3). A
slight simplification in the proof was given in Askey 6, and this result and
the positivity of some Cesaro mean was used in this paper to solve a problem
of Lp convergence of Lagrange interpolation at the zeros of Jacobi polynomials.
F) There are a number of other interesting positive summability methods,
e.g. the analogues of Poisson-Abel summability (Bailey 1); Gauss-Weierstrass
summability (Bochner 2, Karlin-McGregor 3); and the de la Vallee-Poussin
method which uses Bateman's formula D.10.23). Horton 1 used this formula
to prove that the de la Vallee-Poussin method is positive and variation
diminishing.
G) Muckenhoupt-Stein 1 have introduced a number of the deeper functional
from classical Fourier series into the study of orthogonal polynomials and
were able to construct a theory of Hp spaces associated with singular Cauchy-
Riemann equations and prove an analogue of the Marcinkiewicz multiplier
theorem. Another proof of this theorem by mapping the theorem for Fourier
series to Jacobi series was given by Askey-Wainger 2 and Askey 3. Another
extension of the Marcinkiewicz theorem was proved by Bonami-Clerc 1. They
were motivated by work of Coifman-G. Weiss 1.
(8) Hirschman 1 has constructed a theory of variation diminishing trans-
transformations associated with orthogonal polynomial expansions which parallels
Schoenberg's classical theory.

CHAPTER X
REPRESENTATION OF POSITIVE FUNCTIONS
In the present chapter we deal with an extension of Fejer's representation of
non-negative trigonometric polynomials, given in §1.2, to certain general classes
of non-negative functions. In particular, we are interested in the discussion
of this representation if the given function is subjected to certain continuity
conditions. Extensions of FejeVs theorem in this direction are important for
the investigation of the asymptotic behavior of the general orthogonal poly-
polynomials associated with a distribution on a finite real interval or on the unit
circle (Chapter XI). It seemed convenient to separate these considerations
from the subject proper.
Concerning the results of this chapter see Szego 6, 7, 8, 9. See also Grenah-
der-Szego 1, 1.12-1.15.
10.1. Fatou's theorems
Theorem 10.1.1. Let fF) be integrable in Lebesgue's sense, and let
A0.1.1) f(r, 8) = ± I f(t) z o ' flN , . dt
2ir J-r 1 — 2r cos (t — 6) + r2
be the corresponding Poisson integral. Then we have almost everywhere in
— T ^ 6 ^ +7T,
A0.1.2) lim fir, 6) = f(d).
r—¦!—0
See Zygmund 2, p. 54, §3.442.
Theorem 10.1.2. Ijet
A0.1.3) F(z) = Co + ciz + C2Z2 + • • • + cnzn + • • •
be regular for \z\ < 1, and let the integral
A0.1.4) ~ f+'\F(reie)\2dd
Zif J—r
be bounded for r < 1. (This condition is equivalent to the convergence of
A0.1.5) | Co | + | ci | + | Ot | + • • • + | cn I +•••.)
Then we have almost everywhere in —ir ^ 6 ^ -\-ir
A0.1.6) lim F(reie) = F(eie).
r-»l—0
Furthermore, F(elS) is measurable, and \ F(etd) |2 is integrable in Lebesgue's sense.
The Fourier series of F(elS) is obtained by writing z = el° in A0.1.3).
274

[ 10.2 ] GENERALIZATION OF FEjfiR'S REPRESENTATION 275
This theorem is due to Fatou A). We say that F{z) is of the class H2. A
function F(z) which is regular and bounded in | z \ < 1 belongs to this class.
Concerning the more general classes H& , see F. Riesz 2 and Smirnoff 2. In
particular, if F{z) is of the class Hi, the boundary values F{eie) exist (almost
everywhere), | F(eie) \ is integrable in Lebesgue's sense and Cauchy's theorem
can be applied on | z \ = 1. (Cf. F. Riesz, loc. cit., p. 94, c); Smirnoff, loc. cit.,
pp. 337-338.)
10.2. Generalization of Fejer's representation
Concerning this section, see Szego 7.
A) Let gF) be a non-negative trigonometric polynomial not vanishing
identically. According to Theorem 1.2.2, there exists a polynomial h(z) of the
same degree, uniquely determined by the following conditions:
(a) g(d) = | h(z) |2, where z = eie;
A0.2.1) (b) h(z) is different from zero in | z \ < 1;
(c) h@) is real and positive.
We obviously have
A0.2.2) log g(8) = 2ft{log h(z)\, z = eie.
The function log h(z) is regular for | z \ ^ 1 except at the points z = el° which
correspond to the zeros of g{6), at which both functions log g{6) and log h(z)
become logarithmically infinite; log h@) is real. The function 29?{log h(z)\
is regular and harmonic for | z \ < 1 and has absolutely integrable boundary
values log gF). Applying the mean-value theorem of Gauss, we obtain
A0.2.3) i- j^ log g(8) dd = 2«R{log A@)} = 2 log h@),
so that
A0.2.4) {h@)\2 = exp l~ j_ ' log g(e) dd\ = ®(g).
The last expression is called the geometric mean of. the function g{6).
B) The relation A0.2.2) enables us to extend this consideration to arbitrary
non-negative functions/@) (instead of g{6)), defined in [ — w, +ir] and integrable
in Lebesgue's sense, provided ®(/) > 0. This last condition is equivalent to
the existence of the integral
A0.2.5) [+T log/@) dB = I log fF)dd+ f log f(8)d0.
J-r >g/(9)gl Jf(B)>\
The existence of the second integral on the right follows from the integrability
of f{6). Thus, the condition ®(/) > 0 is equivalent to the existence of the
first integral on the right-hand side. This is a restriction on the "nearness"

276 REPRESENTATION OF POSITIVE FUNCTIONS [ X ]
of /@) to 0. A consequence of this condition is that /@) can vanish only on a
set of measure zero.
We now introduce the harmonic function u(ree) defined by Poisson's integral
of *log/@)
A0.2.6) u(reie) = J- [+* log/@ * "/ (ft, 0 ^ r < 1.
47r 7-r 1 — 2r cos @ — ?) + r2
In the case of a continuous function /@) > 0, we know that (see for instance
Zygmund 2, p. 51)
A0.2.7) lim u(reie) = |log/@),
r-*l-0
uniformly in 0. In the general case considered above, this is true only with
the exception of a set of measure zero and without uniformity in general (see
Theorem 10.1.2). If now u is completed to an analytic function u + iv = k{z)
with the condition that k@) is real, then k(z) is uniquely determined. Writing
D{z) = ekW, we obtain the analogue (generalization) of the function h(z) con-
considered before. This function D(z) = D(J; z) has the following properties
(Szego 7, p. 237):
(a') D(z) is of the class i72 (§10.1); almost everywhere in — w ^ 0 ^ +7r,
A0.2.8) lim D(reie) = D(eie) exists, and/@) = | D(eie) |2;
r—1-0
(b') D{z) 9± 0 in \z \ < 1;
(c') D@) is real and positive.
We have again {D@)}2 = ®C0; this is obvious from A0.2.6). Furthermore,
we show that for an arbitrary continuous function FF) of period 2w,
A0.2.9) lim [ ' F(d) \ D(reie) |2 dd = f ' F@)/@) dd.
r-»l—0 J—r J—r
If D(z) = do + dyz + d2z2 + • • • , according to Schwarz's inequality, we have
W")|2-
^ j**(\D{eie)\ - \D(reie)\r dd- J_^ (\D(eie)\ + \D(reie)\Jdd
^ 2 ["{Die'6) - D(reie)\2dd- f+' (\ D(eie)\2 + \ D(reie)\2) dd
J-r ' J-r
r
n—1
for r —> 1 - 0. Here (a') is used. (See also Smirnoff 2, p. 338.)

[ 10.3 ] FURTHER STUDY OF POSITIVE FUNCTIONS 277
It should be observed that the function D(z) is not uniquely determined by
the conditions (a'), (b'), and (c'). For instance, we can multiply it by
exp {-A + z)/(l - z)}. For this point see Szego 7, p. 241.
C) By means of A0.2.6) we can obtain the following explicit representation
of D{z) in terms of /@):
\z\ < 1.
A0.2.10) D(z) = D(f; z) = exp (-L f+' log f(t) ] + Ze~\ dt\,
[ftir J-, 1 — ze~" J
If /@) is an even function, the expansion of D(z) around z = 0 has real coeffi-
coefficients. Let /i@) and /2@) be arbitrary functions satisfying the same condi-
conditions as/@), and let p be arbitrary and complex, p-^0. Then
A0.2.11) D(Ji; z)D(Ja;z) = ?(/i/2; z); [D{f; z))" = D(f; z).
As an example we mention the case/@) = gF) considered in A). We have
A0.2.12) D(g;z) = Hz), D{g-l;z) = {h(z)}-\
In the second formula we assume that g{6) is positive.
A further example is
/(») = 27+8(l - cos 0OA + cos 0)8; D(f; z) = A - «)TA + zI;
7 > -h 8 > -i
10.3. Further study of the representation of positive functions
The derivation of an asymptotic formula for the polynomials orthogonal on
the unit circle requires some further properties of the representation defined
before. In particular, we shall deal with the behavior of the function D(z) on
the unit circle | z \ = 1. In this connection certain restrictive conditions on the
function /@) are necessary.
A) First let us consider again the case of a non-negative trigonometric
polynomial g{6) not identically zero, and let gF) = \ h(z) |2, z = e16, be the
normalized representation defined in Theorem 1.2.2. Then, by A0.2.10), we
have for 0 ^ r < 1,
A0.3.1) D(g; z) = h(z) = h(rete) = exp |i- J_^ log g(t) j ^ re«9-»
Whence
sgn h(reie)
A0.3.2)
Therefore, for all values of 0 with the exception of the zeros of g{6),

278 REPRESENTATION OF POSITIVE FUNCTIONS [ X
sgnh(eie) =¦¦ \h{eid)\~lh{eid) = \g(8))~*h(ea)
= exp <-?- I log git) cot —— dt>
[lir J-r Z )
A0.3.3)
= exp^f I ^ [log git) - log sr@)] cot *—- dt}.
The first integral is taken in the sense of Cauchy's principal value; the second
integral is absolutely convergent.
B) This consideration can easily be extended to a positive continuous func-
function /@) which satisfies certain conditions sufficient for the existence of the
integrals corresponding to those in A0.3.3). Since /@) is continuous, defining
D(z) as in §10.2 we now have, uniformly for all values of 6,
A0.3.4) lim | D(reie) |2 = f(8).
l0
If the integral
e - t
A0.3.5) J^'|log/@ -log/@)|
cot
dt
exists, then
A0.3.6) f+ log f{t) cot t^-1 dt
exists in the sense of Cauchy's principal value. We then show the existence
of the boundary values of
rv «\ / « /"+I l WA 2r sin (<? - 0 m\
sgnD(re ) = exp<— / log/@ ' _ ' 2dt>
do.3.7) ':;
\4tt J-t
+T 2r sin
Indeed, if e is a fixed positive number, we have
lim f [log f(t) - log fie)}
-»i-o J\e-t\x ' 1 —
g g Zfl?!. ,dt
r-»i-o J\e-t\x ' 1 — 2r cos @ — t) -f r2
= f [log f it) -log fid)] cot d-
J\e-t\>t
On the other hand, 1 - 2r cos @ - t) + r2 ^ 2r{l - cos @ - t)}; so
and the last integral is arbitrarily small with e.

[ 10.3 ] FURTHER STUDY OF POSITIVE FUNCTIONS 279
Therefore, under the condition mentioned,
A0.3.8) lim D(reie) = D(eie)
exists, and a representation analogous to A0.3.3) holds for sgn D{eid),
(i C+r ft / 1
sgnD(et9) = exp<— / log f(t) cot —7r- dt >
147T /—x Z
A0.3.9) J ]
= exp ji- / '[log/@ - log/@)] cot*-^-'
In this case we have /@) = | D(eie)
The condition A0.3.5) is satisfied if the function /@) fulfills the Lipschitz-
Dini condition
A0.3.10) |/@ + 8) - f(d) | < L | log 8
-l-X
L and X being fixed positive constants. This is assumed in the present section.
Then A0.3.8) exists uniformly in 6, and D(el6) is continuous.
The preceding exposition could be considerably simplified by using the
theory of conjugate functions (see Zygmund 2, pp. 50, 54, 55, §§3.321 and 3.45).
C) Let m be a positive integer. Then there exists a positive trigonometric
polynomial g{6) of degree m such that
—1 I ^ D/l \—1—X
A0.3.11)
where P is a constant, depending on the minimum and maximum of /@) and
on L and X. This follows by applying Theorem 1.3.2 to the function {/@)}~\
which satisfies the Lipschitz-Dini condition
1/@ + 5)} - WO)} | < {min/@)}-2L | log 8
-l-X
If D(z) and h(z) denote the functions defined in §10.2 which correspond, respec-
respectively, to /@) and gF), we can show that
:—X
A0.3.12) \D(z) - {/i(z)}! < Q(logm)
uniformly for | z | ^ 1. The constant Q depends on the minimum and maxi-
maximum of /@) as well as on L and X.
It suffices to prove this for | z | = 1. An analogous inequality for the differ-
difference of | D(z) | and | h(z) I is trivial (even with (log m)~1-x). We therefore
need a bound only for
sgn h(et0) {sgn D(et0) — sgn [^(e19)]}
A0.3.13) U f+T 0 -t \
= exp<— / {log/@ - log [g(t)] x} cot —-— dt> - 1;
or, what amounts to the same thing, for

280
REPRESENTATION OF POSITIVE FUNCTIONS
X
A0.3.14)
{log/@ - log [git)}-1} cot
To this end, we use Theorem 1.22.1. We have
A0.3.15)
so that
A0.3.16)
gF + 8) - gF) | ^ 5m{max
log g(8 + 8) - log g(d) | ^ 8m
max g(d)
min gF)'
Let E = E{6, m, A) denote the set of ^-values defined by the condition
6 — t \ ^ m~l (log m)~ , and let E' be the complementary set with respect to
[—ir, +7r]. On writing A0.3.14) in the form
A0.3.17)
f {log f(t) - log/@) - log [{/(OF1 + log [g
jB
+ f {log/@ -
JB'
cot
and using A0.3.10) and A0.3.16), we obtain for the first integral
I—1—X
e - t\~xdt + o(m) I \e - t
0A) / \iog\e-t
JE JE
= O[(log m)"x] + O(m)O[m-x(log
On the other hand, A0.3.11) yields the bound
cot
e - t
\dt
2
= O[(log
OKlogm)-1-
cot
e - t
dt = O[(log m)
-X!
for the second integral. This establishes the statement.
10.4. "Local" properties of the representation of positive functions
In this section we prove certain theorems on the representation and approxima-
approximation of positive functions, important for the purposes of Chapters XII and XIII.
A) We have the following theorem:
Theorem 10.4.1. Let fF) be integrable in Riemann's sense, and let it have
the form
_ Zly
z = e*
A0.4.1) /(<?) = <^(») | (z - Zlyi{z - z2y> ¦.
where 0 < A ^ 0@) ^ B, and z, = et9>1 are distinct points on the unit circle,
a, > 0, v — 1, 2, • • ¦ , I. Let f{6) be differentiable at the fixed point 6 = a,
a = e'a 5-± zv, v = 1, 2, • • • , I, and let the following be bounded near 6 = a:
A0.4.2)
fie) -/(a) -f(a)F-a)
F - aY

[ 10.4 ] LOCAL PROPERTIES OF THE REPRESENTATION 281
// D(f; z) denotes the analytic function corresponding to f{6) in the sense of
§10.2, there follows the existence of the limits
lim D{f; reia) = D(f; eia) = D(f; a),
r-»l—0
A0.4.3)
lim !)'(/; reia) = D'(f, O = D'(f; a).
r-»l—0
This is true under more general conditions (see Zygmund 2, pp. 52, 53, §3.44,
and pp. 62, 63, example 13). The following elementary argument is based on
the formula A0.2.10).
If we integrate with respect to t only over a fixed arc not containing a, the
corresponding limits obviously exist. If t is near a, we can write
A0.4.4) log/@ == c + d(e~u - e~ia) + O(l)(t - aJ,
where c and d are certain constants. But if | z \ < 1, then
A0.4.5) ^- / {c + d{e~u- e~ia)} ]+Ze "dt = c - de~ia.
2i"k J—t 1 — ze~%
Hence it suffices to show that if e > 0 is small enough, the function
A0.4.6) / O(l)(t - aJ ZC . dt,
J-t 1 — ze"*'
as well as the derivative of this function with respect to z, is arbitrarily small
if z = reta, r —* 1 — 0. This is true for the derivative since
v< r+< (t-aJ
f+t (t - afdt = f+t
J-t | 1 - rel'<«-» |2 J-t 1 -
dt
2r cos (a - t) + r2
A0.4.7) ,+. ( _ J
< i- / « «> dt.
4r J-f . a - t
sm2 —
The argument is even simpler for the function itself, for in the last denominator
we then have | sin {(a — i!)/2} | instead of sin2 {(a — t)/2\.
B) Now we prove the following theorem:
Theoeem 10.4.2. Let f{6) be integrable in Riemann's sense, and assume
0 < A ^ f{6) ^ B if —t ^ 6 ^ +7r. Also suppose f F) differentiable at the
fixed point 6 = a and the expression A0.4.2) bounded near 6 = a.
If e is an arbitrary positive number, there exist positive trigonometric polynomials
giF) and g2F) such that
A0.4.8) Me) ^ f(e) ^ Me), /i(«) = /2(<*),
where Me) = {gi(e)}-1, Me) = {g^e)}-1; and
do.4.9) rm:m»<-
¦ , e — ex
sin2 —^—

282 REPRESENTATION OF POSITIVE FUNCTIONS [ X ]
Let m and M be the lower and upper bound of A0.4.2) for — 7r ;S 0 < -\-ir.
The function /i@) is greater than a positive number depending only on m, A,
f(a), and f'(a); similarly /2@) is less than a number depending only on M, f(a),
and f'(a).
We observe that A0.4.8) implies
A0.4.10) Ma) = f(a) = Ma), f[(a) = f\a) = f>{«),
so that the integral in A0.4.9) exists; furthermore, A0.4.9) implies
Ao.4.n)
sin''
where e' is arbitrarily small with e.
For the proof we apply Theorem 1.5.4 to the functions
p{6) =
. ,0 - a
sm2
2
A0.4.12) d = ~f («){/(«) \~2>
fF) - f(a) ~ f'(a) Sin@ - a)
sin2
e - a
which are both integrable in Riemann's sense. Therefore, given 8 > 0, there
exist certain trigonometric polynomials P(d) and QF) such that
P(e) zp(fi), ? {P(e) -p(e)}de<s,
A0.4.13) 7+T
g(e) ^ Q(e), J_ {Q(d) ~ q(e)\ de < 5.
Here max PF) and max QF) are less than certain constants depending on
m, A, f(a), f'(a), and on M and /'(a), respectively. Writing
A0.4.14) gi(e) = c + d sin (e ~ a) + P(d) sin2 e~=—, Me) = {gi(e)}~\
we find that {fid)}-1 ^ g^d), or Me) ^ 7@), A (a) = f(a), and that
ao.4.i5) pM^AW*. rmue)g'{e)-mr'<ieiB'S.
sm —^— sm —~
Here/i@) is greater than a positive constant depending on m, A,f(a), and/'(«)•
On the other hand, considering the continuous function

[ 10.4 ] LOCAL PROPERTIES OF THE REPRESENTATION 283
GF)
( ft - I
y(«) + /'(«) sin @ - a) + Q@) sin21—-? \ - c - d sin @ - a)
_ I r__J
A0.4.16)
{Rid)}'1 - c - d sin @ - a) R'1 - c - d sin @ - a)
.2^ Ot -2
sin -T- sm
we have }{&) ^ i2@) < R, where R is a positive constant depending on M,
f(a), and/'(a). Now according to Theorem 1.3.1, a trigonometric polynomial
S@) can be determined such that
A0.4.17) R-
If we write
. 2 ^ — Ot
am -y-
A0.4.18) g,(fi) = c + d sin @ - a) + 5@) sin2 —~, /2@) = {,
we find that
A0.4.19) R~l < gM < {Rid))'1 < gs@) + 8 sin2 6-^.
Furthermore, gt(d) < {/@)}- On account of the last inequality in A0.4.19)
and A0.4.13) we have
r ^2@) 4" 8 sin ——=— )¦ — /@)
dd = I 1 ,-=—+ de
sin«
, .1 J ,s • 2r -1-1
— \ A2F) + 8 sm
sm2
sin2
sm2

284 REPRESENTATION OF POSITIVE FUNCTIONS [ X ]
The last integral is arbitrarily small with 8 (since g2F) > R).
Combining A0.4.15) and A0.4.20), and taking 8 sufficiently small, we ob-
obtain the statement above.
C) Theorem 10.4.3. For the analytic functions D(Ji ; z), D(f; z), D(J2 ; z),
corresponding to the functions /i@), /@), /2@) of Theorem 10.4.2 in the sense of
§10.2, the following inequalities hold:
A0.4.21) | D(f; a) - D(f, ; o) | < e', | D'(f; a) - ?'(/„ ; a) | < e',
where a = eta, v = 1, 2; here e' is arbitrarily small with e.
The symbols D(J; a), D'(J; a) have the same meaning as in A0.4.3).
According to A0.2.10) we have for | z \ < 1, v = 1, 2,
log D(J;z) - log D(f,;z) = 1 /"" {Iog/@ - log/,(«)} \ + **J(' dt,
47r J-t 1 — ze
A0.4.22)
D'(f;z) D'(fy;z) 1 f+'. .fA . f ,AX ze'u
ZD(J^)~Z DU^z) = T, L {logfit) " log/@1
and for z = re1" = ra, r —> 1 — 0,
log Z)(/;o) - logZ)(/,;a) =-L f+' {log/@ -
A0.4.23) ?'(/; a) __ Z)U;a) _ _ J f+T log /«) -
sin -g—
Both integrals are absolutely convergent and arbitrarily small with e (see
A0.4.11)); D(J; a) is a fixed number different from zero.
D) Theorem 10.4.4. LetfF) be a function satisfying the conditions of Theorem
10.4.1. Given an arbitrary positive number e, there exist positive trigonometric
polynomials giF) and g%(e), such that on putting
Me) = ig1(e)r11 (* - 2l)(z - z2) • • • (z - «,) \°, z = ei$,
A0.4.24)
Me) = {g2(e)\ ,
we have
A0.4.25) o ^ Me) ^ f(e) ^ Me), Ma) = Ma),
A0.4.26)
<r {s ?/ie Zeos< e^en number greater than max (<ri , <r2 , • • • , <ri); max M&) is
bounded from above, while min [giF)\~l is bounded from below, and both bounds
are independent of e.

[ 10.4 ] LOCAL PROPERTIES OF THE REPRESENTATION 285
For the junctions D(/i; z), D(J;z), and D(f2; z), corresponding, respectively, to
/i@), /@), and /2@), a statement, similar to that in Theorem 10.4.3 holds.
Remaek. We can choose for <r any even number which is greater than
max (<ri, <r2, • • • , ai); in particular, we can choose an even number divisible
by 4. This is important for certain later purposes (cf. §13.5 B)).
Let the function /@) be identical with /@) except in certain closed intervals
around 6,, v = 1, 2, • • • , I, in which /@) = 1. These intervals are chosen so
small that they do not overlap, they do not "contain a, and in each of them
/@) ^ 1. Furthermore, let
A0.4.27)
sm J —
Then /@) ^ /@) for each 0, and /@) = /@) in a certain neighborhood of a
which can be chosen independent of e provided e is small enough. The func-
function/@) = /(e; 0) satisfies the conditions of Theorem 10.4.2 and depends on e,
although it has an upper bound independent of e. The same is true for the
upper bound M of the ratio corresponding to A0.4.2).
We now determine the trigonometric polynomial gr2@) such that, for /2@) =
\
^ M6); f(a) = }(a) = Ma);
A0.4.28) /•+' log Me) - iog/@) d$
sin2
0 - a 4
We observe that max /2@) is less than a number independent of e. (Here we
use the independence of M of e.)
On the other hand, let /@) be identical with /@) except in certain closed
non-overlapping intervals around 0,, v = 1, 2, • • • , I, not containing a, and
such that in these intervals
A0.4.29) |B - zi)(z - z2) • • • (z - zi)\° ^ | (z - Ziy\z - z2)'2 • ¦ • (z - Zl)°l |,
z = et$. In these intervals we define
A0.4.30) /@) = <K0) |B - zi)(z - z>) ¦ ¦ ¦ (z - 2,)|', 2 = «*.
Furthermore, assume
sm2
2
Then/@) ^ /@), and/@) = /@) near 0 = a. Moreover,
A0.4.32) /@) |B - zi)(z - 2,) • • • B - 2,) I"*, 2 =

280 REPRESENTATION OF POSITIVE FUNCTIONS [ X ]
satisfies the conditions of Theorem 10.4.2 and depends on e, although it
has a positive lower bound independent of e; the same is true for the (not
necessarily positive) lower bound m of the ratio corresponding to A0.4.2).
We determine a positive trigonometric polynomial giF) such that
B -
with equality for 0 = a, and
A0 4 34) ( lQg U{6) I(Z ~ Zl)(z ~ Zz) • • • (z ~ Z'}H ~ log
Then, using the notation A0.4.24), we obtain
sin2
e ~
Addition of the inequalities A0.4.27), A0.4.28), A0.4.31), and A0.4.35) es-
establishes A0.4.26). We observe that min {gfiCfl)}" is greater than a positive
constant independent of e.
The assertion concerning D(ft ; z), D(J; z), and D(/2 ; z) follows as in Theorem
10.4.3.
E) Theorem 10.4.5. LetfF) be an even function which satisfies the conditions
of Theorem 10.4.1; assume that <f>F) =</>(— 6), and that all non-real "zeros" zv of fid)
occur in conjugate pairs with the same "multiplicity." Also, let 0 < a < ir.
Then the functions fi{6) and /2@) of Theorem 10.4.4 can be chosen as even func-
functions, and we have instead of A0.4.26)
A0.4.36)
- cos aJ
An analogous supplement can be made to Theorem 10.4.2. Previous proofs
need only a slight modification. Instead of the first ratio in A0.4.12), consider
{/@)} — c - d (cos 6 - cos a)
A0.4.37) (cos 0 - cos aJ
c = \f(a)}-\ d Sin a = f(a){f(a)}-\
The other ratios occurring in the proof of Theorem 10.4.2 must be similarly
modified. The supplements to Theorems 1.3.1 and 1.5.4 concerning even func-
functions have to be observed. The functions /@) and /@) occurring in the proof
of Theorem 10.4.4 can be chosen as even functions. The inequality A0.4.36)
is equivalent to A0.4.26), since the function
f
f
(cos 0 — cos a)
is bounded from 0 and « if 0 ^ 0 S ir.

CHAPTER XI
POLYNOMIALS ORTHOGONAL ON THE UNIT CIRCLE
A weight function on a given curve having been assigned, we may extend the
definition of polynomials orthogonal on a real interval to the more general
complex domain. The corresponding polynomials are then orthogonal with
this given weight function on the specified curve in the complex plane
(Chapter XVI).
Of all the special instances, that of a circle is most interesting, and in the
present chapter we shall consider the polynomials orthogonal on the unit circle
with a given weight function. It will be seen that these polynomials possess
properties which are in some respects simpler than those derived for poly-
polynomials orthogonal on a real interval. Moreover, there exists a relation between
the case of the circle and that of a real, finite interval which enables us to
apply certain results of this chapter to polynomials orthogonal on a real interval.
Concerning §§11.1-11.4 see Szego 4. See also Grenander-Szego 1, Chapter 2.
11.1. Definition; preliminaries
A) Let/@) be a non-negative function of period 2w, integrable on [ — v, +7r]
in Lebesgue's sense, and assume
r+r
A1.1.1) I f@)d$>0.
We introduce the Fourier constants
i f+ir
A1.1.2) cn = ± I f(d)e-ined9, n = 0, ±1, ±2, •••.
2w J--T
Obviously, c_n = cn, so that the matrix of "Toeplitz' type"
A1.1.3) ; Tn = (c,_M), v, n = 0, 1, 2, • • • , n,
is Hermitian., The corresponding Hermitian form
A1.1.4) Hn=EZc-mu, = ~ I fF) | uo + uiz + u2z2 + • • • + unzn |2dd,
v=0 n—0 Alt J—t
where z = e$, is positive definite and has the positive determinant
A1.1.5) Dn = [c,_J, v, n = 0, 1, 2, • • • , n.
B) Definition. // we orthogonalize the system
"Cf. the last remark in §2.1 D).
287

288
POLYNOMIALS ORTHOGONAL ON THE UNIT CIRCLE
XI]
A1.1.6) {(/(*))**"},
we obtain a system of polynomials
A1.1.7) *,(«), fcOO, fc(«), ••• ,
eie
= eie;n = 0, 1, 2, ¦. • ,
/ie following properties:
(a) 0n(z) is a polynomial of precise degree n in which the coefficient of zn is real
and positive]
(b) the system \4>n(z)} is orihonormal; that is,
A1.1.8)
= Snm, z = e<e; n,m = 0, 1, 2,
Moreover, the system {<j>n(z)} is uniquely determined by conditions (a) and (b).
If f(d) is an even function, that is, if f(d) = /( —0), the coefficients of 4>n(z)
are real.
C) We show (see §2.2 B)) that
A1.1.9) <fin(z) = (Z>n-iZ)n)
Co C_i C_2 • • • C_n
C\ C0 C_i • • • C_n+i
Cn—1 Cn—2 Cn—3 * " * C—i
l z i ¦•• zn
CqZ — C—i C—\Z — C_2
C\ z *~~ Cq Cq z ~*~ c__i
C_n+lZ C_n
_n+2Z C_n+1
Cn—lZ Cn—2 Cn—2 Z Cn—3
C0Z — C_i
n = 1, 2, 3, • • •.
The coefficient of z" in <f>n(z) is
A1.1.10) /<„ = (D^D-y.
The analogues of the representations B.2.10) and B.2.11) can also be readily
derived.
D) We pass now to considerations corresponding to those of A), B), and
D) of §3.1.
Theoeem 11.1.1. Let F(eli) be a given measurable function for which
A1.1.11) [ fF) | F(eie) |2 d6
exists. The weighted quadratic deviation
A1.1.12) ~ l^7@) I Hz) - p(z) Ue, z = eie,
Alt J-TI

[11.2] EXAMPLE 289
where p(z) ranges over the set of all wn, is a minimum if p(z) is the nth partial
sum of the Fourier expansion
F(z) ~ F«h(p) + F^iz) + F202(z) + • • • + Fn4>n(z) + • •. ,
A1.1.13) i r+r
Fn = 2~ J_w MF(zL>n{z) dd, z = ete; n = 0,1, 2, • • -.
As a ready consequence, there follows Bessel's inequality
A1.1.14) | Fo |2 + | F112 + | Fs |2 + • • • + | Fn |2 + • • • ^ -L [+Tf(d) \F(eie) \2dd.
A J
In addition, Parseval's formula (that is, A1.1.14) with the equality sign) holds
if one of the following sets of conditions is satisfied:
(i) F{z) is regular and bounded for | z \ < 1.
(ii) f(d) is bounded and F{z) is of the class H2 (see §10.1).
Concerning a more general condition, see Smirnoff 2, p. 363.
A consequence of Theorem 11.1.1 is the following:
Theoeem 11.1.2. The polynomial K^l4>n(z) minimizes the integral
Jf(d) | n + a*"'1 + + a \2dd z eie
A1.1.15) ~ Jf(d) | zn + ax*"'1 + • • • + an \2dd,
if zn + axzn~x + . • • + an ranges over the set of all 7rn with the highest term zn.
The minimum is kZ ¦
11.2. Example
An important special case in which the system \<i>n{z)} can be calculated
explicitly, except for a finite number of terms, is
(n.2.1) f(e) = {g(e)r\
where gF) is a positive trigonometric polynomial of degree m.
Theoeem 11.2. Letf(d) be defined by A1.2.1), and let g(d) = | h(z) |2, z = eie,
be the normalized representation of g(d) defined in Theorem 1.2.2. Using the
notation of A.12.4), we have
A1.2.2) 4>n{z) = zn~mh*{z) = znh{z~l), n = m, m + 1, m + 2, • • • .
Evidently, condition (a) of the definition in §11.1 B) is satisfied. In order
to show the orthogonality, let p(z) be an arbitrary 7rn_i. Then, if z = e',
we have, according to Cauchy's theorem,
~ [+W f(eL>n(z)pH) dd
2ir J-,
= 1 \ [h{z)h{Z-1) } ^ Z" h{z~l)p{z-1) $
2w Jizi^i iz

290 POLYNOMIALS ORTHOGONAL ON THE UNIT CIRCLE [ XI ]
In addition
h Cm' *-(z) l*de = h 17'h{z) ' |z"* {z~l) |2^ = L
The simplest case is/@) = 1. Then
A1.2.3) 4>n{z) = z\ n = 0, 1, 2, ....
Concerning other cases in which an explicit calculation of 4>n{z) is possible,
we refer to Szego 4, pp. 187-188; and 12, pp. 245-247. See also A1.5.3) and
A1.5.4).
11.3. A maximum problem
Because of the similarity between this problem and that treated in §3.1 C),
we can omit details.
(l) Theoeem 11.3.1. Let p{z) be an arbitrary irn subject to the condition
A1.3.1) h\lj{e) l"B)|2^ = X> * = e"
For a fixed value of a, the maximum of \ p{a) |2 is attained if
A1-3.2) P(z) = e{sn(a, a)}~isn(a, z), \ e | = 1,
where
A1.3.3) sn(a,z) =
v-0
The maximum itself is sn{a, a).
These "kernel polynomials" sn(a, z) can be used for the representation of the
partial sums of the expansion A1.1.13) in form of integrals (see C.1.11)).
B) Theoeem 11.3.2. Fora ^ 0 the polynomials A1.3.2) satisfy the following
identity:
A1.3.4) sn(a, z) = (az)nsn(rl, a'1).
Furthermore,
A1.3.5) sn@, z) =-- E ^ 4
•¦=0
where Kn has the same meaning as in A1.1.10); finally
A1.3.6) 8,@, 0) = Z | 4>M P = «l = Dn-i/Dv
t\
The last formula also holds for n = 0 if D_i = 1.
Introducing p*(z) = r(z) or p(z) = r*(z), we have

[ 11.3 ] A MAXTMUM PROBLEM 291
A1.3.7) I- / fF) \P(z) \2d6 = ~ f f(fi) \r(z) \2d6 = 1, z = eie,
and, for o ^ 0,
A1.3.8) | P(a) |2 - | a"r(O T = I « T I KcT1) |2.
This yields sn(a, a) = \ a [l"sn{aT\ a"), that is, A1.3.4) for z = a, and also
A1.3.9) {sn(a, a)}-^, z) = e[{8n(<T\ a) r*«»(<5~\ z)]*
where e is a proper constant with | e | = 1. (The symbol * refers to the
variable z.) On combining this with the first result, we obtain A1.3.4). In
the limiting case a = 0, the identity A1.3.5) arises, from which, for z = 0,
A1.3.6) follows.
C) Theorem 11.3.3. Let log /@) be integrable in Lebesgue's sense.™ Then
the following limits exist:
00
A1.3.10) lim sn(a, a) = ]C | <t>y(a) |2, \a\ < 1,
n—»oo y=0
A1.3.11) lim sn(a,z) = Z*, <f>,(a)<t>y(z),
A1.3.12) , ,
T4-»oo n-»0
A1.3.13) H_m«,W = 0f |«|<1.
We first consider the special case fF) ^ ju > 0, assuming | a | < 1. Let
p(z) have the same meaning as in A1.3.2). Then we have, because of Cauchy's
inequality (see G.1.4)),
2
P(a) |2 ^
. - OH;
AL3-14)
—i i r+jr —1
p(z) \2dd = :—r2, z = e* ;
consequently, the same inequality holds for n —»• <». Thus A1.3.10) and
A1.3.11) are established in this case. For a = 0, A1.3.6) and A1.3.5) show
that the limits A1.3.12) exist; A1.3.13) follows from the convergence of
A1.3.10).
In order to prove the statement generally, we first observe that the maximum
in the problem of Theorem 11.3.1 is attained for a polynomial p(z) which is
different from zero for | z \ < 1; here again | a \ < 1. In fact, if Zo is a zero of
p(z), | zq | < 1, then
"Cf. §10.2 B).

292
POLYNOMIALS ORTHOGONAL ON THE UNIT CIRCLE
XI]
1 f+r
— / f(&)
2tt /_,
P(z) -
— zoz
z — Zo
|p(o)|*<
— -2
p(a)
1 — Zo a i
a — z0 I
Now assume a = re'*, 0 ^ r < 1, z = e'e; according to A.11.3),
1 = 1
r
1 -r2
1 -r2
-2rcos@ -
z = e .
je
The last integral is Poisson's integral of the harmonic function 29? log [p{z)\,
which is regular for | z \ < 1. The last exponential factor is therefore equal to
exp {29? log [P(a)}} = \P(a)\2;
this establishes the boundedness of max | p(a) |2 = sn(a, a).
The further formulas follow as before. Later (§12.3 F)) we shall calculate
the limits A1.3.10)-A1.3.12).
11.4. Algebraic properties
A) Let a be fixed, \a\ < 1. The above considerations show that the zeros
of sn(a, z) lie in | z | 2; 1. The same is true, of course, for \ a\ ^ 1. From
A1.3.5) we find that the zeros of 4>n{z) lie in | z | ^ 1.
We now prove the following more informative statement:
Theorem 11.4.1. For \ a \ < 1 the zeros of sn(a, z) lie in \ z \ > 1,/or | a | > 1
in | z | < 1, and for | a | = 1 on | z | = 1. The zeros of 4>n(z) lie in | z | < 1.
Let Zo be an arbitrary zero of sn(a, z). If we put
sn(a, z) 2
A1.4.1)
hie) = f{e)
z —
and consider all the linear functions p(z) with
A1.4.2)
± fM\P(z)\2de = i,
t'9
z = c ,
it is clear that max | p(a) |2 is attained for p(z) = const., (z — zo). It therefore
suffices to discuss the case n = 1. From A1.1.9)
A1.4.3)
with the zero
A1.4.4)
h(a, z)
— <f>0\(l) 0O\^) T" 01 (^) 01'
= JJo ~r JUo L)\ {Cod —
C\ (X — Co
-? .
Co a, — Ci
ci)(coz -

[ 11.4 ] ALGEBRAIC PROPERTIES 293
This establishes the statement concerning sn(a, z) since | Ci | < c0 (see A1.1.2)).
The statement concerning <j>n(z) follows from A1.3.5).
B) Theoeem 11.4.2. We have the identity
A1.4.5) sn(a,z) = ~~
1"
— az
and the "recurrence formulas"
(H-4.6) KnZ4>n(z) = Kn+l(f>n+l(z) — <?n+l@)<?*+i(z),
(H-4.7) Kn(f>n+1(z) = Kn+1Z<t>n(z) + <t>n+1(O)(f>*n(z).
The first identity corresponds, in some respects, to the Christoffel-Darboux
formula C.2.3). The proof can be given along the same line as the proof in
§3.2 C). As in the case of a real interval, we can characterize sn(a, z) by the
equation
A1.4.8) — J_^ f(d)sn(a, z)p{z)dd = P(a), z = eiS,
which holds if p(z) is an arbitrary irn . But
27CJ-,
A1.4.9)
1) *
1
2tt
1
2,
*-n(aHl
/ fid
J-"
L *
1
— dz
l(a)<?n-f
«^.
.i(aH»+iO
-i(z) - 4>T
1 — dz
-- P(z) de
,+i(aH»+i(z)
!»n+i(aHn+1 .
p(z)
1
— p(a)
— az '
z = eie.
The last integral vanishes, for if we write p{z) — p(a) = (z — a)r{z), wo have
f{6L>l+l{z)zr{z) d6 = j f(d)cf>n+l(z)zr(z) U8 = 0, z = e'e.
Therefore,
1 — az
where c is independent of 2. Interchanging a and z and taking the conjugate
complex values of both sides, we see that c is also independent of a. Writing
z = a = 0, we obtain
A1.4.12) /c2n+i ~ I 0»+i(O) |2 = c Z I 0,@) i2,
>¦=<)
so that c = 1, by A1.3.6).

294 POLYNOMIALS ORTHOGONAL ON THE UNIT CIRCLE [ XI ]
Comparison of the coefficients of d" in A1.4.5) leads to A1.4.6). By taking
the reciprocal polynomials of both sides of A1.4.6), and eliminating <?*+i(z),
we find A1.4.7).
11.5. Relation to polynomials orthogonal on a real interval
A) Theorem 11.5. Let w(x) be a weight function on the interval —1^x^+1,
and let
A1.5.1) f{0) = w(cos 6) | sin d |.
Further let {pn(x)\ and \qn(x)\ be the orthonorrnal sets of polynomials which
belo?ig respectively to w(x) and A — x2)w(x) in —1 ^ x 5= +1, and {4>n{z)\ the
orthonormal set associated with f{6) on z = e'e. Then, on writing x = %(z + z~x),
we have for n ^ 1
Pn(x) = BT)
A1.5.2)
n W\ — @/ VUl
<ln\XJ — \6/1T) U
K2n
2n-f2@I Z U fa+liz) — z" 4>{z )
r ¦
K2n+2 J Z - Z~X
i I ^»2n+2@)\ Z H(f>2n+l(z) — z"^2n+l(z )
«2n+2 J Z — Z'1
See Szego 6, pp. 204-206. The second equation follows from the first one,
and similarly, the fourth follows from the third, by means of A1.4.7). The
function f(d) is even; so the polynomials <t>n{z) have real coefficients.
The constant factors in these equations are different from zero (see A1.3.6)).
These formulas, except the second one, hold also for n = 0.
B) The right member of the first equation represents a 7rn in x; the property
of orthogonality can be expressed in the form
/
pn(cos 9) cos p6-w(gos 6) | sin 6 \ dd = 0;
J ~v
or, for z = e",
t^ {z-n<hn(z) + z^niz'1) 1 [z + z-'\fF)d6 = 0, , = 0, 1, 2, • • • , n - 1.
This is the case because
*4>2n(z)(zn~' + zn+v)f(d)dd = 0,
and 4>2n(z) = <hn(z). Moreover,

f 11-5 ] POLYNOMIALS ORTHOGONAL ON A REAL INTERVAL 295
dd
= 7T + 7T +
J z~n<t>2n(z)- z"cf>2n(z-i)f(d) de
dd) = 2tT + 2,T ^^ .
J
For the third formula the proof is similar, except that it is convenient now to
express the orthogonality in the form
gn(cos 0) *—~—-—L sin2 d-w(cos d) \smd\dd = 0,
•T SIII V
v = 0, 1, 2, • ¦ •, n - 1.
A new proof for the formulas B.6.2) and B.6.3) can be derived from the first
and third formulas in A1.5.2) by use of A1.2.2).
If in the last two equations of A1.5.2) we replace n by n — 1, we can express
z~'V>2n(z) andz~"+102,,-i(z) as linear combinations of pn(x) and A — x2yrXqn_{{x),
where x = §(z + z). (See Szego 18, pp. 9-11.) These relations enable us to
calculate the polynomials 4>n{z) associated with
A1.5.3) fF) = | A - z)y(l + z)s |2 = 2r+5(l - cos d)y(l + cos 6)s, z = eie,
in terms of Jacobi polynomials. We find
—n i /^ i p(y—^,5—J) f w i^ —1\ 1 i^ r>(' „—1\ r)(T+J.5+J) t 1 / i^ „—1\ )
Z (p2n \Z) ^ -^1 jt n I  \Z ~p Z y I ~p Zj\Z — Z ) L n \ i2\^ ~l~ ^ / ( j
A1.5.4) z-n+V2n-i(z) = CP^^'^MKz + z)!
+ D(z - z-1).
where A, B, C, D are proper real constants.

CHAPTER XII
ASYMPTOTIC PROPERTIES OF GENERAL ORTHOGONAL
POLYNOMIALS
The following sections deal with the asymptotic properties of polynomials
orthogonal on the unit circle, or on a real, finite interval, when the degree
n of these polynomials becomes infinite. In both cases the weight function
will be restricted merely by certain conditions of continuity and boundedness.
Two important problems appear in connection with polynomials orthogonal
on the unit circle. These are (a) the asymptotic behavior exterior to the unit
circle, (b) the behavior on the unit circle itself. For a weight function iden-
identically unity, the system in question is {zn\. This latter instance is typical to a
certain extent.
The corresponding problems for polynomials orthogonal on a finite segment
are (a') the asymptotic behavior in the complex plane cut along the given seg-
segment, (b') the corresponding question on the segment itself. (See Chapter
VIII.) We give the following illustration as characteristic:
Tn(x) = \{zn + O, x = \(z + z~l).
Problems (a) and (a') are simpler, and relative to them comparatively general
results will be obtained. It is only recently that problems (b) and (b')( which
are much more difficult than (a) and (a'), have been treated by S. Bernstein
and G. Szego. We remark that the weight function conditions in case (b) are
more restrictive than those in case (a). A similar comment may be made
concerning (a') and (b').
The results of Chapter X are applied in discussing the above problems. Our
investigation is first concerned with questions (a) and (a'). As regards (b')
we may state that S. Bernstein's main result is obtained in a new way which
is shorter than his original argument. Then there follows Szego's older method
as applied to (b').
12.1. Results
A) Let G denote the class of functions /@) ^ 0, defined and measurable in
[— 7T, +7r], for which the integrals
A2.1.1) f+ f(d)dd, f+ \ log f F) \d6
exist with the first integral supposed positive. With such a function f{6) we
associated in §10.2 a uniquely determined analytic function D(f; z) = D{z),
regular and nonzero for | z j < 1 with Z)@) > 0. The conditions for the class G
imply the existence of the "geometric mean" ©(/) = {Z)@)|2 of f@).
296

[12.1] RESULTS 297
Theorem 12.1.1 (Asymptotic formula for the polynomials orthogonal on the
unit circle, considered for z exterior to the unit circle). Letf(d), belonging to the
class G, be a weight function on the unit circle z = eie. If {<t>n(z)\ denotes the
corresponding orthonormal set of polynomials, then in the exterior of the unit circle
A2.1.2) 4>n(z)^zn{D(z-1)}-1, \z\> 1.
This holds uniformly for \ z \ 2; R > 1.
Theorem 12.1.2 (Asymptotic formula for the polynomials orthogonal on the
segment [ — 1, +1], considered for x exterior to this segment). Let w(x) be a
weight function on the interval —1^x^+1 such that w(cos 0) | sin 0 \ = f(d)
belongs to the class G. If D(f; z) = D(z) denotes the analytic function corresponding
tof(d) in the sense mentioned, the orthonormal polynomials {pn(x)\, associated with
w(x), possess the asymptotic formula
A2.1.3) pn(x) ^ BTv)-izn{D(z-1)}-1.
Here x is in the complex plane cut along the real segment [ — 1, +1], and x =
%(z + z), where \ z \ > 1. Formula A2.1.3) holds uniformly for \ z\ ^ R > 1.
B) In order to obtain the deeper asymptotic formulas valid on | z-| = 1 and
on—1^x^+1, respectively, we must impose certain further restrictions on
f{6) and w(x).
Theorem 12.1.3 (Asymptotic formula for the polynomials orthogonal on the
unit circle, considered for z on the unit circle). Let f(d) be a positive weight
function on the unit circle, which satisfies the Lipschitz-Dini condition
A2.1.4) | f(9 + 5) - f(9) | < L | log 5 | -^
where L and X are fixed positive numbers. Then we have, for \ z \ = 1,
A2.1.5) 4>n{z) = z^Diz'1)}-1 + en(z) = zn[D{?)}-' + en(z),
where limn_oo en(z) = 0, uniformly for \z\ = 1. More precisely,
A2.1.6) | en(z) | < C(log n)~\-
the positive constant C depends on L, X, and the minimum and maximum of f(d).
Theorem 12.1.4 (Asymptotic formula for the polynomials orthogonal on the
segment [ — 1, +1], considered for x on this segment). Let w(x) be a weight
function on the interval —1 ^ x ^ -\-l, x = cos 6, such that the function
w(cos 0) | sin 0 | = /@) satisfies the conditions of Theorem 12.1.3. Putting
A2.1.7) sgn D(eie) = | D(eie) p1^) = eiyW,
we have uniformly on the segment —1 ^ x S -\-l or 0 ^ 6 ^ t, x = cos 0,
A2.1.8) A - z2)W))*pn(z) = B/t)*cos {nd + y(9)} X

298 ASYMPTOTIC PROPERTIES OF GENERAL POLYNOMIALS [ XII ]
The constant factor in the O-term depends only on L, X, and the minimum and
maximum off(d).
C) Finally, we prove two theorems similar to Theorems 12.1.3 and 12.1.4
under conditions possessing a kind of "local" character.
Theorem 12.1.5 (Asymptotic formula for the polynomials orthogonal on the
unit circle for | z \ = 1, and with weight function subject to a "local" condition).
Letf(d) satisfy the conditions of Theorem 10.4.1. Then A2.1.5) holds for z = a =
e'a, in the less informative form
A2.1.9) 4>n{a) = an{D(a)r + e» , e» -» 0.
From this we obtain the following theorem of "local" character, which cor-
corresponds to Theorem 12.1.4:
Theorem 12.1.6 (Asymptotic formula for the polynomials orthogonal on the
segment [—1, +1] for x on this segment, and with weight function restricted
by a "local" condition). Let w(x) be integrable in Riemannfs sense, and let it
have the form
A2.1.10) w(x) = t(x) | x - xi |Tl | x - x2 T2 • • • | x - xi | ",
where 0 < A ^ t{x) ^ B, and — 1 ^ xi < x2 < ¦ • • < xj ^ 1, t, > 0, v = 1, 2,
¦ • ¦ , I.59 Further let w(x) be differentiate at the fixed point x = ?, where — 1 < ?
< +1, and ? ?* x,, v =¦¦ 1,2, ¦ ¦ . ,1, and let
w(x) — w(i;) — (x ¦— ?)w'{?)
A2.1.11)
- ay
be bounded if x is near to ?.
Then A2.1.8) holds in the less informative form
A - i?)\w(®)hPn@ = {2/ir)h cos [na + 7(a)} + en ,
A2.1.12)
? = cos a, 0 < a < 7T, lim en = 0,
n—»oo
where 7 (a) has the same meaning as in A2.1.7).
12.2. Remarks
A) Theorems 12.1.2, 12.1.4, and 12.1.6 follow readily from 12.1.1, 12.1.3,
and 12.1.5, respectively. We observe that D(z) = D(z) in A2.1.3); that is,
D(z) is in this case a "real" function.
In Theorems 12.1.3-12.1.6 the function D(z) has boundary values at the point
considered (in Theorems 12.1.3 and 12.1.4 even continuous boundary values
on the whole unit circle \ z\ = 1). This follows from the considerations of
Chapter X.
The important function 7@), defined by A2.1.7), is completely determined
68 For x\ = —1 it suffices to assume n s? —1/2; similarly for Xi = +1.

[ 12.2 ] REMARKS 299
save for an integral multiple of 2tt. If we choose (see A0.3.9))
A2.2.1) 7@) = -^ J*' [log f(t) - log/(<?)] cot 6-^ dt,
then 7@) is continuous. In the cases occurring in Theorems 12.1.4 and 12.1.6,
7@) can easily be expressed in terms of the weight function w(x). Let
A2.2.2) /@) == w(cos 0) I sin 0 I = W(cos 0).
From A0.3.9) we obtain (see S. Bernstein 2, p. 132),
7@) = _L I {log W(cos t) - log W(cos 0)} cot -ZJ dt
= -i / {log W(cos t) - log W(cos 0)} (cot 6~.~ + cot 6-t
A2.2.3) k
0 - log W(cos (9)} s-^l— dt
COS I — COS 0
~ 2x7-1
In this case
A2.2.4) 7(-<?) = -1F).
For the functions {(/@))^n(z)}, z = e*9, n = 0, 1, 2, • • •, which form an
orthonormal system in the usual sense (see the Definition in §11.1 B)), we
obtain from A2.1.5) the simple asymptotic expression znetyW = e'fn9+1'(9>)
B) Theorem 12.1.1 is a direct consequence of Theorem 12.1.3 provided con-
condition A2.1.4) is satisfied. In fact, the function z~n(f>n(z) — {5(z~x) p1 is regular
for | z | > 1 and continuous for | z \ ^ 1. In this special case, A2.1.2) follows
in the more informative form
)-X
A2.2.5) z'nUz) = {Diz'1)}'1 + O{(logn)
uniformly for | z | ^ 1.
C) Concerning Theorems 12.1.1 and 12.1.2, see Szego 6. Under more re-
restrictive conditions than those in Theorem 12.1.2, Faber D) proved
A2.2.6) Mm | pn(x) |1/n = \z\, \ z \ > 1.
n—»oo
This less informative statement suffices for various applications, for instance,
for the purposes of §12.7 B) and C). Theorem 12.1.3 is new, while 12.1.4 is
due to S. Bernstein B). The proofs of Theorems 12.1.3 and 12.1.4, given in
§12.4 and 12.5 B), respectively, are based essentially on an idea of S. Bernstein,
used in his original proof, and on another idea similar to that used in connection
with the method of Liouville-Stekloff (§8.61). As mentioned, this arrange-
arrangement seems to be simpler than S. Bernstein's original line of argument. The-

300 ASYMPTOTIC PROPERTIES OF GENERAL POLYNOMIALS [ XII ]
orem 12.1.5, and Theorem 12.1.6 which is its consequence, are due to Szego (8).
The conditions required in the present treatment are slightly more general than
those in the paper just cited.
S. Bernstein assumes B, p. 132, A8))
W(x + 5) - W(x) | <L | log 5 | ~1~x; -1 ^ x ^ +1, -1 ^ x + 5 S +1,
instead of A2.1.4). These conditions are, however, equivalent since
log {| cos 0i — cos 62 p1}
is bounded from 0 and °° if 0i and 02 are arbitrary in [0, tt], | 0i — 02 | ^ ^.60
12.3. Proof of Theorem 12.1.1; applications
A) Before we proceed to this proof, let us consider the special case /@) =
{gr@)p1, where gr@) is a positive trigonometric polynomial of degree m. Let
h(z) have the same meaning as in §10.2 A). According to the second formula
in A0.2.12) we have D(f; z) = {h(z)}'\ On the other hand, by A1.2.2)
A2.3.1) 4>n{z) = znl{z'1) = zn{T>{z'1)\~\ n > m.
The limit relation A2.1.2) can therefore be replaced by one of equality in this case
provided n ^ m.
B) Now let /@) again be an arbitrary function satisfying the conditions of
Theorem 12.1.1. Let p(z) = zn + a\Zn~x + • • • + an be an arbitrary irn with
the highest term zn. According to Theorem 11.1.2, the minimum /;„(/) of
'z)\2 dd, z = e*
A2-3.2) f /@)
is Kn~~2, attained for p(z) = K^4>n(z).
Lemma. Let /@) satisfy the conditions of Theorem 12.1.1, and let Mn(/) have
the previous meaning. Then
A2.3.3) lim Mn(/) =
where ©(/) is the "geometric mean" o//@).61
If p(z) is any one of the polynomials considered, then zp(z) is a irn+i with the
highest term zn+\ and | zP{z) \2 = | P(z) \ 2 if z = eie. Thus Mn+i(/) ^ Mn(/).
(This follows also from A1.3.6).) Consequently, limn_oo Mn(/) = m(/) exists,
and m(/) ^ 0. We must show that n(f) = ©(/).
C) Using the inequality for the arithmetic and geometric means, we have
for z = eie,
60 If 0i — 02 | = 5, & ^ tt/2, is given, the maximum and minimum of cos 8\ — cos 02 |
are 2 sin (8/2) and 2 sin2 F/2), respectively.
61 Cf. §10.2.

12.3
PROOF OF THEOREM 12; 1.1; APPLICATIONS
301
A2.3.4)
expjl
^- / log
P*@) |2 =
according to Jensen's theorem (see, for example, Titchmarsh 1, p. 125). There-
Therefore, Mn(/) ^ ©(/), and n(f). ^ ©(/).
On the other hand, let T{6) be a non-negative trigonometric polynomial of
degree k, not vanishing identically, and let T{6) = \ P{el°) | 2 be the correspond-
corresponding normalized representation in the sense of Theorem 1.2.2. Then ©G1) =
{P@)}2. The highest coefficient of {P@)} ~XP*(z) is 1, so that
. 1 f+
A2.3.5)
By Weierstrass' theorem, the inequality
A2.3.6) ©(/) ^ Kf) ^ {©(T7)} T" / f@)T@)d0
z = e
holds for an.arbitrary positive and continuous function 7X0) which has the period
2tt. For the special case in which /@) is positive and continuous, the lemma
follows from this by writing T{6) = {f(d)}~\
D) Now in the general case assume first /@) ^ fi > 0, and let e be an arbi-
arbitrary positive number. By Theorem 1.5.3,-. we can find a trigonometric poly-
polynomial (f>F) such that <?@) ^ fi and
A2.3.7)
We then have
A2.3.8)
hence ®(d) ^
A2.3.9)
so that
A2.3.10)
o1- / \m -
log
- log/@) ^ m'1 I m - f(d) I;
In A2.3.6) we write T{6) = @@) }~\
^u-1
r\i
Then
—1
e,
whence, since e is arbitrary, /i(/) = ©(/).
In the general case we use the obvious inequality n(f) ^ n(f + e) =
®(f + e)> e ^ 0) from which /i(/) = ©(/) follows again. This establishes the
proof of the lemma.

302 ASYMPTOTIC PROPERTIES OF GENERAL POLYNOMIALS [XII]
E) For the proof of Theorem 12.1.1 let us now consider the function
A2,3.11) D{zL>l{z) - 1 = [.D@K - 1] + dnlz + dn2z2 + ¦¦¦ ,
which is regular for | z | < 1. If r < 1, we have
1 \+^ \D(re*L>l{reie) - 1 \2dd = ± f+' \D(reie) |2 ^re") \2d6
A2.3.12)
1 / ^/ iS\ , * ,
~j_^ D(re"L>l(re") dej.
The third term is evidently -23?[?>@)tf)*@)] = -2D@)Kn . As r -» 1 - 0,
we obtain, by use of A0.2.9),
lim ±- r\D{reieL>*n{reie) - 1 |2d9 = 1 fV(^) |^(e<J) |2d0 + 1 - 2Z)@)/cn
r_l_0 2,-K J-x 27T 7_t
c» = 2 - 2Z)@)/cn
or in. another form,
i D(o)Kn -1 r +1 d»i r +1 dn212 +1 d* i2 +..
A2.3.13)
-2-
In consequence of Cauchy's inequality this yields, for \ z\ < 1,
A2.3,4)
1 -|z|2
and as n —+ oo, the last expression tends to 0 uniformly in z for | 2 | ^ r < 1.
The same is true for | Z)(O)/cn — 1 |2. Theorem 12.1.1 now follows at once
from A2.3.11).
F) As an application we calculate the limits occurring in Theorem 11.3.3.
The second limit in A1.3.12) is given by A2.1.2); the special case z = oo yields
A2.3.15) lim Kn = lim lim {z~B(z)} = {D@)} ' = {©(/)}"*.
n—*oo n—*oo z~*oo
The same formula A2.1.2) furnishes, for | z | < 1,
A2.3.16) lim <?*+i(z) = lim zn+1 tn+iiz'1) = {?>(*)}~\
n—»oo
, *
und a similar formula holds for <j>n+i(a), \ a\ < 1. Since <j>n+i(a) —> 0 and
^B+i(z) -» 0 (see A1.3.13)), we have from A1.4.5)

[12.4] PROOF OF THEOREM 12.1.3 303
A2.3.17) lim s»(o, z) = E *ZflL>M = —— =L — .
»-« "=° \ — az D(a) D(z)
For instance,
A2.3.18) | ,¦.(,) f--^^, |.|<1,
and in particular (see A1.3.6))
A2.3.19) ± | *,@) |2 =-- lim K2 = lim %:> = -J— = J- .
From the last equation A2.3.15) follows again.
12.4. Proof of Theorem 12.1.3
A) Let g(d) be a positive trigonometric polynomial of degree m, determined
as in §10.3 C), and let D(g; z) = h(z), as there, so that Dig'1; z) = \h{z))~\
Let {\pn(z)\ be the orthonormal set of polynomials associated with the weight
function {gid)}^1 on the unit circle. We have, by Theorem 11.2,
A2.4.1) *n(z) = znh{z-1), n^m.
In close relationship with an idea of S. Bernstein B, p. 158, G5)), we shall
now express the polynomial <?m(z), associated with fF), in terms of the poly-
polynomials \py(z), corresponding to {gid)}'1:
A2.4.2) = *m+m{z) + j_x
+ s- / f(t)<t>M)\ HMMt) \dt,
ZTV J-* ^ J
The last term vanishes because of the orthogonality of <j>m(z).
Let Km and /cm = h@) be the highest coefficients of <j>m(z) and ^m(z), respec-
respectively; then am = KmKnT1 = Km[h@)}~1. Furthermore, by §12.3 B), we have
Ue
z = e";
and because of A0.3.12) this equals
{M0)P2U + O[(logm)-X]} = (Z)@)}2 + O[(logm)-x],
so that
A2.4.3) Km = {DiO)}-1 + O[(logm)-X], am = 1 + O[(log m)~

304 ASYMPTOTIC PROPERTIES OF GENERAL POLYNOMIALS
B) We shall next try to find a bound for
A2.4.4) max | <f>m{z) \ = M = M(m).
Using A0.3.11), we find from A2.4.2) that
A2.4.5) M ^ 0A) + O[(log m)"x]-M-max
XII
m-1
dt, r =
Jt
Because of A1.4.5), the sum under the integral sign can be represented in
the form
A2.4.6)
We now show that
A2.4.7)
v=0
1 -
= O(logm),
r =
uniformly in z for | z | = 1. Indeed, the numerator is a irm in z, which vanishes
for z = f. Therefore, the theorem of S. Bernstein (see M. Riesz 1, especially p.
357) furnishes O(m) for the integrand. Thus the contribution of the arc | f — z
^ to is 0A), while the complementary arc | f — z | > mT1 supplies
0A)
Returning to A2.4.5), we obtain
M ? 0A) + O[(log m)
so that M = 0A). Hence, in view of A2.4.3), A2.4.1), A0.3.11), and A2.4.7),
equation A2.4.2) yields
A2.4.8)
O[(log
O[(log m)-1-A]O(l)O(log m).
The assertion of Theorem 12.1.3 now follows because of A0.3.12). The con-
constants of all 0-terms of this section depend only on L, X, and the minimum
and maximum of fF).
12.5. Asymptotic formulas for the polynomials on a finite segment; proof
of Theorems 12.1.2 and 12.1.4
Theorems 12.1.2 and 12.1.4 follow from Theorems 12.1.1 and 12.1.3, respec-
respectively, almost immediately by using A1.5.2). It suffices to use the first formula.
A) If x and z have the same meaning as in Theorem 12.1.2, we have
lira z-2ntf,2n(z) = {^(z)}; Km ^(z) = 0;
A2.5.1)
lim <?2n@) = 0; Km K2n = k > 0.

[ 12.6 ] THE ASYMPTOTIC PROBLEM UNDER LOCAL CONDITIONS 305
Here we took into account A2.1.2), A1.3.13), and A1.3.12). If A1.5.2) is
taken into consideration, this establishes Theorem 12.1.2.
B) For the proof of Theorem 12.1.4, let n be an arbitrary integer, m = n — 1,
and let gF), as well as h{z), be determined as in §10.3 C). Then by A2.1.5)
and A2.1.6),
-1 - h(z)]de
A2.5.2)
O[(logn)-X], z = e<{
The first integral on the right is O[(log n) x], while the second vanishes, since
h(z) is a irn_i. Therefore,
A2.5.3) 0,@) = O[(log n)-x].
The sequence {/c2n} is bounded from 0 {k\ ^ | <ft>@) |2 ^ {max^^)}, accord-
according to A1.3.6)), so that
pn(x) = Br)-J{l + Ofdognr^^^lz-IDCi)]-1 + 0[(logn)~x]}
A2.5.4) = B/*)* | /)(**) I cos {n^ + 7(«)} + O[(log n)~\
x = cos 6, z = e*e,
which is identical with A2.1.8). The assertion concerning the constant of the
0-term is immediate.
The same result is obtained from the second formula in A1.5.2).
12.6. The asymptotic problem under "local" conditions; proof
of Theorems 12.1.5 and 12.1.6
In this section essential use is made of the approximations given in Theorems
10.4.4 and 10.4.5.
A) First the following problem will be considered:
Problem. Let X, n, and a be arbitrary complex numbers, and let fF) be an
arbitrary weight function on the unit circle. We intend to determine the maxi-
maximum of
A2.6.1) | Xp@) + Ma) I2
when p{z) ranges over the set of all irn which satisfy the condition
J- / /-/n\ i / \ |2
A2.6.2) ~ J_^ f@) | p(z) \2dd=l, z =
Concerning the special case X = 0, m = 1, see §11.3.
We write
P(Z) = Uo^o(z) + Ul<t>l{z) + ¦ ¦ ¦ + Un4>n(z),

30G
ASYMPTOTIC PROPERTIES OF GENERAL POLYNOMIALS
XII
where {<?n(z)} is the orthonormal set associated with /@). Then Z"=o
= 1, and with the notation of A1.3.3), according to Cauchy's inequality,
n
*Z
u,
A2.6.3)
X|2sn@, 0)
2 n
»»@, a)}
n(a, a).
The expression in the right-hand member is the desired maximum.
B) For clarity we now write sn(f; a, z) instead of sn(a, z) and use an analogous
notation for the orthonormal polynomials 4>n(z) = <j>n(f; z), associated with/@),
and their highest coefficients «„ = /<„(/). Then the preceding solution of the
maximum problem shows immediately that for the functions /i@), /@), /2@) of
Theorem 10.4.4,
I X |2«n(/i ; 0, 0) + 23?{.XMsn(/1 ; 0, o)} + | m |2«n(/i ; a, a)
A2.6.4) ^ | X |2«»(/; 0, 0) + 29?{XMSn(/; 0, a) ] + | n |2 sn(f; a, a)
^ I X |2sn(/2 ; 0, 0) + 29?{Mn(/2 ; 0, a)} + | M |2sn(/2 ; a, a).
In particular, we have for arbitrary a
A2.6.5) sn(/i ;a, a) ^ sn(/; a, a) ^ sn(/2 ; a, a).
Furthermore, we find
I «»(/; 0, a) - sn(/2 ; 0, a) |2
A2.6.6) ^ {«»(/; 0, 0) - sn(/2 ; 0, 0)}{sn(/; a, a) - sn(J» ; a, a)}
^ {«»(/i ; 0, 0) - sn(/2 ; 0, 0)} {«»(/i ; a, o) - sn(/2 ; a, a)}.
Then, by virtue of A1.3.5) and A1.3.6),
{sn(/i ; a, a) - sn(/2 ; a, a)}.
Here <$>n denotes the polynomial reciprocal to <?„ . If | a | = 1, the same
inequality holds for <?„ , that is
A2.6.7)
»(/; a) - Kn
; «)
; a, a)}-
Now, if n is sufficiently large,
A2.6.8) Uf2 ; a) = a"|D(/i;o)r'
(see Theorem 11.2). Next, according to A2.3.15),
A2.6.9) lim Knif) = {©(/)}"* = exp \~ [+* log f@) dd\;
n-.oo ^ 41T J-r )

[ 12.6 ] THE ASYMPTOTIC PROBLEM UNDER LOCAL CONDITIONS 307
and similar relations hold for /i, /2 . By Theorem 10.4.4,
A)}- {©OO
P1
-jT [log/2@) -
<
Thus
lim sup
^ (ee/2jr - 1) lim sup {sn(/i; a, a) - sn(/2; a, a)},
-1 |2
or (see Theorem 10.4.4)
lim sup | <?„(/; a) - ^{Dif^a)}'1
n—»oo
A2.6.10)
< e" + {e1 r — 1) lim sup {sn(/i; a, a) - sn(/2 ;a, a)},
where e" is arbitrarily small with e. This reduces the proof of the statement to
the discussion of the difference
Sn(/i ; a, a) - sn(/2 ; a, a).
C) Let us use the abbreviation DC/71; z) = h(z). We find from A1.4.5)
and Theorem 11.2 that
A2.6.11)
Sn(/2 ;a,z) =
h(a)h(z) -
1 - az
provided n exceeds the degree of {/2@)} x = g2F); consequently using l'Hospi-
tal's rule, a = exa, we obtain
Sn(/2; a, a) == (n + 1) | h(a) |2 - 2®t{ah(a)h'(a)}
A2.6.11')
--= {/2(a)P1<n + 1 +29?
r aDu1ayr\
L D(/2;o)J/-
D) The discussion of sn(Ji ; a, a) is slightly more complicated. From
A0.4.24) it follows that
A2.6.12) D(/x ; z) =
Now let p(z} range over the set of the irn satisfying the condition
A2.6.13) 1 l+*f1(e)\p(z)\2dd= 1,
^T J-T
which can be written in the form
*0
z = e ,

308
ASYMPTOTIC PROPERTIES OF GENERAL POLYNOMIALS [ XII
A2.6,4) s/>Wn(A-
-ziz)V'2p(z) \2d6 = 1,
z = e .
A2.6.15)
Therefore, by Theorem 11.3.1, if V = al/2,
+i>(gT1; a, a) ^ max | A - zia)(l - z\a) ¦ ¦ ¦ A - z,a) |' | p(a)
= I A - 2io)(l - 22a) • • • A - ha) |' SnM I a, a).
Making use of the previous result, we find that
SnM I a, a) ^ | A - zia)(l - z\a) • ¦ • A - zta)
A2.6.16)
+ »[«}.
provided n + Z' exceeds the degree of gri(^). Now from A2.6.12) we have
y1; a) D'ifi ;a) _ay -
A2.6.17)
D{grl;a)
2t=[
az,
y
; a) 2t=[\-zPa'
Consequently, the important inequality
A2.6.18) Sn(/i; a, a) ^ {Ma}}'1 |n + 1 + 29? [a
holds provided n is sufficiently large. Since/x(a) = /2(a) = /(a), we have
lim sup [snifi] a, a) - sn(/2; a, a)}
A2.6.19)
^2{/(a)}
-l
DVi;a) D'{h;a)
D{U;a) DM', a)
But this expression is arbitrarily small with e, which establishes the proof of
Theorem 12.1.5.
E) Under the assumptions of Theorem 12.1.6, the function
A2.6.20)
f@) = w(cos 0) | sin 0
satisfies the conditions of Theorem 12.1.5 or of Theorem 10.4.1. Indeed, we
have, if x, = cos 0,, 0 ^ 6P ^ t, e " = f»,
A2.6.21)
»-i
(z -
z = e ;

112.7 ]
whence
A2.6.22)
MB)
APPLICATIONS
c%~~Tj,™-T2~~' ' '— ri~ 1 ,t
— & 61
309
6).
We also observe that/@) is differentiable at 6 = a and that the ratio A0.4.2)
is bounded near 6 = a.
As in §12.5, we use Theorem 11.5, particularly the first formula in A1.5.2),
and find that /c2n = «2n(/) tends to a positive limit and \imn^M<fan@) = 0. Thus
P»($) = P»(cos a) = B/t)}{1 + e^na^Dif^}-1}, lim en = 0.
By use of the function 7(a) discussed in §12.2 A) we obtain
lJ^j1 a)r81(a'«w) = {/(a)} cos \na + y{a)}.
This establishes Theorem 12.1.6.
The validity of the asymptotic formula A2.1.9) has been extended lately
by G. Freud C).
12.7. Applications
A) By means of Theorem 12.1.2 we can readily derive certain asymptotic
formulas for the highest coefficients of the orthonormal polynomials {pn(x)\.
Theorem 12.7.1. Let w(x) be a weight function on the interval — 1 ^ x = +1,
satisfying the conditions of Theorem 12.1.2, and let
A2.7.1) pn(x) = knOxn + fcmz"-1 + kn2xn'2 + • • • , n = 0,1,2, ... ,
be the associated orthonormal system. Then as n —* °o,
1 f+\ , x dx
A2.7.2) fcno SI ir J2n exp j-— jf^ log w(x) - _¦-_,,
and
A2.7.3) f i ,,
Concerning these formulas, see Shohat 2, p. 577. If the right-hand member
of A2.7.3) vanishes, we read A2.7.3) as follows: lim^oo 2~nkni = 0. There is no
difficulty in deriving corresponding formulas for the later coefficients knv, v fixed.
(See Problems 54, 55, 56.) From A2.7.2) we readily derive an asymptotic
formula for Dn/Dn_i, where Dn-is the determinant of Hankel's type defined
by B.2.7) (see B.2.15)). The first asymptotic investigation of these deter-
determinants is due to Szego A, p. 517). The coefficient kno was denoted by kn in
Chapter II (see B.2.15)).
The proof follows immediately from A2.1.3). If
D(z) = d0 -\
we have

310 ASYMPTOTIC PROPERTIES OF GENERAL POLYNOMIALS [XII
lim 2~n (l + A ~ X V) " (kn0 + Kix-1 + kn2x'2 + • • ¦)
\ ? /
n-»oo
do1 -
Now we use Titchmarsh 1, p. 95 and A0.2.10). We note that on account of
the mean-value theorem of Gauss,
r i r ) r i r
exp {—-7-\ log sin 6 dd} = exp < — — / log
1 -z2
dd
= exp (-1 log 0 = 2*,
z = e
Combining Theorem 12.7.1 with the formulas C.2.2), we obtain for the
coefficients An , Cn of the recurrence formula C.2.1),
A2.7.4) lim An = 2, lim Cn = 1,
n—»oo n—»oo
provided the conditions of Theorem 12.7.1 are satisfied. If we compare the
terms with xn~x in C.2.1), we obtain
A2.7.5) Bn = ~
Thus, under the same conditions,
A2.7.6) lim Bn = 0.
n—»oo
The formulas A2.7.4) and A2.7.6) are noteworthy from the point of view
of a classical theorem of Poincare on recurrence formulas (see Blumenthal 1,
p. 16).
B) For the further applications we need only the less informative form A2.2.6)
of our asymptotic formula.
Theoeem 12.7.2 (Distribution of the zeros). Let the weight junction w{x)
satisfy the conditions of Theorem 12.1.2, and let
A2.7.7) Xin , #2n , • • • , Xnn
denote the zeros of the orthogonal -polynomial pn(x). Let xvn = cos 0,n , 0 < 0,n < tt.
If F(d) is an arbitrary Riemann-integrable function, we have
(H7.8) Hm *¦(*¦¦¦> + "(>'.) +¦¦¦+
n
Using the terminology of Weyl, we say that the values {6Pn} are equally
distributed in the interval [0, tt] (see P6lya-Szego 1, vol. 1, p. 70). This
result is proved under slightly more restrictive conditions by Szego A, p. 531).

12.7
APPLICATIONS
311
It is rather remarkable that the asymptotic nature of the distribution is inde-
independent of the weight function. As a consequence of A2.7.8) we obtain the
following result: Let [a, b] be a subinterval of [—1, +1], and let a = cos a,
b = cos j3, ir ^ a > j3 ^ 0. If N = N(n, a, b) denotes the number of the zeros
of pn(x) in the interval [a, b], we have
A2.7.9)
lim
n—»oo
N(n, a,b) a-p
n
Thus the "density" of the zeros xPn towards the end-points of the interval
[—1, +1] becomes large.
For the proof of Theorem 12.7.2 we write A2.2.6) in the following form (see
A2.7.2)):
lim ri {log 1 pn(x) 1 — log 1 /Cnoz" |} = lim n 1 2 log
A2.7.10)
- Xvn
X
= log
2x
= IT
-1
log
1 -
COS0
X
dd.
Here \z\ > 1. The last formula follows from Gauss's mean-value theorem,
since
1
z +
(r - *)(r -
2zz
= eV9, |z| >1.
Thus,
so that (see Titchmarsh 1, p. 95)
n
A2.7.12) lim n'1 E (cos 0,n)k = v'1 / (cos 6)kd6, k = 0,1, 2.
This establishes the statement (see P61ya-Szego 1, loc. cit.).
C) Further applications are the following theorems:
Theorem 12.7.3 (Expansion of an analytic function in terms of orthogonal
polynomials). Let the weight function w(x) satisfy the conditions of Theorem
12.1.2, and let {pn(x)} be the associated orthonormal system of polynomials. Let
f(x) be an analytic function regular on the segment [—1, +1], and let
f(x) ~ foPo(x) + fiPi(x) + • • • + fnPn(x) + ' ' ' ,
A2.7.13)
+!
fn = / f(x)pn(x)w(x) dx,
-i
n = 0, 1, 2,
be its Fourier expansion. Let R be the sum of the semi-axes of the largest ellipse
with foci at ±1 in the interior of which f(x) is regular. Then the Fourier expan-
expansion A2.7.13) is convergent (with the sum f(x)) in the interior and divergent in

312 ASYMPTOTIC PROPERTIES OF GENERAL POLYNOMIALS [ XII ]
the exterior of this ellipse. The convergence is uniform on every closed set lying
in the interior of the ellipse. Moreover
A2.7.14) lim inf |/n |~1/n = R.
n—»oo
Theorem 12.7 A. Let the weight function w(x) satisfy the conditions of Theorem
12.1.2, and let
A2.7.15) fopo(x) + flPl(x) + /2p2(a;) + • • • + fnPn(x) + ¦¦¦
be an infinite series proceeding in terms of the orthonormal polynomials pn(x)
associated with w(x). Let
A2.7.16) lim inf \fn\~lln = R.
n—*oo
Then the series A2.7.15) is convergent in the interior of the ellipse with foci at ±1,
whose semi-axes have the sum R; it is divergent in the exterior of this ellipse. The
series represents an analytic function which is regular in the interior of the ellipse
and. has at least one singular point on the ellipse itself. The expansion associated
with this function is identical with A2.7.15).
Thus the "ellipse of regularity" coincides with the "ellipse of convergence";
A2.7.14) and A2.7.16) are the analogues of the Cauchy-Hadamard formula (see
(9.1.4)). Concerning these theorems, see Szego 1, p. 538; 6, p. 193. Theorem
12.7.4 must be modified in an obvious way if R S. 1, or if R = <x>.
As a consequence of A2.2.6), we see that the domain of convergence of series
of type A2.7.15) is always an ellipse | z \ = const. To prove Theorem 12.7.3,
we use Theorem 1.3.5. We can find a xn_i , say p{x), such that
A2.7.17) \f(x) - p{x) | <
Here e > 0 is arbitrarily small, and M = M(e) is independent of n. Thus,
r+i r+i
/» = / f(x)pn(x)w(x) dx = / {f(x) - p(x)\pn(x)w{x)dx,
J-l J-i
so that
|/» |2 ? M\R~l + eJn f+1 {Pn(x)\2w(x) dx l+' w(x) dx;
j—i j—i
whence lim sup,^*, |/B ]1/n ^ R~x. This furnishes the statement of Theorem
12.7.3 concerning the convergence. Now the last relation must be an equality,
for otherwise, the expansion would be uniformly convergent in the interior of
an ellipse larger than that of the ellipse of regularity, and f(x) would be regu-
regular therein. This argument also furnishes the divergence in the exterior of the
ellipse of regularity.
Theorem 12.7.4 can also be established without difficulty. It is quite re-
remarkable that the domain of convergence of expansions of type A2.7.13) and
A2.7.15) is independent of the weight function w(x) (compare Theorem 9.1.1).

CHAPTER XIII
EXPANSION PROBLEMS ASSOCIATED WITH GENERAL
ORTHOGONAL POLYNOMIALS
We shall now prove four theorems of the equiconvergent type in the sense
of Chapter IX (see the introduction). Two of them, Theorems 13.1.2 and
13.1.4, deal with expansions of a preassigned function on a finite segment in
terms of polynomials orthogonal on this segment. The other two Theorems,
13.1.1 and 13.1.3, are concerned with the expansion of the boundary values of
an analytic function, regular in the interior of the unit circle, \z\ < 1, in
terms of polynomials orthogonal on the unit circle. In all cases the expansions
in question are compared with trigonometric and power series expansions, and
the weight functions are subject to conditions similar to those in the asymptotic
theorems of Chapter XII. The function developed is very general and merely
satisfies certain integrability conditions.
The basic idea of the method used in the proof of Theorems 13.1.1 and
13.1.2 is due to Szego (see 9 where only the case of a segment is considered).
Our present treatment of these theorems is slightly different from and more
general than in Szego 9. The two other theorems are new. It is noteworthy
that no direct use is made of the asymptotic results of the previous chapter.
The methods, however, are very closely related.
After having finished the manuscript, I came into possession of three im-
important papers of Korous C, 4, 5).
In 3, Korous deals with the expansion problem of Theorem 13.1.2. His con-
conditions are of "local" character but less restrictive than those of Theorem
13.1.2. Also1, his method is entirely different from that used by SzegQ in 9 or
from that of the present treatment.
In 4 and 5, Korous proves two other equiconvergence theorems generalizing
the Laguerre series of Theorem 9.1.5.
13.1. Results and remarks
A) Theorem 13.1.1 (Equiconvergence theorem on the unit circle | z | = 1
if the weight function is subject to "local" conditions). Let f{6) be a weight
function on the unit circle which satisfies the conditions of Theorem 12.1.5
(= 10.4.1). Let F(z) be an analytic function regular in \z\ < 1 and of the
class H2 (§10.1).
If sn(z) denotes the nth partial sum of the expansion of the boundary values
F(z)> I z I = 1, w terms of the orthonormal polynomials {4>n(z)\ associated with
f(d), and if sn(z) is the nth partial sum of the ordinary power series expansion of
F(z), we have
313

314 EXPANSION PROBLEMS FOR GENERAL POLYNOMIALS [XIII]
A3.1.1) ' lim {sn(a) - sn(a)} = 0.
n—»oo
Here a = eia has the same meaning as in Theorem 12.1.5.
Theorem 13.1.2 (Equiconvergence theorem on a finite real segment if the
weight function is .subject to "local" conditions). Let w{x) be a weight function
on the segment [—1, +1] subject to the conditions of Theorem 12.1.6. Let $(x)
be an arbitrary real-valued function, measurable in Lebesgue's scjise, for which
the integrals
A3.1.2) j \Hti}2w(x)dx, f"l\Hx) | A - x2)~hdx
exist.
If sn(x) and sn(x) denote the nth partial sums of the expansions of $(x) in terms
of the orthonormal polynomials \pn(x)} associated with w(x) and of the Tchebichef
polynomials {cos nd\, cos 6 = x, respectively, we have
A3.1.3) lim {sn@ -sn(?)} = 0.
n—>oo
Here — 1 < ? < +1, and ? has the same meaning as in Theorem 12.1.6.
It is remarkable that in both cases a wide class of expansions display the
same convergence behavior. The expansion into a series of cos nd occurring
in Theorem 13.1.2, is, of course, the ordinary cosine expansion of $(cos 6). A
comparison of sn(?) with other special expansions can also be readily made.
In making such comparisons, existence of certain other integrals must be re-
required. By the use of theorems thus obtained, the customary convergence
and summability theorems of the classical Fourier expansions can be jeasily
extended to the general expansions in question. Theorem 13:1.2 can be easily
extended to an arbitral finite segment instead of the segment [ — 1, +1].
These theorems hold under the conditions of "local" character of Theorems
12.1.5 and 12.1.6.
B) The following theorems correspond to the conditions of "S. Bernstein's
type" occurring in Theorems 12.1.3 and 12.1.4.
Theorem 13.1.3 (Equiconvergence theorem of S. Bernstein's type op. the
unit circle). Let f(d) be a weight function on the unit circle z = e% which satisfies
the conditions of Theorem 12.1.3 with X > 1. Let F(z) be an analytic function,
regular and bounded for \z \ < 1. Employing the same notation as in Theorem
13.1.1, we have
A3.1.4) lim \sn(z) - sn(z)\ = 0, \z\ S 1,
uniformly in the whole closed unit circle \z \ ^ 1.
The function F(z) has integrable boundary values F{e%e) (for almost all; 6).

[13.2]
MAXIMUM PROBLEM OX UNIT CIRCLE
315
Theorem 13.1.4 (Equiconvergence theorem of S. Bernstein's type on a finite
real segment). Let w(x) be a weight function on the interval — 1 ^ x ^ +1,
x = cos 6, which satisfies the conditions of Theorem 12.1.4 with X > 1. Let
$(x) be an arbitrary bounded function, which is measurable in Lebesgue's sense.
Employing the same notation as in Theorem 13.1.2, we have
A3.1.5) lim {sn(x) - sn(x)} = 0,
~1 < x < +1,
uniformly in the interval [— 1 + e, 1 — e], 0 < e < \.
In the proofs of Theorems 13.1.3 and 13.1.4 essential use is made of Theorems
12.1.3 and 12.1.4, respectively.
13.2. A maximum problem on the unit circle
A) Let f(d) and F(z) have the same meaning as in Theorem 13.1.1. We
write G(d) = Bw)~1f(d)F(eie), so that
A3.2.1)
\G(d)\d6
is convergent. In what follows we may vary/@), but G(d) is supposed to be a
fixed function for which A3.2.1) exists.
Problem. Let X and fi be arbitrary complex numbers, and let \a\ = 1. We
intend to determine the maximum of
A3.2.2)
XP(a)
G(d)P(z) dd
for p(z) ranging over the set of all irn which satisfy the condition
A3.2.3)
~ rf(e)\p(z)\2dd = i,
Z"K j—r
z =fc e .
Concerning the special case X = 1, y. = 0, see §11.3. Compare also with §12.6
A). We write again
• • • + un<t>n{z),
u,
= 1, and
+»
| i ^ j
G{6)p{z) ddl = E uA\4>,(a) + n / G{dL>,(z) dd>\
I 0 I ' J j
A3.2.4) p(z) = Uo<po(z) + Ui4>i(z) -f
where {<t>n{z)\ has the usual meaning. Then
\p(a) +
A3.2.5)
GWL>,(z)dd
Sn(a))+U|2#n,
Here, as in Theorem 13.1.1, sn(a) is the nth partial sum of the expansion of
z =* e ¦

316
EXPANSION PROBLEMS FOR GENERAL POLYNOMIALS '¦¦ [ XIII ]
2irG(d){/@)} in terms of the polynomials {4>n(z)}, and
A3.2.6)
En =
= V
+r
de
z = e
i$
The right-hand member of A3.2.5) is the maximum in question.
B) For clearness, as in §12.6 B), we now write sn(f; a, z), sJJ) a), 4>n(f; a),
Hn(J) for the expressions involved in the previous considerations, whjch are
associated with the weight function /@). We note that «„(/; a) is the nth
partial sum of the expansion of 2xGf@){/@)}~1 in terms of the polynomials
4>n(f; z) associated with /@). We add that in all subsequent considerations,
even if /@) is replaced by some other weight function, we shall keep the func-
function G(d) used before.
Applying Theorem 10.4.4 again, we obtain !
| X |2sn(/i J a, a) + 29? {XjZs»(/i ; a)} + | M |2ff
A3.2.7) ^ | X |2sn(/; a, a) + 2${X/isn(/; a)) + \ fi \2Hn(
= I ^ |2«n(/2 ; a, a) + 23?{X/isn(/2 ; a)} + | /x \2h
In particular, we have
A3.2.8) s«(/i ; a, a) ^ «„(/; a, a) ^ sn(f2 ; a, a),
A3.2.9) Hn{h) ^ HJJ) ^ Hn(f2),
«n(/; a) — sn(/2 ; a) |2 ^ {#„(/) — Hn(f2)\ {sn(f; a, a) — sn(/2 '; a, a)}
A3.2.10)
g Hn(f){sn(fi ;a, a) - sn(/2 ; a, a)}.
13.3. Proof of Theorem 13.1.1
A) Bessel's inequality enables us to obtain
A3.3.1) #„(/) ^ 2
Whence A2.6.19) shows that
lim sup | sn(f; a) - s«(/2; a) |2
A3 3 2)
- h I-
"^'' a' a) ~ s"<-fo a' a) ^
+T
1 —1
D{f2;a)
The right-hand member is arbitrarily small with the positive number e occurring
in Theorem 10.4.4.
B) Next we discuss the behavior of the partial sum sn(/2 ; a) as n t-» °°.
As in §12.6 C), we use the abbreviation Dift1; z) = h(z). We find thalt if n

[ 13.3 ) PROOF OF THEOREM 13.1.1 j 317
is sufficiently large (see A2.6.11)),
A3.3.3) " ' +r _""" _n+i_ *
2x /_»• 1 — az ' ~
Comparing this latter expression with
A3.3.4) sn(a) = i- (+*F{z)~
y — az
we obtain
z =
03.3.5)
j
J 1 — az
The second integral approaches zero as n —¦> oo, since
,_;.-.
^ fF)h(a~)h(z) - 1 ;
.oj : , z =
1 — az
has a limit at 0 = a. The first integral can be written as follows:
. ^ .. - -- ^ .. 1 — az
A3.3.7)
1 — az
since (see the remark at the end of §10.1)
A3.3.8) f F(z) l- ~ h{a)/hSz) dz = 0.
J\z\-i a — z
By virtue of Schwarz's inequality,
4x2 lim sup | sn(f2;a) — sB(o) |2
4 sin2 °
-*d$
2
+jt r+T f
sin2 —-r—

318 EXPANSION PROBLEMS FOR GENERAL POLYNOMIALS [ XIII J
Combining this result with A3.3.2), we find Km sup*.,*, | «„(/; a) — sB(a) |
to be arbitrarily small with the positive number e occurring in Theorem 10.4.4.
This establishes the statement.
13.4. A special case of Theorem 13.1.2
A) In this section we consider the special case
A3.4.1) w(x) = A - xY^pix)},
where p(x) is a ttj which is positive in the interval [— 1, +1]. The corresponding
orthogonal polynomials (for 2n > I) have been calculated in §2.6. We shall
verify the validity of Theorem 13.1.2 in this special case. To this' end it
suffices to assume the existence of the integral
A3.4.2) f^ \$(x)\(l- xY^dx,
a condition which is more general than that required in Theorem 13.1.2. As
in that theorem we have ? = cos a, — 1 <?<-{-l,0<a:<7r.
By virtue of the Christoffel-Darboux formula C.2.3)
Kn+\ J-\ X — ?
or
¦7- sn(cos a)
/10 a o\ fK *,( n\ Pn+i(cos 0)pn(cos a) — pn(cos 0)pn+i(cos a)
A3.4.3) == / <?>(cos 6) —— ^— i—^ ^ -i
> cos 0 — cos a
•{p(cos0)}~1d0.
Assume now 2n > 1. Use formulas B.6.2) and write B/ir)ie<neh(ew) *= un@)
+ wn@). Then
pn(cos 0) = un{6),
A3.4.4)
pn+i (cos 0) = 9?{e [mb@) + wn@)]j = mb@) cos 0 - vn(d) sin (?,
so that
pn+i(cos 0)pn(cos a) — Pnjcos 0)pn+i(cos a)
COS 0 — COS a
\un(d) cos 0 — vn(d) sin 6}un(a) — un(d){un(a) cos a. — vn(a) sin a}
cos 0 — cos a
A3.4.5)
= un{6)un{a) — vn{d)Un{a)
cos 0 — cos a
un(d)vn(a) — vn{d)un{a) ct.
cos 0 — cos a

[ 13.4 ] A SPECIAL CASE OF THEOREM 13.1.2 319
In view of Riemann's lemma, this leads to
'sn(cos a)
kn+l
K
n
Now
{p(cos e^iitnifivnia) - vn(d)un(a)\
= -{p(cos d^-'SiU
7T
Repeated application of Riemann's lemma shows that h{ea){h{ei6))~l can be
replaced by 1, and
) sin a 12 sin °LJ1 sin a ~
sin a (cos 0 ~ cos a) = sin a 12 sin °LJ-1 sin
2 "" 2
i-i
by {2 sin (a - 0)/2} \ Thus the right-hand member of A3.4.6) assumes the
form
1 f' *.t n\ sin n@ — a) ,. , /
- / *(cos d) ~—- dd + o(
ir Jo . 6 — a
*(cos e)
( *(cos e) „, + 0A)
sin-^-
Finally kn/kn+i — ^ when n is sufficiently large (see B.6.5)). This establishes
the statement.
B) In the same special case we intend to calculate Kn(?, ?), where Kn is the
kernel polynomial denned by C.1.9). We have, by C.2.4),
A3.4.8) Knit, *) = ~- {p'n+l(Z)Pn® ~ Pn(*)Pn+l(*)}.
Kn+1
Now, by A3.4.4), for a sufficiently large n,
sin a pi (cos a) = — MB(a),
A3.4.9) sin a p^+1(cos a)
= —un(a) cos a + »n(«) sin a -\- MB(a) sin a + «B(a) cos a,

320 EXPANSION PROBLEMS FOR GENERAL POLYNOMIALS [ XIII
and kn/kn+i — \. Thus,
2 sin a Kn(cos a, cos a) = [ — u'n(a) cos a -\- v'n{a) sin a
+ un(a) sin a + vn(a) cos a}un(a)
-f- un(o:){urt(Q:) cos a — wB(a) sin a}
= sin a S\[(
hence
2 sin a „ , \ • r* fwB (a) -\- iv'n (a)
i^« (cos a, cos a) = sin a$ '
| un(a) + %Vn{a) \2 \un(a) + tvn(a)
sin a + i3 {f^tl^
II () + () |2
Observing that
• . /' \ f v 1 ~7'
we get the important formula
/Tf , I
/i(a) I
{Y6A.W) .__
+ B-sin aT'S an+l h^ , ? = cos a, a = e'a, 0< a < x.
This holds for sufficiently large values of n.
13.5. Preliminaries for the proof of Theorem 13.1.2
A) Let w(x) and <?>(x) have the same meaning as in Theorem 13.1.2. Write
G(x) = w(x)$(x), so that the integral
A3.5.1) f+1\G(x)\dx
is convergent. In subsequent considerations we may vary w(x), but G(x) is
supposed to be a fixed function for which the integral A3.5.1) exists.
B) In what follows we apply Theorem 10.4.5 with /@) = w(coa 0) | sin 0 |.
This function satisfies the conditions of the theorem mentioned (see §12.6 E)).
We define the functions Wi{x) and w2(x) by
A3.5.2) /,@) = w\,(cos 0) | sin 0 |, v = 1, 2.
Obviously, ? = cos a, 0 < a < x,
A3.5.3) 0 ^ wi(x) g w(x) ^ w2(x),

13.5
PRELIMINARIES FOR THE PROOF OF THEOREM 13.1.2
321
Moreover, w2(x) is of the form A3.4.1), while Wi(x) = k(x)u(x) = {r(x)\2u(x).
Here u(x) is of the form A3.4.1), and r(x) is a polynomial of degree aZ/4. (See
the remark to Theorem 10.4.4.)
C) Problem. Let X and fi be arbitrary complex numbers, and assume that
— 1 < ? < +1. We intend to determine the maximum of
A3.5.4)
+i
n / G(x)P(x) dx
-i
when p(x) ranges over the set of all xn which satisfy the condition
A3.5.5)
—i
\p(x) \2w(x)dx = 1.
This is the problem corresponding to that of §13.2. We substitute again
A3.5.6) p(x) = -uopo(x) + uxpi{x) + ... + unpn(x),
where (p»(x)} is the orthonormal set of polynomials associated with w{x).
Then 2^T=o | uv |2 = 1, and according to Cauchy's inequality
4- m I i G(x)P(x) dx
Z ur{\p>(?)
4-AtJi G{x)Vv{x)dx\
A3.5.7)
+1
-i
G{x)pv(x) dx
= iXfXnft) 4-29?[XMSn(?)] 4- I At \2Hn.
Here Kn(?) = Kn(w; ?) has been written for the "kernel" Kn(?, ?), and
A3.5.8) sn@ = sn(w; 0 = E pX
G(x)p,(x) dx = I G(x)Kn(Z, x) dx,
-i 7-i
A3.5.9)
Hn = Hn{w;0 =
-i
We notice that sn(?) is the nth partial sum of the expansion of $(x) =
[w(x)}~1G(x) in terms of the polynomials pB(x) associated with w{x).
The right-hand member of A3.5.7) is the maximum in question. Hence
X f Kn{Wl; 0 4-
A3.5.10)
^ | X
^ | x
n(w; f) 4-
; 0 4-
; {)] +

322 EXPANSION PROBLEMS FOR GENERAL POLYNOMIALS [ XIII ]
In particular
A3.5.11) Knto J 0 ^ Kn{w; ?) ^ Kn(w2 ; ?),
A3.5.12) Hn{Wl) ^ Hn(w) ^ Hn(w2).
Furthermore,
I sn(w; ?) - sn(w2 ; ?) |2 ^ (ffn(w) - #„(«*) j {#„(«,; ?) - #»
A3.5.13) ^ #„(«,) {Xn(Wl ; f) - #»(«* ; 0}
"+1
< / \$>(t} !2WrWr ( K (?/>, • ?\ — K
-i
if we observe Bessel's inequality. We next show that the last difference is
arbitrarily small with «¦, uniformly in n.
13.6. Proof of Theorem 13.1.2
The main tool of this proof is the special equiconvergence theorem of
§13.4 A.) and the representation A3.4.10) of the kernel. The latter yields
immediately the representation
A3.6.1) - -^''a)
since {^2@:)} = /2(a) = /(«)• We derive, on the other hand, an inequality
for Kn(wi ; ?) analogous to that in A2.6.18). Using the notation introduced
in §13.5 B) and writing \al = V', we obtain as in §12.6 D),
A3.6.2) Kn+Vl2{u; $) ^
Also, u(x) = m(cos 6) =-- {gi(e)} \ sin 6 \~\ so that from A3.4.10)
ir\g,(a)ylKn+vl2{u; 0 -- n + V/2
- dt\ a
L
4
; a)
^4
1; a)
where, as before, ? = cos a, a = eia. Now we can use A2.6.17). Moreover
from A2.6.12)
;a) g(gr^) A

[13.7] PROOF OF THEOREM 13.1.3 323
also toiCa)}^) = /i(a) = f(a), so that from A3.6.2)
f; *) S n + \ + ft Fa
A3.6.5) L
On comparing this with A3.6.1), we have
Kn(wi; ?) - Kn(w2; ?)
+ {27r/(a)sina}
-l
DUi,a) D(f2;a)
The left-hand side is non-negative, while the right-hand aide is arbitrarily small
with e.
Returning to A3.5.13), we notice that sn(w2 ; ?) is the wth partial sum of the
expansion of \w2(x) \"lG(x) = \w2{;x)\ w{x)^{x) in terms of the polynomials
associated with w2(x). Since {^(a;)}^^) ^ 1, and since the second integral
in A3.1.2) exists, the result of §13.4 A) can be applied. Thus sn(w2 ; ?) =
sn(w2 ; cos a) can be replaced by the nth partial sum of the corresponding
Fourier expansion with an error o(l), n —* *>. Also {w2(x)}~1w(x) = 1 for
x = ?, so that the partial sum can be replaced by that of $(x), which is sn(?).
This completes the proof of Theorem 13.1.2.
Remark. It is easy to show that the difference
A3.6.7) Hn{w) - Hn(w2),
occurring in A3.5.13), is also arbitrarily small with e. If this fact were used,
it would suffice to obtain an upper bound for the first difference in A3.6.6).
This remark furnishes a slight variation of the preceding argument.
13.7. Proof of Theorem 13.1.3
A) It is sufficient to discuss the statement for | z \ = 1. The partial sums
in question are
A3.7.1) 1 rF^)sn^,z)f(x)dx and 1 fFtf1 ~ ^fdx, f = eix,

324 EXPANSION PROBLEMS FOR GENERAL POLYNOMIALS [ XIII ]
where sn(f, z) has the usual meaning. We introduce the difference of the
"kernels"
A3.7.2) An(r, z) = sn(r, z)f(x) -
-ft
and show that
A3.7.3) lira / Fit)Ani{,z)dx = 0, f = eix.
uniformly on the unit circle \ z \ = 1.
B) First, as a consequence of the Lipschitz-Dini condition A0.3.10) assumed
for fid), we prove that a similar condition (with the exponent —X instead of
-1 - X) holds for D(J; z) = D(z); that is,
A3.7.4) | Diei(e+i)) - DieiB) \ < U \ log 8 \~\
where U is a positive constant. Let m be an arbitrary integer. By use of
A0.3.12) we obtain
D{eue+i)) - Die*)
The last term, according to Theorem 1.22.2 is equal to «50(m); whence the
bound C[(log m)~x] -f 8O(m) follows. Putting m ~ h~l \ log 8 j~x, we have
(log m)~ '^ ) log 8 \~ , and 8m ~ ) log 8 \~ , which establishes the statement.
This furnishes, for the function 7@) = 3{log W")) denned by A2.1.7), also
the relation
A3.7.5) I 7@ + 8) - 7@) \<L"\ log 8
i-x
where L" is a positive constant. These results hold for an arbitrary X > 0.
By the same argument the following more general inequality follows:
A3.7.6) I D(zi) - Diz2) I < Z/ I log I zi - z2 \\~\
where zi and z2 are arbitrary in the unit circle \ z\ ^ 1.
C) Let I z I = I f I = 1, z ?* f. From A1.4.5) and Theorem 12.1.3 it is seen
that
- 1-~?^1 + I 1 - U \-lO[i\og n)"x]
A3.7.7) l rz
tf)]-1 ~ 1 ^-1
Now, according to Cauchy's theorem,

[ 13.7 ] PROOF OF THEOREM 13.1.3 325
A3.7.8)
- z
since the integrand is a function of f which is of the class lh (see the remark
at the end of §10.1). Here we used the condition X > 1. And by Riemann's
lemma,
A3.7.9) Km f (fz)rt+1F(r) Ml^H^Zli dx = 0, f = e
J|f|1 1 fZ
|=1
(Concerning the uniformity cf. below.) Therefore we obtain, if e is an arbi-
arbitrary positive number, E = E(z, n, e) is the set on | f | = 1 with | f — z | ^
en, and 2?' the complementary set, the relations
+O{(logn)
-X|
0{(logn)-x|0(logrO
On the other hand, the numerator of A1.4.5) is bounded in the present case
so that Theorem 1.22.2 yields (see §12.4 B)) sB(f, z) = 0{n), and hence also,
An(f, z) = O(n). This holds uniformly for \z\ = | f | = 1. Therefore, we
have
r
n{X, z)dx = O(n)tri~l = eO(l), t = e™,
where 0A) is independent of e and n. This establishes the statement.
D) The uniform validity of A3.7.9) needs some explanation. According to
A3.7.4) (see the remark at the end of B)),
¦ 1 - fz
is a function of f which is of the class //i. If pn(r) denotes the nth Cesaro
mean of the partial sums of the power series expansion of F{?), we have
f r+1F({)K(?)dx= f fn+1{F({) -Pn(fi\K(f)dx = f + [.,

320
EXPANSION PROBLEMS FOR GENERAL POLYNOMIALS [XIII
where El is the set on | f | = 1 with | f - 2 | ^ e, and E[ the complementary
sot. The integral over i\ as n —> 00 is
0A)
- pB(f)|rfx = 0A)
- pB(f)
The integral over E[ (since F(f) - Pn(f) is uniformly bounded) is equal to
0A) f,\K@\dx,
J*i
that is, it is arbitrarily small with e.
13.8. Proof of Theorem 13.1.4
A) We show that
A3.8.1) lim f *E)An(«, x)(l - f)^ = 0,
uniformly in—1 + e^xgl — e; here a'b(^, x) is the difference of the
"kernels"; that is, according to Christoffel's formula
7.**, x) = A -
A3.8.2)
n+l X —
sin Bn + 1) ^-±J
sin Bn + 1)
sin
sin
j
= cos <f>, x = cos 8.
The symbol &„ is used for the highest coefficient of pn(x).
B) We first notice that from A1.5.2) for n ^ 1
kn = ^W
A3.8.3)
*
l
follows so that on account of formulas A2.4.3) and A2.5.3) (see also Theorem
12.7.1)
kn = B«-)-*{l + O[(logn)-x]j2B{[D@)]-1 x
A3 8 4)
B)
k
O[(logn)-X].
n+l
From A2.1.8) we conclude, using the notation A2.2.2), that

13.8
PROOF OF THEOREM 13.1.4
327
An(?, x) = An(cos0, cos0)
= ir~l(x- g)~1\W(x)\~~i{W(?)\i{cos[(n+ 1H
- cos [nd + 7@)] cos [(n +1H + 7
A3.8.5)
cos [n0
sin Bn + 1) t+1 sin Bn + 1) tr
sin
sin
+ I x -
I
0[(log n)""x]
= 4»@, 0) - BB@, B) + \x- S I"' O[(log n)
The first term can be written in the form
in^-JJ {W(cos 6) \^{W(cos
AnD>,6) = B7r)-1
¦ {cos [n0 + (n + 1H + 7@) + yF)} + cos [n0 - (n + 1H + 7@) - yF)]
- cos [(n + 1H + nd + 7@) +.7@)] - cos [(n + 1H - n6 + 7@) - 7^
= < 2tt sin
. 0 -
sin ——
•/sin [Bn + 1) ^
+ sin [Bn + 1) I
+
J sin
- 7@)] sin
2ir
sin [
Bn + 1)
+ 7@) + y(e)
• 0 + 0
olli ^r
sin[Bw
-7@)
.0-0
sin z—,?:—
C) The integrals
A3.8.7)
0)ri{^(cos 0)!* cos [7@) ± 7@)] - 1
+*
. 0 ± 0
sin —7:—
sin [7@) ±7@)]
. 0 rb 0
sin ——-
d4>,
d<f>

328
EXPANSION PROBLEMS FOR GENERAL POLYNOMIALS [XIII
exist (see A2.2.4), A2.1.4), and A3.7.5)), so that by Riemann's lemma,
r+*
A3.8.8) lim / <f>(cos <t>){AnD>, 8) ~ BnD>, 6)\ d<t> = 0,
lim /
I—»0O /~JI
uniformly in 8. Let 77 be an arbitrary positive number, E = E{8, n, rj) the
set I eo.s <f> — cos 8 | ^ r/n, and E' the complementary set. We have
I <J>(cos </>)a1(cos </>, cos 8) d<t>
J
A3.8.9) + / *(cos 0)A'n(cos </>, cos 6) d<t>
Je'
/ ) {4n@, 8) - Bn{4>, 8)\d<t>
O[(logn)
"A]
Now
-ifmaxFd)!1 f , , sin{T(^)
A3.8.10)
sin{7@) - l(8)\
2 sin -
- e
= O(n) + 0A) I 4> - 8 T I log I 4> - 9 \ \~\
\BnD>,6)\ g 7rBn + l),
so that the first term in the right-hand member of A3.8.9) is
O(n) f d<t> + 0A) f \4> - 8 I-' I log \4> - 8 1 pX d* = vO{t) + (log n)~x+l 0A),
A' U1
where the bound of 0A) in the term r\O{\) is- independent of 77. The third
term is O[(Jog n)~x]O(log n). Finally, in the last term we apply Theorem
1.22.3. Since the polynomial Pn+i(x)pn(^) — pn(x)pn+i(^) is uniformly bounded,
— 1 ^ x S +1, —1 ???;§= +1, we have An(?, x) = O(n), uniformly in x and
?, confined to an interval of the form [—1 + 6,1 — e]. Thus, for the term
in question, the bound r)vrl0{n) = rjO(\) can be obtained. Here 0A) is again
independent of -q. This completes the proof of Theorem 13.1.4.

CHAPTER XIV
INTERPOLATION
In this chapter we shall consider certain problems of interpolation related to
the theory of orthogonal polynomials. In particular, we are interested in inter-
interpolations whose abscissas are the zeros of the orthogonal polynomials pn{x)
associated with a given distribution of the type da(x) or w(x)dx. We deal' with
the ordinary Lagrange polynomials and with the "step polynomials" (Treppen-
polynome) introduced by Feje'r. This topic is naturally very closely related
to that of the next chapter on mechanical quadrature.
Concerning the subject matter of this and of the next chapter, see the
recent monograph of Feldheim D).
14.1. Definitions; problems
A) Let [a, b] be a finite or infinite interval, and let
A4.1.1) Sn: xln < x2n < ¦ ¦ ¦ < xnn , xm ^ a, xnn ^ b
denote a set of n distinct points of this interval. Let l(x) be a irn , not iden-
identically zero, vanishing at x = xvn , v = 1, 2, • • • , n; it is determined save for a
constant nonzero factor. When there is no ambiguity, we shall write x, instead
of Xvn ¦ The polynomials
A4.1.2)
are called the fundamental polynomials of the Lagrange interpolation cor-
corresponding to the set Sn ¦ They have the property
A4.1.3) Kx,) =8^, v, m = 1, 2, • • • , n.
Let /i, f2, • • • , /„ be arbitrary values. Then the expression
n
A4.1.4) Ln(x) = Y,f,l{x)
represents the uniquely determined Trn-\ which assumes the value /„ at x = xv.
This is the nth Lagrange polynomial corresponding to the abscissas Sn ¦ It is
readily seen that
A4.1.5) h(x) + k(x) + • • • + ln(x) = 1.
B) Now let Sn , n = 1, 2, 3, • • • , be a sequence of sets of abscissas. If f(x)
is a given function defined in [a, b], we can consider the sequence of the cor-
corresponding Lagrange polynomials Ln(x), n = 1, 2, 3, • • • , defined by A4.1.4)
with/,, = f(x,n). Various convergence and divergence properties of this sequence
have been studied under proper conditions of continuity concerning/(x). In
329

330 INTERPOLATION [ XIV ]
what follows we shall be interested exclusively in the case in which the abscissas
Sn are the zeros of the orthogonal polynomials associated with a preassigned
distribution. Different types of "convergence" can then be considered; for
instance:
(a) ordinary convergence: limn_c» Ln(x) = f{x);
(b) convergence in mean: lim^*, Jba \Ln(x) — f(x) |2dx = 0;
(c) generalized convergence in mean, arising from (b) by replacing the ex-
exponent 2 by p, where p is positive;
(d) quadrature convergence: limn^x Jba{Ln(x) — f(x)}dx = 0.
The last type is particularly important from the point of view of Chapter XV.
Of course, a certain fixed weight function can be introduced into the integral
conditions.
C) Let a and b be finite. If only the continuity of f{x) is assumed, the
behavior of the Lagrange polynomials is rather irregular. Faber B; see also
Feje"r 11, pp. 450-453. and Marcinkiewicz 1) proved that for a given arbitrary
sequence {Sn \ there exists a continuous function f(x) such that the sequence
\Ln(x)\ is not uniformly convergent. S. Bernstein D) has even proved the
existence of a continuous function for which Ln(x) is unbounded at a preassigned
point x0. According to Helly's theorem (§1.6), this is equivalent to the un-
boundedness of the sequence of "Lebesgue constants"
n
A4.1.6) 22 I ^(^o) | , as n —> °o.
In the case of the special sequence
__ (./-vkjf2v i "i -IL |, _ i o ... »i
a = —l,b = +1, that is, for the zeros of Tn(x), much more is known. Griin-
wald A) and Marcinkiewicz B) proved the existence of a continuous function
f(x) for which the sequence of Lagrange polynomials corresponding to these
x,n is everywhere divergent, even everywhere unbounded.
D) In order to obtain convergent sequences of interpolation polynomials,
it is necessary to introduce additional restrictions concerning either: (a) the
function f(x), particularly, restrictions concerning its modulus of continuity
(see Theorem 1.3.2), or (b) the interpolation polynomial, such as conditions
concerning its derivative, and the like.
We introduce the polynomials
k(x) = ji - y^ (x - x
f)r(x) = (X — X,)\1,(X)\2,
called the fundamental polynomials of the first and second kind of the "Hermite

[14.11 DEFINITIONS; PROBLEMS 33L
interpolation" corresponding to the set Sn . These 7r2n_i are completely deter-
determined by the conditions
h^xj = 8vli , h'^x,) = 0; %{x») = 0, &,'(zM) = 8^ ,
A4.1.8)
v, m = 1, 2, • • • , n,
and for any given set of values /„, f',,
A4.1.9) Wn(x) =
I—1
represents the uniquely determined 7r2n_i for which
A4.1.10) Wn(x,) = /,, W'n(x,) = /,', v = 1, 2, • • • , n.
E) Again, let Sn , n = 1, 2, 3, • • • , be a sequence of sets of abscissas. If
f(x) is a given function having a derivative in [a, b], we can take /, = f(x,n),
/,' = f'{x,n) and consider the corresponding "Hermite polynomials2 Wn(x),
n = 1, 2, 3, ¦ • • , defined by A4.1.9). If only the continuity of f(x) is known,
we may choose/,' arbitrarily; for instance, we may take/,' = 0. If f, = 0, we
call Wn(x) the step polynomials corresponding to {Sn\. In the more general
case |/, | < A, where A is a constant independent of v and n, they are called
generalized step polynomials.
The simple and generalized step polynomials Wn(x) display a more regular
behavior as n —> °o than do the ordinary Lagrange polynomials Ln{x). They
coincide with the given function/(x) at the same points as do the corresponding
Lagrange polynomials, but they satisfy certain additional restrictions concerning
their first derivatives. Their degree is In — 1 instead of n — 1. We shall
show that for certain sequences {$-„} the step polynomials (even the generalized
step polynomials) are uniformly convergent if f(x) is an arbitrary continuous
function.
The step polynomials and their generalizations have been introduced and
investigated by Feje"r A0, 11, 13, 16). The trigonometric analogue for f', — 0
had been previously considered in the simplest case of equidistant abscissas by
Jackson D, p. 145, Theorem VI).
For the fundamental polynomials A4.1.7) the following important relations
hold:
h(x) + h2(x) + • ¦ • + hn{x) = 1,
A4.1.11) » »
yl 0™ /) ( 0™ 1 ¦ l.i. x n ( 0™ 1 — o*
An important consequence of the first identity may be pointed out for the step
polynomials, that is, when/, = 0. Let the set ($„) be such that
A4.1.12) Kix) SO, a ^ x ^ b, v = 1, 2, ¦ • • , n.
62 We write "Hermite polynomials" in quotation marks in order to avoid confusion
with the Hermite polynomials of Chapter V.

332 INTERPOLATION [ XIV ]
Then A4.1.9) implies
A4.1.13) min/, ^ Wn(x) ^ max/,, a ^ x S b, n = 1, 2, 3, ¦ ¦ ¦ .
We observe further that A4.1,12) is equivalent to the fact that the linear
functions
A4.1.14) v,(x) = 1 - Z^ (X - x,)
I \Xv)
do not vanish in the open interval a < x < b, or, what amounts to the same
thing, that the "conjugate points"
A4.1.15) • *' + Z^' *=1,2, ...,n,
lie outside of this interval. (See Fejer 13, 16.)
14.2. Fundamental polynomials of the Lagrange interpolation
A) Theorem 14.2.1. Let da(x) be an arbitrary distribution on the interval
[a,b], {pn(x)\ the associated orthonormal polynomials, and li(x), h{x), ¦ •• , ln{x)
the fundamental polynomials A4.1.2) of the Lagrange interpolation corresponding
to the set of zeros of pn(x). Then we have
A4.2.1) / l(x)Ux)da(x) = Xr5rjl, v, M = 1, 2, • • • , n,
Jo.
where \, are the Christoffel numbers defined by C.4.1).
This follows immediately from C.4.1), since lv(x)l,,.{x) vanishes at the zeros of
pn(x) if v ?± n. For v — fx the zero xv is the only exception, and we have \lv{xv) }2
= 1.
Also, by A4.1.8),
I hv{x)da{x) = \v, I h',(x)da{x) = I xh[(x)d<x(x) = 0;
/ t)y(x)da(x) =0, / i)'y(x)da(x) = \y, / xi)l(x)dx = \xv;
Ja Ja Ja
v = 1,2, ••• ,n.
B) We obtain as an important consequence the following result:
Theorem 14.2.2. Let Kn(x0, x) have the same meaning as in C.1.9). Then
the following identity holds:
n—\ n
A4.2.3) tfn-iOco, x) = Z) pMp,(x) = E ^lAx^Ux).
63 Erdos-Turiin 1.

14.3 ] CONVERGENCE IN MEAN OF LAGRANGE POLYNOMIALS
333
Indeed, let p(x) be an arbitrary irn^ . The polynomial Kn^(x^,, x) is uniquely
determined by the condition C.1.12). But by A4.1.4) and A4.2.1),
b ( n
= p(x0),
and this establishes the statement.
On setting x0 = x = xv in A4.2.3), we obtain a new proof of C.4.8).
C) In view of A4.2.1), A4.1.4) shows that
A4.2.4)
Ln(x)
K-l
Furthermore, let /(x) be an arbitrary complex-valued function for which the
integrals jbaf(x)x1'da(x), v = 0, 1, .. • , n - 1, exist. Then A4.2.3) yields the
identity
A4.2.5)
n— 1
E
f(x)p,(x) da(x)
Thus, according to Bessel's inequality (see C.1.5)),
A4.2.6)
Ex:1
da(x)
^ \f(x)\2da(x),
provided the last integral exists.
14.3. Convergence in mean of Lagrange polynomials
A) Theorem 14.3.1. Let da(x) be an arbitrary distribution on the finite inter-
interval [a, b], and let \pn(x) \ be the corresponding set of orthonormal polynomials. For
the complex-valued function f{x) let the Riemann-Stieltjes integrals
A4.3.1) [bf(x)xnda(x), (l\f{x) | 2da(x), n = 0, 1, 2, • • • ,
Ja Ja
exist. Then if Ln{x) denotes the Lagrange ¦polynomial of degree n — 1 which
coincides with f(x) at the zeros of pn(x), we have
A4.3.2)
lim
f(x) - Ln(x) \2da(x) = 0.
The convergence in mean in the usual .sense follows from A4.3.2) for any f(x)
integrable in Riemann's sense in the case where da(x) = w(x)dx, w(x) ^ m > 0.
For the proof wo use A4.2.4), A4.2.6), and the later Theorem 15.2.3 on
quadrature convergence. Now
64 Cf. Erdos-Tur&n 1 and Shohat 8.

334 INTERPOLATION [XIV
f |/(z) - Ln(x) \2da(x) = f \f(x) \2da(x) + Ex, \f(xr) |2
Ja Ja v*=\
x7) f f(x)l(x)da(x)
J
\f(x)\2da(x) + E
A4.3.3) Ja
"
b\f(x)\2da(x)
J X, \f(x,)
Hence, using Theorem 15.2.3, we obtain
fb rb
A4.3.4) limsup / |/(x) - Ln(x) \2 da(x) ^ 4 / \f(x) \2 da(x).
n-*<x> Ja Ja
Next let e be an arbitrary positive number, and p{x) a polynomial for which
(Theorem 1.5.2)
A4.3.5) f \f(x) - P(x)\2da(x) < e.
If Ln(f)x) denotes the Lagrange polynomial of degree n — 1 corresponding to
f(x), we have
A4.3.6) f(x) - Ln(f;x) = f(x) - P(x) - Ln(f - P;x),
provided n exceeds the degree of p{x). This establishes the statement.
B) In case da(z) = A - x^dx, a = - 1, b = +1 (that is, for the Tchebichef
abscissas of the first kind) Erdos and Feldheim showed that we may even assert
the validity of the following theorem:
Theorem 14.3.2.65 Let p be an arbitrary positive number, and let f(x) be a
continuous function. Then for the Lagrange polynomials corresponding to the
Tchebichef abscissas of the first kind,
r+i
A4.3.7) lim / \f(x) - Ln(x) \p A - x*)"* dx = 0.
n—»oo J—l
It suffices to show this for even integral values of p. In the proof which is
based on induction with respect to the even integer p essential use is made of the
property of the Tchebichef abscissas formulated in Probhem 57 (see below).
66 Erdos-Feldheim 1; cf. also Feldheim 2 and 3, pp. 33-36.

[14.4 ] LAGRANGE POLYNOMIALS FOR JACOBI ABSCISSAS 335
Feldheim points out B, p. 330) that A4.3.2) is not true in general if da(x) =
A — x2)hdx, a = —1,6 = +1 (that is, for the Tchebichef abscissas of the second
kind), and if the exponent 2 is replaced by p = 4, For the same abscissas it is
also not true that
r+i
A4.3.8) lim / \f(x) - Ln(x) \2 dx = 0.
n->n J—l
In both cases the superior limit of the integrals in question may be + °o if f(x)
is a properly chosen continuous function.
C) From Theorem 14.3.1 the mean convergence in the sense of A4.3.8) fol-
follows immediately for the Jacobi abscissas, that is, for the zeros of the Jacobi
polynomial Pi"J)(x), provided max (a, /3) ^ 0 (cf. Erdos-Turan 1); here f{x)
is an arbitrary continuous (or even Riemann-integrable) function.
Recently, A. Hollo A) proved this mean convergence for the Jacobi ab-
abscissas with max (a, /3) < \ provided f(x) is continuous. The bound \ is
the precise one, on account of the last result of Feldheim mentioned in B).
Hollo also investigated the validity of
,+i
A4.3.9) lim / | f(x) - Ln(f) \dx = 0,
n-»oo J—l
where f(x) is continuous, and showed that A4.3.9) holds for max (a, /3) < f.
The latter bound is again the precise one, at least in the sense that for
max (a, /3) > f and for a proper continuous f(x), the statement A4.3.9) does
not hold. (This follows from the second part of Theorem 15.4.)
14.4. Lagrange polynomials for Jacobi abscissas
For the following discussion Theorem 8.9.1 will be found invaluable, while
the bounds of P{na'?)(coa 6) obtained in §7.32 will also be used.
A) Assume a > — 1, /3 > — 1, and let Xi > x2 > ¦ ¦ ¦ > xn denote the zeros
of the Jacobi polynomial Pia'?)(x) in decreasing order. Here xv = cos #„,
0 < 6V < ir. Then we may assert the following theorem:
Theorem 14.4. Let f(x) be continuous in [—1, +1] with the modulus of con-
continuity u>(<5),66 Then the Lagrange polynomials coinciding with f(x) at the zeros
of P^"' j(x) converge uniformly to f(x) in every interval [—1 + e, 1 — e], where
0 < e < ^, provided that a>(<5) = o( I log 8 I ~J). The same holds in the interval
1 fi7
[—1 + e, I] if either a ^ — \ and u>(<5) = o{ \ log 8 | ~ ) or n — \ ^ a < m + \
and f(x) has a continuous derivative of order // with modulus of continuity w^E)
satisfying the condition u^S) = oE"~''+'), fx = 0, 1, 2, ¦ ¦ • .
In case a < —\ the Lagrange polynomials are convei ent at the point x = +1
if f{x) is an arbitrary continuous function.
There exist functions continuous in [ — 1, +1 ] whose Lagrange polynomials are
60 Cf. Theorem 1.3.2.
07 For fx = 0 the equality sign, that is, the ease a = —1/2, is excluded.

33E INTERPOLATION [ XIV ]
divergent (unbounded) at a preassigned point x0, — 1 < x0 < +1; the same is true
at the point x0 = +1 provided that a. ^ — •§.
Similar statements hold in the interval [— 1, 1 — e] and at the point x = — 1
if we replace a by /3. The convergence is uniform in the whole interval [—1, +1]
provided max (a, /3) ^ — \ and u>(<5) = o( | log 8 | ~'), or ^ — \ ^ max (a, /3) <
M + •§ and f(x) has a continuous derivative of order fx with modulus of con-
continuity uv(<5) = o(8max(a' ~'J+ ), ^t = 0, 1, 2, • • • . Again the equality sign in
case m = 0, that is, the case max (a, /3) = — ^, is excluded.
Compare Fejer 13, pp. 22, 24, 27; Shohat 5, p. 146. The present results are
more precise than those obtained in these papers.
B) We shall start with a discussion of the "Lebesgue constants"
A4.4.1) E I Uxo) | ,
where —1 < x0 < +1. Let 8 be a fixed positive number, 8 < 1 — | x0 \.
ThenP^^ixo) = 0{n~h), so that
A4.4.2) '*" *ol> '*' Xol>S
I-1
According to (8.9.2)
A4.4.3) E \P\a'')r{x,) p1 = 0A) E ^+5n~"~2 + 0A) E ^+ln'""-2 = 0{nh).
Consequently,
A4.4.4) E Ur(xo)| = 0A).
On the other hand, assume | x, — xa | S 8. For a fixed v we have (see (8.8.2))
A4.4.5) Z,(z0) - O(n"*) ^-.J^--_iJL_W
3"f — Xq
so that if a;0 = cos 60 , 0 < 60 < ir,
A4.4.6)
p
where 8' is a fixed positive number. According to (8.9.1) the last expression is
O(log n). The same bound holds for A4.4.1), and does so uniformly if the con-
condition — 1 + e ^ Xo ^ 1 — e is satisfied.
We can also show tha,t

[ 14.4 ] LAGRANGE POLYNOMIALS FOR JACOB I ABSCISSAS 337
(H.4.7) ? m*a)[~logn,
if n tends to + °o over a proper sequence of integers. It suffices to choose n so
that | cos(Nd -f 7) | = cos e, where N and 7 have the same meaning as in Dar-
boux's formula (Chapter VIII), and e = \ min @O, v — 0O). This is obviously
possible since from
(to + \)tt - e < N60 + 7 < (m + 1)tt + e,
?n integral, the inequalities
(m + |Or + e < (N + lHo + 7 < (m + I)* - e
follow. Now the formula of Darboux coupled with the previous argument
and (8.9.1) gives us the desired proof.
C) As a consequence of the last result, we may conclude by means of Theorem
1.6 (Helly's theorem) the existence of a continuous function/(x) whose Lagrange
polynomials are unbounded at the interior point x = xo. On the other hand,
let /(x) have the modulus of continuity co(<5) = o( | log <5 | ~x), and let Ln(f; x)
be the Lagrange polynomial of degree n — 1 corresponding to/(x). Approxi-
Approximate /(x) by a 7rn_! = p(x) such that
A4.4.8) f{x) - p(x) = oKlogn)], -1 ^ x ^ +1
(Theorem 1.3.2). Then (see A4.3.6))
I Ln(f; x) - f{x) I = I Ln(f - p; x) - {f(x) - p(x)}
A4.4.9)
Ul)-1} O(logn) = o(l).
D) Assume 1 — 8 ^ x0 ^ 1. On putting n = n ¦+¦ 1 — v, m = O(n), we
have, by Theorem 7.32.2 and (8.9.2),
E 1 uxo) 1 -.= o(D 1 piaj)(x0) 1 e 1 ^a'3)/(x,) r1
\xy-xo\>i \xr—xo\>&
A4-4.10) = 0(na) E M3+!n-^2 = O(na+i),
a = max (a, -^%).
We now pass to the determination of an upper bound for
I
== E
COS 0o — COS i
A4.4.11) =
e.)
(since @o — 02)/(cos 0o — cos 0) is bounded). In what follows we use both
bounds A4.4.11). We have by D.21.7),

338 INTERPOLATION [ XIV
A4.4.12) cos0o - cos0, ' - '^'
= O(D
where r is between 0o and 0V .
Let 0o = n"'?, and consider first the case ? = 0A). Let v = 0A); then
A4.4.12) becomes 0{l)na+V+in~a'1 = 0A). On the other hand, if v is larger
than a sufficiently large fixed positive number, the second expression in the right-
hand member of A4.4.11) becomes (see the second bound in G.32.5))
A4.4.13) O(l)na ? va^na'2Q~2 = 0A) ? v"^ = 0(naH), 0 (log n), 0A),
according as a > —\, a = — %, or a < — |.
Now let ? be "large" and ? — v-k = 0A), so that the number of these values of
v is bounded. Then we obtain for A4.4.12) (see the first bound in G.32.5))
A4.4.14) 0(l)T~a~ln"i«'tt+in~a = 0(l)(v/nya~in~iva+in~a~l = 0A).
Finally, assume that both ? and ? — v-k are "large". Then the second ex-
expression in the right-hand member of A4.4.11) becomes (see the first bound in
G.32.5))
A4.4.15) 0(C~*) ? |^^T| = ?i + ?2 + ?a,
where the summation is extended over v-k ^ ?/2, ?/2 < v-k ^ 3?/2, v-k > 3?/2,
respectively. Here we have to take into account the fact that the ratio
0(l)}:
has a positive lower bound. In the second sum | ? — vir | is larger than a fixed
positive constant. We find now
i = o(r~h)o(f2)
A4.4.16) Z2 = Oft-^O^*) E 3?/2 [€ - w r = OOogf) =O(log n),
E
/
""^ = 0(na+i), O(log n), 0A),
according as a > —§, a = — \, or a < — |.
Recapitulating, we obtain for the Lebesgue constant A4.4.1), the bounds
0(na+i) and O(log n), uniformly in — 1 -f- € ^ x0 ^ 1, according as a > — \ or
a ?= —I- A similar result is found for the interval — 1 ^ x0 ^ 1 — e by re-
replacing a by /3. In the whole interval —1 ^ x0 S= +1 we obtain the bounds
0(n7+*) and O(log n) for y > — I and 7 ^ — §, respectively, where we have
y — max (a, /3).
Now an argument similar to that in C) must be used. Assume first that

U4.5] STEP POLYNOMIALS IN CLASSICAL CASES 339
-1 + e ^ x0 ^ 1, and let /(x) satisfy the conditions of Theorem 14.4.
According to Theorem 1.3.3 a irn^ = p(x) exists such that
f n-"o(n-a+"-J) = o{n~a~h),
A4.4.17) f(x) - p(x) = .
H(logn) '], -1 rg x ^ + 1,
according as a > -|or« ^ — §. This establishes the statement concerning
the convergence in the interval [—1 + e, 1]. The proof is obviously the same
for [-1, 1 - e] and for [-1, +1].
E) There remain to be discussed the "Lebesgue constants" for x0 = +1,
that is, the expressions
A4.4.18)
n"
'
The positive zeros xv furnish a contribution ~ na E (v/n) 2v"+in
=¦ ^ v
which is O(na+i), O(log n), Oil), according as a > -\,a= -\,a< -i The
contribution of the negative zeros is
na D | Pia')f(xr) r'^n-j; vMn-''2 - na+\
And now, one more step, namely, the application of Helly's theorem, yields the
desired proof of Theorem 14.4.
14.5. Preliminary discussion of the step polynomials in the classical cases
We calculate the linear functions vr(x) occurring in A4.1.7) for the Jacobi,
Laguerre, and Hermite abscissas, respectively. We assume a > — 1, /3 > —1
in the first and a > —1 in the second case.
A) From D.2.1) we obtain in the Jacobi case, since l(xv) = 0,
, l"{xv) q-/3 + (q + /3 + 2)x,
.l.J jry r — r .
l'(x,) 1 - x\
Whence
O\ f \ 1 - x[a - ff + (a + ff + 2)xv] + (a - /3)x, + (a
2; V{x)
.2; Vv{x)
1 - xl
In particular,
„,(_!) := A +«)(! +x,) -0A -x.)
1 — x,
A4.5.3)
A
The zeros x, are everywhere dense in [—1, +1] (Theorem 6.1.1) if n is large.
Thus v,( — 1) is non-negative for each v and n if and only if 0 ^ 0. Similarly,
vr{ + \) ^ 0 if and only if q ^ 0. Since vv{x) is linear, we*obtain the following
result:

340 INTERPOLATION [ XIV ]
Theorem 14.5. The fundamental polynomials h,(x) of the first kind associated
with the Jacobi abscissas, are non-negative in —1 rg x ^ -f-1 for all values of v and
n, if and only if
A4.5.4) -1 < a ^ 0, -1 < 0 ^ 0.
B) In the Laguerre case (cf. E.1.2), first equation)
n, r r\ / \ xv{xv — a) -\- x(a -f 1 — z,,) , ,
A4.5.5) v,,(x) = — ——-—¦ ; v?@) = xv — a.
xv
Here vv(x) changes its sign for all values of a if v and n are properly chosen.
In the Hermite case (cf. E.5.2), first equation) we obtain
A4.5.6) v,(x) = 1 - 2xvx + 2x1.
14.6. Step polynomials and "Hermite polynomials" for Jacobi abscissas
We again assume a > —1,/3> —1 and use the same notation as in §14.4.
Theorem 14.6. Let f(x) be continuous in [—1, +1]. The generalized step
polynomials A4.1.9) [/„ = f(x»), \f» \ < A] converge uniformly to f(x) over every
interval [—1 + e, 1 — e]. The same holds over the interval [— 1 + e, 1] provided
a < 0. The step polynomials are in general divergent at x *= -\-\ if f{x) is merely
continuous and a ^ 0.
The "Hermite polynomials" A4.1.9) [/, = f(x,),f, = f'{xv)] converge uniformly
to f(x) in [— 1 + e, 1] if a < \ and f(x) has a continuous second derivative, or if
n/2 ^ a < (m + l)/2 andf(x) has a continuous (m + l)st derivative with modulus
of continuity av+i(<5) satisfying the condition co^+iE) = o(<52a~"), m = 1, 2, 3, • • • .
Similar statements hold in [— 1, 1 — e] and at x = — 1 if we replace a by /3,
and in the whole interval [— 1, +1] if we replace a by max (a, /3). (Cf. Shohat 5,
pp. 138-139; Szego 14.)
A) We start with the discussion of the convergence for —1 < x0 < +1.
Here again Theorem 8.9.1 is virtually indispensable.
If x = xv, the numerator of vv{x) in A4.5.2) is
1 _ Xv[a - 0 + (a + 0 + 2)xv] + (a - /3)z, + (a + 0 + I)*,2
A4.6.1)
= 1 - xl > 0.
Therefore, vv{xQ) is positive if | x, — xo | is sufficiently small, that is, | xv — xo |
^ 8. The same is of course true for K(x0). Furthermore, vv(xo) has for these
v a positive lower bound which is independent of 5. We therefore obtain, on
account of the first identity in A4.1.11),
A4.6.2) X) l| ? X)
\XV—XO\>&
and from A4.1.7) we see that

14.6 ] STEP AND HERMITE POLYNOMIALS FOR JACOBI ABSCISSAS 341
A4.6.3) E I Uxo) \?6 E ^^ = 0AM
iXr-Xol^S 11,- "' ~
Here the bound 0A) is independent of 8.
We shall now find a bound for the corresponding sums if | xv — x0 | > <5. From
A4.5.2) and (8.9.2) we have
E |^(xo)| E
|x,,-xo|>5 \xy-xo\>S
't\x0)}2 e a - xiri{Pia''h(xv
\xr-x0\>i
A4.6.4)
E
= 0(n"x).
Moreover, we notice
E |^
||>J
K—1
"x)
E \
|xv-xo|>«
(){Pi°^(x0)}2 E
A4.6.5) \x,-xo\>i
= Oin'1).
This yields the convergence of the generalized step polynomials of a continuous
function if —1 < x0 < +1- Indeed, according to A4.1.9) and A4.1.11),
| Wn(Xo) - /(Xo) | ^ E |/(Xv) - /(Xo) | | K(XO) \ + A~
rg max \f(x,) -
A4.6.6) '""
+ 2max|/(x) __ _
\x,—xo\>S \xy-xo\>S
= max | /(z,) - /(xo) | 0A) + 50A) + O^) + 0(n~l).
The factors 0A) in the last expression are independent of 8.
B) Now assume a < 0 and 1 — 8 ^ x0 ^ 1. The second formula in A4.5.3)
shows that vr(xo) and ftv(xo) are again positive and vv(xa) is bounded from zero
if | xv — xo | ^ 8, provided 8 is sufficiently small. (We have v,{xo) ^ vv{-\-\)
ifa-/3 + (a + /3 + 2)z, > 0.) Then the analogues of A4.6.2) and A4.6.3)
follow immediately. In A4.6.4) and A4.6.5) a slight modification is necessary
due to the fact that in this case P<na^\xo) = 0(na), a = max (a, — f). (The
formulas corresponding to A4.6.4), A4.6.5) hold for arbitrary a > —1; this
remark is used in C).) Since a < 0, the conclusions of A) remain valid; this
establishes the convergence of the generalized step polynomials if a < 0 and/(x)

342 INTERPOLATION [ XIV ]
is continuous. (Of course, the same is true for the "Hermite polynomials" if
fix) is bounded.)
C) We postpone the discussion of the step polynomials at x = +1 if f(x) is
continuous and a ^ 0, and we pass on to a discussion of the "Hermite poly-
polynomials" for 1 — 5 ^ oro = 1 with a arbitrary but greater than — 1. First we ob-
observe that the numerator of A4.5.2) vanishes for x = xv = +1; thus we have
for | xv — x0 | ^ 5,
A4.6.7)
where eE) -> 0 if 5 -> 0. Now,
E \h,(xo)\ = e(8) E A -xly'
A4-6.8)
\x,-xQ\<,i ^ Xo — X,
By use of the notation and argument of §14.4 D), we obtain for the sum above:
(a) O(n2) if ? = 0A), v = 0A);
(b) 0A) E via~\v/nY2 = 0{n2a), 0{n log n), 0(n), according as « > 1,
a = 1, a < 1, if ? ¦= 0A), and v is "large";
(c) 0A) (v/n)~2 = 0(n2) if ? is "large" and I - v* = 0A);
(d) O(l)r2a-1 E -2°l+3a2 - -V2)-2 (v/n)-2 = Ei + E2 + Ea if both ? and
% — vir are large. The summations in the last three sums are extended over
the same values of v as in A4.4.16). We have
v2a+\v/nT2 = O(C3n2) = 0{n2),
A4.6.9) E2 = OA){/,2<?<3{/2(* ~ ^)("A)~2 = O(r2n2) = 0{n2),
E ,2a-Vn)-2 = O(n2a), O(n2 log n), O(n2),
according as a > 1, a = 1, or a < 1. To these cases the bounds
A4.6.10) E I h,(x0) I = eE)O(n2°), eE)O(n2 log n), eE)O(n2)
i
correspond.
In order to obtain the analogous bounds for f)v(z0), we cancel in A4.6.8) the
factor A — a;,) ~ (v/n)~ . Thus, we readily obtain
A4.6.11) E I Uxo) I = 8O(n2a), SO(logn), 50A),
according as a > 0, a = 0, or a < 0.
The corresponding bounds for | x, — Xo | > 5 are O(n2a) (cf. the remark made
in B)). Thus, A4.1.9) and the first formula in A4.1.11) give the result

14.6 ] STEP AND HERMITE POLYNOMIALS FOR JACOB I ABSCISSAS 343
Wn(x0) - f(x0) | ^ max
A4.6.12)
(O(n2a)
-/(xo) | eE) | O(n2 log n)
{O(n2)
iO(n2a)
+ max
S{
O(log n)
+ max | f(x) | O(na)
10A) J
+ max |/'(x) | 0(n2a);a = max (a, — |).
In the first term we have the alternative a > 1, a = 1, or a < 1, in the second
term a > 0, a = 0, or a < 0. The O-expressions are independent of /(x) and 5.
We now apply the usual argument. Let Wn(f] x) be the "Hermite poly-
polynomial" corresponding to/(x), and p(x) an arbitrary 7r2n_i. Then
A4.6.13)
Wn(f; x0) - /(xo) =
- P; x0) -
- P(x0)}.
Under the condition mentioned in Theorem 14.6 we can determine p{x) (cf.
Theorem 1.3.3) such that
A4.6.14)
fix) - p(x) = oin'2),
f(x) - p(x) = o(n~2a~1),
f'{x) - p'ix) = oin'1) if a < i
fix) - p'ix) = o(n-2a),
^a< Km + 1), a* = 1, 2,3, ••• .
This establishes the statement concerning the "Hermite polynomials."
D) Finally, we discuss (cf. 14.5.3))
A4.6.15)
The part of this sum defined by x, ^1 — 5 is, for a
A4.6.16) ~n2a E A - xy2{P{naJ
n
E
xv>.\—i
which is of the order na or 1, according as a. > 0 or a < 0. This shows
that the step polynomials (and also the generalized step polynomials) of a
continuous function are in general divergent at x = +1 if a > 0. (The con-
convergence for a < 0 has been proved in B).) The possibility of divergence in
case a = 0 follows by choosing /(x) = 1 — x. The corresponding step poly-
polynomial is in fact
E
A4.6.17)
E
E
1-2
\y/n) v
-2.2/3+3-2/3-4
It

344 INTERPOLATION [ XIV ]
so that it cannot tend to /(I) = 0. Still simpler is the proof of the divergence
of the step polynomials of /(x) = A + 0)A — x) — a(l + x) if a > 0, since
/(I) = _2« < 0.
14.7. Step polynomials for Laguerre abscissas
Assume a > — 1, and let Xj < x% < ¦ • ¦ < xn denote the zeros of the Laguerre
polynomial Ln(a)(x). We prove the following theorem:
Theorem 14.7.68 Let f(x) be continuous for x ^ 0 and /(x) = O(xm) if
x —» + °° ; here m is an arbitrary but fixed positive number. The generalized step
polynomials A4.1.9) (/„ =f{xv), \f, \ < A) converge uniformly to fix) over every posi-
positive interval e ^ x ^ w. The same holds in the interval 0 ^ x ^ u> provided a < 0.
The step polynomials are in general divergent at x = 0 if f(x) is continuous and
a ^ 0.
For the proof we shall need considerations similar to those in the Jacobi case
(§14.6). In particular, we shall use Theorem 8.9.2. Some modifications in the
argument are necessary due to the fact that the zeros are unbounded. The
mechanical quadrature appears as an important new tool (cf. A5.3.5)).
A) Assume 0 < e 5= Xo ^ w. If vv(x) has the same meaning as in A4.5.5),
the values «v(x0) and A,(xo) are positive, and vy(xo) is bounded from zero and
infinity provided | x, — x0 | is sufficiently small. Thus for small 5,
A4.7.1) E \K{xa)\= E K(xo) = 1 - E hv{xa),
A4.7.2) E I Uxo) | = 50A) E K(xo).
\x,-xo\SS |i,,-io|gJ
Here 0A) is independent of 5.
If xv is small, vv{xti) = O(xVl); if x, is large, ^(x0) = O(x,,). Therefore (cf.
G.6.8))
E
\xy-x0\>S
=O{n~') E xT'iL^'M^ + Oirr) E
E x
Xy>XQ+S
But combining A5.3.5) with C.4.5), we find
A4.7.4) ± *:¦ |l<-" w r = L("
T(n + a + 1)
which yields the bound 0{na~lH(n~a) = O{n~h) for A4.7.3).
"8 Cf. Shohat 5, p. 139; Szego 14, p. 597.

[ 14.8 ] LAGRANGE POLYNOMIALS FOR GENERAL CLASSES OF ABSCISSAS 345
More generally, we find from A5.3.5) and C.4.1), if m is a positive integer,
m ^ 2n — 1, that
.o; 2-,x, \Ln \x,)\
vi I (n + a + 1)
Hence the same argument as before leads to
A4.7
6)
m fixed.
A4.7
7)
We
have
also
\x,-xo\>5
E
\x,-xo\>5
h.
(xo)
(x0)
Equations A4.7.1), A4.7.2), A4.7.6), and A4.7.7) establish the uniform con-
convergence of the step polynomials in the interval e ^ x0 ^ u> (cf. A4.6.6)).
B) Now assume a < 0 and 0 ^ x0 ^ 5. If 5 is sufficiently small, and
| xv — x0 I ^ 5, both t;»(x0) and A»(xo) are positive, and vv(x0) is bounded from
zero. Hence A4.7.1) and A4.7.2) are again valid. In A4.7.3) only the bound
of \Lla\x0)}2 must be changed. According to G.6.11) this will be 0{na)
where a = max {\a — \, a). Therefore,
A4.7.8) E \K(xo)\ =O(n2a-a),
\r.y-xo\>S
and the same bound holds for the sums in A4.7.6) and A4.7.7). Since the ex-
exponent 2a — a = max ( — \, a) < 0, these sums tend to zero. From this the
uniform convergence in 0 ^ x ^ w follows.
C) The case x0 = 0, a SO, can be readily disposed of by choosing f(x) =
x — a. We have
I)}
 D
D (
K-l \ Xv
Since this expression is positive, it cannot tend to /@) = —a if a is positive.
If a = 0, the last expression in A4.7.9) is 1 (cf. A4.7.5), m = 1), and/@) = 0.
14.8. Lagrange polynomials for certain general classes of abscissas
A) Let xln > xin > ¦ ¦ ¦ > xnn denote the zeros of the nth orthogonal poly-
polynomial pn(x) associated with the weight function w(x) in the interval—1 ^
x ^ +1. We consider two classes A, B of weight functions characterized by
the following conditions:
A. There exists a positive number y. such that
A4.8.1) w{x) ^ n, -1^x^
B. There exists a positive number n such that

346 INTERPOLATION [ XIV ]
A4.8.2) w{x) ^ mA - x2)~\ -1 < x < +1.
By using this notation we prove the following statement:
Theorem 14.8. Letf(x) be defined in —I ^ x ^ -f 1. Let \Ln(f; x)\ denote
the sequence of the Lagrange polynomials coinciding with /(x) at the zeros xvn
of the orthogonal polynomials pn(x) associated with the weight function w{x),
— 1 ^ x ;§= +1- Then liiru-oo Ln(J) x) = f(x), uniformly in the interval [—1, +1],
provided thai w(x) belongs to A and f(x) has a continuous derivative in [—1, +1].
The same conclusion holds if w{x) belongs to B and wE) = oE*). Moreover
limn-oo Ln{f; x) = f(x), uniformly in the interval [—1 + e, 1 — e], where
0 < e < 1, provided that w{x) belongs to A and wE) = oE*).
Compare Shohat 7, Griinwald-Turan 1. Here, as before, wE) is the modu-
modulus of continuity of/(x) in [ — 1, +1].
B) We show that for the fundamental polynomials of Lagrange interpolation
(O(n), -1 < x < +1,
A4.8.3) E I U*) I = \ 0{nh), -1^x^+1,
[O(n*), -l + e^x^l-6,
where w(x) belongs to A in the first and third case, and to B in the second case.
From A4.8.3) the statement follows by reference to Theorems 1.3.2 and 1.3.3.
Let x be fixed, e, = sgn lv{x). We write in the case A
n n—1
A4.8.4) p(t) = Ec>Jr(<) = Ec,P,(<),
where Pv(t) is the j'th Legendre polynomial. Then
p\x) = 2s \ l-w \ = 2-j c,r,{x) ^ s 2s —r
.=0
A4.8.5) = jjT [p(t)
Now, according to A4.2.4),
A4.8.6) / w(i)[p{t)f dt = EXr[p(z,)]2 = X) ^l = E ^ = / w(t) dt.
J-l, y=-l i—l y=~l 7-1
Statement A4.8.3) is readily derived by using G.21.1) and G.3.8).
The only essential modification of the proof in the case B is that we write
n—1
A4.8.7) P(t) = E«rU<) = 22d,T,(t),
i—l i—0
where Tv{t) is Tchebichef's polynomial of the first kind.

[14.9] FURTHER RESULTS ON INTERPOLATION 347
14.9, Further results on interpolation
We indicate briefly a few more recent results on interpolation.
A) We employ the notation of §14.8. Let the weight function w(x) be con-
continuous with a positive minimum in [ —1, +1] and let e > 0. Then
max |?,,(a;)| —> 1 as n —> oo
where the maximum is taken in the interval [ — 1 + e, 1 — e] and the limit
relation holds for any sequence v = v(n) for which the corresponding zero
x, = xvn is in [ — 1 + e, 1 — e]. (Erdos-Lengyel 1.)
B) The result of Griinwald A) and Marcinkiewicz B) mentioned in §14.1 C)
has been deepened by Erdos-Griinwald and Erdos. We consider the zeros of
Tn(x) as the set of abscissas.
Erdos-Griinwald A) have shown the existence of a continuous function
f(x) = /(cos 6) the Fourier series of which is uniformly convergent and at the
same time the sequence of the corresponding Lagrange polynomials Ln(x) is
everywhere divergent, even everywhere unbounded.
Erdos B) has shown that if x0 = cos (pir/q), p and q odd, there exist continuous
functions f(x) for which Ln(x0) —> oo. Erdos-Turan A) have shown previously
that this cannot hold for any other points x0; see also Erdos 2, p. 313.
C) Important results have been obtained by Erdos for the "normal" sets of
abscissas introduced and investigated by Fejer. They are characterized by the
property that the "conjugate points" A4.1.15) lie outside of the interval [a, b]
of interpolation.
Fejer showed that for a normal set x, — xv_x tends to zero as n —> oo. Erdos-
Turan B) have sharpened this result and Erdos A) proved that
x, - z_! = -A -z5)-* + O.(n-3)
TO
provided that x, = xvn is restricted to a fixed interval [ —1 + e, 1 — e].
Erdos A) solves the following remarkable extremum problem. We consider
all normal sets {xvn\, v and n fixed. (We follow the notation A4.1.1).) What
is the minimum and maximum of x,n1 They are given by z, and — zn-, where
z, = zvn are the zeros of Pn(z) + Pn-i(z).
D) Griinwald B) and Webster A) have studied a certain type of "sum-
mability" for Lagrange polynomials similar to a procedure of W. Rogosinski for
the partial sums of Fourier series. The abscissas used are the zeros of the
Tchebichef polynomials of the first and second kind, respectively.
Griinwald C) gives a survey of divergence properties of the Lagrange poly-
polynomials. This paper contains also a discussion of the convergence of the
Lagrange and Hermite interpolation polynomials under the condition that the
abscissas form a normal set.
E) Balazs-Turan A, 2) and Suranyi-Turan A) have investigated various
properties of certain polynomials of interpolation connected with ultra-
spherical abscissas. The novel feature of this investigation is the study of inter-

348 INTERPOLATION [ XIV ]
polation polynomials for which/(x) and/"(x) are prescribed.
F) Erdos C) has investigated the "Lebesgue function"
V = 1
see A4.1.6). Denoting the maximum of this function in the interval —1
Sx^l, by Mn, he proved the following inequality
Mn> B/7r)logn -c.
This is a rather deep result. It is known that for the zeros of Tchebichef
polynomials
Mn<B/w)logn+c.
Here c is a positive absolute constant.
G) The more difficult analogous problem concerning the fundamental
polynomials fyv(x) of Hermite interpolation has been solved by Erdos-Turan
D), who prove that
n
max ?
i—i
1/2 \
^ - ( - logn -c2loglogn 1 .
This is "asymptotically best possible" as the case of Tchebichef abscissae
shows.
The paper Erdos-Turan 4 also contains a proof that
n 2
max ? |/,(x) | > - logn -c3loglogn,
-lSxg+l v^i TV
which is weaker than the result cited in F) above, but preceded it in time.
(8) If Ln(x) is the Lagrange interpolation polynomial of a continuous
function f{x) with interpolation at the zeros of Pt'^{x), a,/3^ — \, then
A4.9.1) lim f \Ln(x) -f(x)\p(l-x)a(l+x)l)dx=O
n—»a> J — 1
when p<minD(o + l)/Bo + l),4(j8 + l)/Bj8 + lH =A(o,j8) and for p >
A(a,ff) there is a continuous function f(x) for which A4.9.1) fails. See Askey 6.
(9) By simple methods of interpolation Egervary-Turan 1 have proved the
following beautiful identity
where the xv are the zeros of the Legendre polynomial Pn(x). From this they
derive a new proof of the inequality G.21.1). By similar reasoning one can
obtain a new proof of the inequality G.21.3). See Egervary-Turan 2.

CHAPTER XV
MECHANICAL QUADRATURE
The reader will recall that the Gausw-Jacobi mechanical quadrature was
studied in §3.4. In the present chapter we turn to other mechanical quadra-
quadrature problems which are also connected with the theory of orthogonal poly-
polynomials.
15.1. Definitions
A) Let [a, b] be a finite or infinite interval, and let
A5.1.1) Sn : Xm < X2n < ¦ ¦ • < Xnn , O ^ Xln , Xnn ^ 6,
denote a -set of n distinct points in [a, b\. Furthermore, let
A5.1.2) AB : \m, X2n, ••• , XnB
be a set of real numbers. If /(x) is an arbitrary function defined in [a, b], we
write
A5.1.3) Qnif) = Z)X,n/(z,»).
We call the numbers xrn the abscissas and the numbers Xvn the Cotes numbers
of the "mechanical quadrature" Qn(J). Having been given the sequences of
corresponding sets {Sn\ and {A,,}, n = 1, 2, 3, • • • , we are interested in the
convergence properties of the sequence
A5.1.4) &(/),&(/), ••• ,Qn(f), •••
associated with a given function/(x).
As in Chapter XIV we write xv and X,, instead of xvn and \vn when there is
no ambiguity.
B) An important special case is the following. Let the set \&n\ be an
arbitrary set of distinct numbers in [a, b], and let u{x) be a given non-decreasing
function. We shall define the Cotes numbers Xv by requiring that
A5.1.5) Qnif) = [ fix)duix)
Ja
shall hold if f(x) is an arbitrary irn-i . Obviously, uAder such a condition,
A5.1.6) X, = / L(x)du(x), v = 1,2, ••• ,n,
Ja
where h(x) denote the fundamental polynomials A4.1.2) of the Lagrange inter-
349

350 MECHANICAL QUADRATURE [ XV ]
polation corresponding to the abscissas {Sn\. In this case Qn(f) is called a
quadrature of the interpolatory type.
The Gauss-Jacobi quadrature is seen to be a special case of the interpolatory
type when the abscissas {Sn} are the zeros of the orthogonal polynomials
associated with the distribution du(x). Another remarkable case of the inter-
interpolatory type is u(x) = x with {Sn} arbitrary. The integral A5.1.5) is then
the ordinary integral of /(x) over the interval [a, b], which is assumed to be
finite in this case.
We shall use the previous notation throughout the whole chapter.
15.2. A general convergence theorem on mechanical quadrature;
theorem of Stekloff-Fejer
A) Theorem 15.2.1.69 Let [a, b] be a finite interval, and let the system {Sn\
(in [a, b}) and {An} be arbitrary; let Qn(f) be defined by A5.1.3). Denote by u(x) a
non-decreasing function, and assume the "quadrature convergence"
A5.2.1) lim Qn(f) = ff(x)du(x)
for an arbitrary polynomial f(x). Then a necessary and sufficient condition for
the validity of A5.2.1) for an arbitrary continuous function /(x) is the bounded-
ness of the sequence of the "Lebesgue constants"
A5.2.2) | Xnl | + | Xn21 + • • • + | A™ |, n -> oo.
This theorem is an immediate consequence of Theorem 1.6 (Helly's theorem).
It is to be noted that condition A5.2.1) holds for an arbitrary polynomial/(x)
if the quadrature is of the interpolatory type (cf. §15.1 B)).
B) As an application we shall prove the following important theorem of
Stekloff and Feje"r:
Theorem 15.2.2.70 Let [a, b] be a finite interval, and let the Cotes numbers
{An} be non-negative. If the quadrature convergence A5.2.1) for an arbitrary
polynomial /(x) is assumed, the same convergence can be stated for an arbitrary func-
function for which the Riemann-Stieltjes integral in the right-hand member of A5.2.1)
exists.
The expression A5.2.2) remains bounded in the present case, since
n n
A5.2.3) Z I X»» I = Z X»» = Q»(D
»-i »-1
has a limit as n —> <». Therefore, A5.2.1) holds for a continuous/(x). The
68 Cf. P61ya 4, p. 267, Theorem I.
70 Cf. Stekloff 2, pp. 176-179; Fejer 15, p. 291; P61ya 4, p. 282, d); Shohat 7, pp. 474-476.

[ 15.2 ] THEOREM OF STEKLOFF-FEJER 351
extension to Riemann-integrable functions is made by means of Theorem 1.5.4.
Theorem 15.2.2 holds without change if the Cotes numbers {An} are non-
negative only for sufficiently large values of n.
C) A remarkable special case of Theorem 15.2.1 is obtained by considering
an arbitrary quadrature of the interpolatory type (defined by a monotonic
non-decreasing function u(x)), and by choosing for the set {Sn\ the zeros of the
orthogonal polynomials {pn(x)} associated with a preassigned distribution
da(x). Here a(x) is in general different from u(x). Then A5.2.1) holds for an
arbitrary polynomial f(x).
The Gauss-Jacobi quadrature is derived as a special case by taking a(x) =
u{x). The Cotes numbers are then identical with the Christoffel numbers of
§3.4 and are all positive. Applying Theorem 15.2.2, we obtain the following:
Theorem 15.2.3. Let da(x) be an arbitrary distribution on the finite interval
[a, b], and let Qn(J) be the corresponding Gauss-Jacobi mechanical quadrature
(that is, x,n are the zeros of the orthogonal polynomials pn{x) associated with da(x)
and Xm the corresponding Christoffel numbers). Then the "quadrature con-
convergence"
A5.2.4) lim Q»(/) = / f(x) da(x)
n—»oo Ja
holds for an arbitrary function f(x) for which the Riemann-Stieltjes integral in the
right-hand member exists.
D) Another remarkable special case arises upon choosing u(x) = x with
a(x) arbitrary. Such a choice permits us to assert the following theorem:
Theorem 15.2.4.n Let x, be the zeros of the orthogonal polynomial pn(x) asso-
associated with the arbitrary distribution da(x) on the finite or infinite interval [a, b].
If we define the mechanical quadrature Qn(f) of the interpolatory type by the re-
requirement in §15.1 B) with u(x) = x, the corresponding Cotes numbers can be
represented as follows:
/. f n rt \ fcn+1 -K-nKXy) 1 o
A5.2.5) ^y=- v = l,2,.-.,n,
Kn Pn\Xv)Pn+l\3-v)
with
/b fb r n ^ n rb
Kn(x, t)dt= / \ E P,{x)p,{t) \dt = ? p,(x) / p,(t) dt.
Ja {vO ) ,-0 Ja
Here we use the symbols kn and Kn(x0, t) in the sense explained in B.2.15) and
C.1.9); x, and X, stand for x,n and X,n , respectively.
For the proof we write A5.1.6) as follows:
,.,„« . i '( \ /VM-1 r Pn(t)pn+l(Xy) - pn+1(t)pn(Xy)
A5.2.7) X, = {Pn{Xy)Pn+AXy)\ / dt.
Ja I Xv
71 Szego 17, p. 94.

352 MECHANICAL QUADRATURE [ XV )
Comparison of this formula with equation C.2.3) establishes the statement.
According to C.3.6) we have
A5.2.8) sgn X, = sgn Kn(x,).
In particular, if Kn(x) ^ 0 in [a, b], the Gotes numbers are non-negative. Then
the Stekloff-Fejer theorem can be applied.
The polynomial Kn(x) may be characterized by the following condition:
A5.2.9) / Kn{x)p{x) da(x) = f p{x)dx,
Ja Ja
where p(x) is an arbitrary 7rn . In case da{x) = w(x) dx, we see that Kn(x) is
the nth partial sum of the expansion of {wix)}'1 in terms of the poly-
polynomials pn(x).
E) In the next sections we consider the special cases previously mentioned,
namely:
(a) u(x) = *(x) (cf. C)),
(b) u(x) = z(cf. D));
here the abscissas {*Sn} are the zeros of the orthonormal polynomials associated
with the distribution d<x(x) with the additional specialization that ?hese poly-
polynomials are the classical ones.
15.3. Cotes-Christoffel numbers in the case u(x) = a(x) (Gauss-Jacobi
quadrature) for the classical abscissas
A) We use the representation C.4.7) of the Christoffel numbers Xv and
arrange the zeros xv in decreasing order in the Jacobi and Hermite case and in
increasing order in the Laguerre case. Concerning the following results see
Winston 1. The representation A5.3.5) has already been used in §14.7.
By use of D.3.4) and D.21.6), the second formula D.5.7) furnishes for the
Jacobi abscissas
, 9«+/3+i T(n + a + l)r(n + /3 + 1) n 2n-i, p<«,/3),/ n>-2
A5.3.1) ' T(n+l)T(n + « + p + l){1-X>} {P" (*')} '
v = 1, 2, ••• ,n;a > -1,0 > -1,
and, in particular, for the ultraspherical abscissas (cf. D.7.1) and A.7.3))
A5.3.2) r(» + !)
v = 1,2, ••• ,n;X > ~i X ^ 0.
For X = 0, that is, in the "Mehler case" w(x) = A — re2)"*, we find
A5.3.3) X, = -. v= 1,2V ••• ,n,

[ 15.3 ] COTES-CHRISTOFFEL NUMBERS IN THE CASE u(x) = «(*) 353
that is, Xi = X2 = • • • = Xn . For X = 1, that is, in case w{x) = A — x2)\
we obtain
A5.3.4) X, ^l) i2
X, = ^l*,)-—*!! __, ,= iJ, ,,
For the Laguerre abscissas we have from E.1.10) and E.1.14)
A5.3.5) X, = *> + + t) " ^^fc)}-2, v = 1, 2, • • • , n; a > -1,
whereas for the Hermite abscissas, from the second identity of E.5.10),
A5.3.6) X, = T^nHH'nix,)}-2, v = 1, 2, • • • , n.
B) In the Jacobi case the argument of §7.32 B) shows that, for a > — ^ and
(8 > — 2, that part of the sequence Xi, X2, • • • , Xn corresponding to the zeros
xv > Xo (cf. G.32.1)) is increasing, and that part corresponding to xv < Xo is
decreasing. The opposite is true if a < — \, 0 < — ?. If a > — \, 0 < — \,
or a < — i, (8 > — \, the whole sequence in question is increasing or decreasing
according as the first or second pair of relations holds.
In the ultraspherical case this argument furnishes
A5.3.7) Xx < X2 < • • • < X[(n+1)/2] if X > 0,
and
A5.3.8) Xx > X2 > • • • > X[(n+1)/2] if X < 0.
The symmetry relation (Problem 11) yields the analogous statement for the
other Xv. The values A5.3.3) correspond to the case X = 0. In the Legendre
case we have A5.3.7).
In the Laguerre and Hermite cases we use the argument of §7.6 A). In the
Laguerre case the sequence Xi, X2, • • • , Xn is increasing for xv < a + 7, and
decreasing for xv > a + ?. In the Hermite case we have (xj. > x2 > • • • )
A5.3.9) Xi < X2 < ... < X[(n+1)/2].
C) In the Jacobi case, if xv = cos 6V, and 6V belongs to a fixed interval in the
interior of [0, x], Darboux's asymptotic formula (cf. (8.21.10) and (8.8.1)) yields
9«+/3+l / a \ 2«+l / a \ 2/3+1
A5.3.10) X, = X,n ^ Z—^ [sm|J [cos |J
Here, for a = |8 = — \ the symbol ^ can be replaced by =, according to
A5.3.3). The same is true for a = (8 = +% (see A5.3.4)) if we replace n by
n + 1. On the other hand, if v is fixed and n —•> «>, we obtain, by use of equa-
equation (8.1.1),
/1EO 11\ \ _ \ r~u r»«-t-P+i/.- /r>\2«( T' 11 \ 1 -2_—2o—2

354 Mechanical quadrature [ xv ]
where j, is the vth positive zero of Ja(z). (Here the symbol ^ can be replaced
by = if a = (8 = — ^.) Therefore, on account of the monotonic property
proved in B), we have uniformly in v,
A5.3.12) X, = 0{n~l) or O(na~2) if 0 < 0, S x - e,
according as a ^ — \, or a ^ — §. Here e is a fixed positive number.
By use of the argument and notation in §7.32 C), we readily see that for
increasing v, v ^ v0,
/ a \ -2"-1 / * \ -2/5-1
A5.3.13) 0@.) sin -v) cos - ) X,
V 2/ \ 2/
is increasing or decreasing according as a2 ^ \ or a2 ^ J, and 0 < 0V ^ 8,
where 6 is a sufficiently small positive number. (If n is sufficiently large,
<K0v) > 0, v ^ vo.) Thus A5.3.13) has uniformly the "order" n if 0 < 0, ^ 5.
In view of A5.3.10) and A5.3.11) for 0 < 0, ^ x - e this yields (see (8.9.1))
/ip n i a\ \ rfla+1 —1 2a+l —2a—2 r\ . Q <r-
A5.3.14) Xv '>>' 0V n '~ v n , 0 < 0V g x — e,
that is to say, the ratio of these expressions is uniformly bounded from zero and
infinity if v and n are arbitrary, 0 < 0V ^ x — e. The expression in the right-
hand member of A5.3.14) attains its greatest value for v ~ n or v ~ 1 according
as a ^ -^ or a ^ — \. This again furnishes A5.3.12).
Similar results can be obtained if 0V is confined to the interval e ^ 0V < x.
D) In the Laguerre. case, if e ^ xv ^ co, we obtain (cf. (8.22.1) and (8.8.4))
A5.3.15) Xv = Xvn ^ t* "'^° i i n —> oo,
where e and co are fixed positive numbers. On the other hand, if v is fixed and
n —-> oo, we find from (8.1.8)
A5.3.16) X, = \,B as (
where ,/„ has the previous meaning.
Applying an argument similar to that in the Jacobi case to the fourth equa-
equation in E.1.2), and taking into consideration the argument of §7.6 B), we find
that the sequences
f 4n + 2a + 2 - a:, + ^-^
A5.3.17)
Un + 2a + 2 - x, +
are monotonic as v increases provided a2 ^ j and 0 < xv ^ co. In case a > J
the same sequences consist of two monotonic parts (v ^ v0). On account of
A5.3.15) and A5.3.16) we obtain for the second sequence A5.3.17) the uniform
"order" n*, so that

[ 15.4 ] QUADRATURE OF INTERPOLATORY TYPE WITH u(x) = x 355
A5.3.18) X, ~ s"+V"*, 0 < x, ^ co.
Because of (8.9.10) we may write
A5.3.19) X, ~ v2a+1n-a-\ 0 < xv ^ co.
The condition a;,, = 0A) is equivalent to v = 0{n*).
15.4. Quadrature of interpolatory type with u(x) = x for Jacobi abscissas
A) In this section we prove the following theorem:
Theorem 15.4. Assume a > — 1, /3 > —1, and let the mechanical quadrature
Qn(f) be defined by the following conditions:
(a) PiaJ)(xv) = 0, v = 1, 2, • • • , n,
A5.4.1) t » t r+i
(b) Qn(xk) = 22 \,xl = xkdx, k = 0, 1, 2, • • • , n - 1.
"-I 7-1
max (a, (8) ^ 3/2. Then for an arbitrary function f{x) which is con-
continuous in [—1, +1],
A5.4.2) lim Qn{f) = f+1 f(x) dx
J
f
—l
holds. If the numbers a, (8 are such that max (a, 0) > 3/2, there exist continuous
functions f(x) for which A5.4.2) is not true.
Notice the difference between this statement and that of Theorem 15.2.3.
Apart from the special case max (a, |8) = 3/2, a ^ /3, this theorem has been
proved by Szego A7, pp. 102-108). The present proof is simpler; it is based
on the bounds G.32.5) of P^"l/3)(a;), and on Theorem 8.9.1 concerning the zeros
of Pia^(x). The notation of §14.4 is used.
B) First assume a ^ 3/2, /3 ^ 3/2. We must show the boundedness of the
sum A5.2.2) if Xv = Xvn is defined by A5.4.1). We discuss Xv only in case
0 s; xv < 1, or 0 < 9, ^ 7r/2. For the remaining values, D.1.3) can be
used. According to (8.9.2)
A5.4.3) X, -V^n—2 f P"/(cosg) Slned6.
Jo cos 6 — cos dv
We decompose the last integral into five parts I, II, III, IV, V, corresponding
to the intervals
o ^ e ^ 0iA 0i/2 ^ e s e,/2, ev/2 s e ^ 3o,/2,
A5.4.4)
30,/2 S d ^ 3tt/4, 3t/4 S 6 ^ 7T,
and we note that @2 — 02)/(cos 6 — cos 0,,) is bounded in all these integrals.
Since 0X = Ofa),

350 MECHANICAL QUADRATURE [ XV
A5.4.5) I = O(na) I ' d~2ddd = O(na-2)(v/nT2 = 0{v~2na).
Jo
Further, applying (8.21.18), we obtain
— a W— cos (Nd + y) sin 0 dd
!/2 COS 0 — COS 0v
A5.4.6)
+ 0A)
: e; - v
The first integrand can be written as follows:
A5.4.7) rTh ~ 2 4>{dv, 0) cos (A^0 + 7),
where 4>F,, 0) and its partial derivative with respect to 0 remain bounded in
the interval considered (uniformly in v). Therefore, integration by parts gives
A5.4.8) 00
Here the symbol [/@)]a means f(b) + /(a). A simple discussion gives for the
first term of A5.4.8) (cf. (8.9.1))
A5.4.9) O(n~!07a"!) + O(n~hTa+i972) = 0(v~a~\a) + 0(v~2na),
and for the second term
A5.4.10) O(n-'
t + 4, + _
- 02 0j - 02 @^ - 02)
We readily see that A5.4.10) can be combined with the second integral of II.
On putting 0 = dvx in this integral, we obtain
re,\2 a-a-i rh -«-j
A5.4.11) n~* / ~ 2d0 = OV-tn-l f -dx.
Jti/a o; - e J l - x2
The lower limit of integration is 0i/B0v) ~ l/v, so that the last integral is
O(v"~*), O(log v), or 0A) according as a > ^, a = ?, or a < \. Consequently,
A5.4.12) II = 0{v~2na), O(v"V log v), or 0(v"a~!n"),
according as a > ^, a = ^, or a < ^.
Now by (8.8.2),
III - rr/1n-»ib(« cos(iV0 + 7)-co
A5.4.13) ^8'/2 cos0-cos0v
Here n~JA;@,) = O(n~i07"~i) = O(v~"~*na), and the same bound holds for the

[ 15.4 ] QUADRATURE OF INTERPOLATORY TYPE WITH u{x) = x 357
second term. Furthermore,
in N —-—
sin
dd
sin
A5.4.14)
ej2 sin N
e -
sin
inH
- sin ( N
0A)
since | sin 0 - sin @ + 6,)/2 \ < \ (d - 0,)/2 |, and 5,{sin @ + fl,)/^} is
bounded. The term with cos (N0v + 7) vanishes, since the integrand is an odd
function of 0 — 0V. The denominator in the last integral can be replaced by
@ - 0,)/2. Hence A5.4.14) is bounded, and
A5.4.15)
Now, as in II,
3*74
3r/4
A5.4.16) IV =
and the first integral is
[Q-a+i -|3ir/
A5.4.17)
III =
n-* —....^(^ e) Cos (NO + y)dd + 0A)
'39,12
3W4
@2 - el)
dO
3—~td0.
S$r/2 V — Or
On putting 0 = 0va;, we have
/¦3W4 fl-a-J
A5.4.18) /
zeji
2
0 —
| r
a;2 — 1
so that
A5.4.19)
IV =
Finally, the second mean-value theorem gives (compare D.21.7) and also
§7.32 B))

358 MECHANICAL QUADRATURE [ XV
re'
V = - (cos 0, - cos [37T/4]) / P^(cos 0)sin 0 d6
A5.4.20) 0(n_1} ( pu-i,H
e>) _ P^+Ti./»-i)(c08 37r/4) ,
= O[nmax(*-2'-!)] = O(n~»), 3tt/4 ^ 6' ^ t.
Here we used the assumption /8 ^ 3/2. Hence
=
In all cases we obtain
A5.4.22) 53 |X,| = 0A),
and this establishes the boundedness of A5.2.2) and the first part of the
statement.
C) Now assume a ^ /3, a > 3/2. We assume that dv lies in a fixed interval
[a, b], 0 < a < b < ir; then v ~ n, and the number of zeros 0V satisfying this
condition is also ~n. Let e and co be fixed positive numbers, e < min (a, t — b).
Decompose the integral in question into the parts I, II, III, IV, V, corresponding
to the intervals
0 ^ 6 ^ ta/n, w/n ^ d ^ 6V - e, 0, - e ^ (9 ^ (9, + e,
A5.4.23)
0, + e^0^7r — co/n, X — co/n ^ 0 ^ x.
Then,
I = A - COS0,)-1 / PlaJ)(cose)sm9de
A5.4.24) 7o ,„,„
+ 0A) / |Pi"^(cos0)|002d0.
The first integral can be calculated according to D.21.7), and we obtain (cf;
G.32.5))
I _ ft _ oos»,)- _iL_ {,<-*"<« - Pitr-" (cos ?)}
A5.4.25) + Oin")
~ r(«) 1 -cos0, ' V ' ' V '¦
The bound of the term 0(n"~2) is independent of co. In the same way we find
Jo cos 0
A8.4.26)
,_,

[ 15.5 ] AN ALTERNATIVE METHOD IN THE ULTRASPHEKICAL CASE 359
where O(nfi~2) has a bound independent of w. (For (8 = 0 the first term
vanishes.)
Furthermore,
/0,—e rx—w/n
In jev+t
Here the same remark applies to the O-terms as above.
Finally, according to (8.8.2),
III - »-*(*.) r ^lM±^^^^L±21 s[n9d9 h
Jet cos 6 — cos 6,
A5.4.28)
= 0(n"*)
»,+«
sin N
9 - dv
sin
dd
2
The last integral is O(log n) (cf. Zygmund 2, p. 172); whence
A5.4.29) III = O(rT* log n).
Now assume a. > /3, a. > 3/2; then the previous results give
I + II + III + IV + V = n"~21~ + ,
A5.4.30) \r(«)l-cos0,
where 0A) is independent of co. Hence, by A5.4.3)
A5.4.31) Z) | X, | — nH.
In the more complicated case a = 0 > 3/2 we find
I + II + III + IV + V
A5.4.32) _2 f 2
= n
where again 0A) is independent of co. This furnishes the same result A5.4.31).
15.5. An alternative method in the ultraspherical case
In the ultraspherical case a = (8 the convergence theorem A5.4.2) follows
from the following theorem:
Thkokkm 15.5. Let —1 <a = (8^f. Then the Cotes numbers Xv, defined
by A5.4.1), are positive provided that n is chosen sufficiently large. In the canes
— 1 <a = (8^0 and \-, ^ a = /3 ^ 1 the statement is true for all values of n.

360 MECHANICAL QUADRATURE [ XV )
In view of the Stekloff-Fejer theorem (Theorem 15.2.2; cf. the last remark
i^ §15.2 B)) this leads indeed to another proof of the convergence theorem of
§15.4, even for Riemann-integrable functions. It is very probable that X,. > 0
for all values of n provided —1 < a = ? ^ •§.
Concerning this section, refer to Szego 17, pp. 95-99, 109-110, where the
positiveness of X, is proved for —5^a = (8^0 and for certain other special
cases.
A) We prove X, > 0 by showing that the function Kn{x) = K^\x) of §15.2
is positive for —1 < x < +1; here da(x) = A — z2)x~* dx with X = a + |
(cf. A5.2.8)). We can assume that n is even, since the last integral in A5.2.6)
vanishes if n is odd, and this assumption will be made throughout this section.
According to D.7.15), D.7.14), and D.7.3), we have (cf. A.7.3))
IX)(t) - 23~n ?^LVX + m
K(t) 2 p(r)
A5.5.1) ~ T(X + *)r(X - f) ^ BX + m - l)(m +T) Fm W
= T,d™P™(x), m = 0,2,4, ••• ,n.
Let X < 2. From a remark made in connection with A5.2.9), we have by
Theorem 9.1.2, for -1 < x < +1,
A5.5.2) lim K^(x) = A - x2)^,
n-»oo
uniformly in— l + e^a;^l — e.
B) First assume 0 < X < i Then
A5.5.3) <ioX) > 0, <iiX) < 0, m = 2,4, 6, • • • .
In addition, PiX)(cos &), as a cosine polynomial, has non-negative coefficients
(cf. D.9.19)). Therefore iflX)(cos 6) attains its minimum for 6 = 0, and
this minimum decreases as n increases. Now
r(x + |)r(x -
Y2X
m=o,2,4,--- \2X + m — 1 m +
TBX - 1)
05.5.4) " r<x+i)r<x-»
' '2X + m - 2\ /2X + ro - l\
/ + V m+1 /
rBX - D
/2X + m - 2\ =
V m /
- " r(x + |)r(x - |),
and this establishes the statement for 0 < X < f. If X = §, we have d^ = 0,

[ 15.5 ] AN ALTERNATIVE .METHOD IN THE ULTRASPHERICAL CASE 361
m = 2, 4, 6, •• ¦ , and K^\x) = 1. In the limiting case X —> 0, we find (cf.
D.7.8))
lim KiM (cos 8) = 2- - - ? (-L- L-) cos m8,
A5.5.5) x-*o t 7T *¦" \m - 1 m + 1/
m = 2, 4, • • • , n.
Therefore, this case can be discussed as before. (See Szego 17, p. 96.)
C) Next, assume that — \ < X < 0. In this case d^ > 0. According to
D.9.19), the coefficients of the trigonometric polynomial PiX)(cos 8) are non-
negative except for the highest coefficient, that is, that of cos m8, which is
negative, m > 0. Therefore, the coefficient of an arbitrary term cos m8,
m > 0 even, in ifiX)(cos 8) appears as a sum of the form — u0 + ux + w2 +
• • • + /U; with non-negative u0 , ux , ¦ ¦ ¦ , ui where I = (n — m)/2. If m is
fixed, n —* <», I —> oo, this expression tends to the coefficient of cos md in
A — ?2)}x = I sin 8 |J~2\ But this is negative (cf., for example, Polya-
Szego 2, pp. 31-32), so that the same is true for the partial sums
— ¦Mo + U\ + Ui + • • • + ui. Consequently, ifiM(cos 8) is again of the same
type as in the previous case, and it attains its minimum for 8 = 0. The terms
<iiX)-PiX>(l) are negative, except for m = 0. This establishes the statement.
D) In case 1 < X ^ f, we start with the following identity:
A5.5.6) A - x2)KlX)(x) = K(tl\x) + axP'^2l\x), n even,
where a\ is a proper constant. We readily show (we can write x = 1, or com-
compare the highest terms, cf. D.7.9)) that a\ < 0. For later purposes we give
the value of a\ :
7, _ 2J-2X TBX)n + 2
To prove A5.5.6), let p(x) be an arbitrary xn . Then, by A5.2.9),
A5.5.8) (+l {A - x^iK^ix) + axPL«x)(a;)] - \)P(x)dx = 0,
where ax must be determined so that the right-hand member of A5.5.6) vanishes
for a; = 1.
According to the result in B), the coefficient of cos md, m > 0, in the right-
hand member of A5.5.6) is again non-positive, and the minimum is attained
for 8 = 0, or x = 1; whence A - x)K(n\x) > 0, -1 < x < +1.
This argument needs only a slight modification for X = 1. (See Szego 17,
pp. 96-97.)
E) We now prove Kln\x) > 0, -1 < x < +1, for sufficiently large values
of n, provided \ < X < 1, or f < X ^ 2.
We may assume n even, 0 < x < 1. First let X < 1. By G.33.6), we have
for 8 = en
A5.5.9) PiX)(cos 8) = 8~xO{n*-1).

3(J MECHANICAL QUADRATURE [ XV ]
Now from A5.5.2), n even, we obtain
A5.5.10) A*,(X)(cos 0) ^ (sin 0)^2X - ? | d™ I | P^icos 0) |.
m-n+2,n+4,- • •
The last sum is (cf. A5.5.1)) equal to
A5.5.11) 0A) ?m~Vy"' = 0~xO(wx~1),
since n~ > m~l and X < 1; this furnishes
Kn (cos 0) > |(sin d)l~~x > 0,
provided rt is sufficiently large and 0 > en"', where c is a proper positive con-
constant.
This argument also holds for X = I with the following modification. We
have (cf. D.7.2))
R'<1V o\ 4 V * sin (m + 1H 1 4 ^ 1 sin (m + 1H
7T m + 1 sin 0 sin 0 x wi + 1 sin 0
Here m is even; in the first sum m ^ n, in the second sum w > n. By A.11.6)
we find
A5.5.12) ^"(cos 0) = (sin 0) + 0O(n~1) > 0,
provided 0 > cn~\ c > 0.
In the case | < X < 2 we use A5.5.6) and A5.5.7); taking the previous
results into account, we find
Kix~v(aos 0) > Ksin 0I-2(X~1), (rxP^'tcos 0) = d^Oin^2),
0 > cn~\
which again shows that A"lX)(cos 0) is positive provided 0 > c'n", where c'
is a proper positive constant. For X = 2 we use A5.5.12). Since o-2 =
-2/x + O(n-1), we obtain for 0 > cV1,
sin2 6 Ki2)(cos 6) > A - 2/7r)(sin 0)
A5.5.14) ,
+ d~2O{n~l) + O^) (sin d)~l > 0.
F) Finally, we assume 6 > 0, 0 = ©(n), n even. According to D.7.1),
we have, X = a + \,
cos ^ - F + r(m+2X)
cos ^^ _ z
where m = 0, 2, 4, • • ¦ , n. By (8.1.1) we obtain for a > 0:
0A), m = 2,4, • • • , n.

[ 15.5 ] AN ALTERNATIVE METHOD IN THE ULTRASPHEKICAL CASE 363
If n —> oo, the first term of the right-hand member is
uniformly in i, 0 < x 5? xo, where
A5.5.16) x~2afa(x) = x~2a I'f~lJa{t)dt = i\2a-l{{tx)-aJa{tx)}dt.
Jo Jo
The second term is equal to o(n2a), a > 0.
Obviously, limI_+o x~2afa(x) exists and is positive. We shall show that the
integral function fa(x) > 0 for x > 0 provided 0 < a S |.
The representation (Watson 3, p. 170, C))
2a+l —i /"oo ,—1 .
holds for — | < a < +|. Assuming 0 < a < §, we can integrate by parts.
We thus obtain the following representation, valid for 0 < a < |:
/,KK 1Q\ ,«-i /¦ /,n 2V~i /"°° (iw) sin iw — cos tu „
A5.5.18) t Ja{t) = -y- r / — — d,U, t > 0.
T(| — a) Ji u{u2 — I)"-*
This reduces the statement to the discussion of the special case
A5.5.19) Gr/2)Vi(z) = / (r1 sin t ~ cos t) dt = / T1 sin tdt - sin a;.
Jo Jo
This function increases for 0 < x < ti and decreases for tx < x < t2, where
h and t2 are the least positive roots of sin t — t cos t = 0; h < 2r < t2. Thus
we need only prove /§ (x) > 0 if x ^ 2tt. We have, however, for x 2: 2tt
A5.5.20) / r1 sin <dt = x/2 - / f1 sin tdt.
According to the second mean-value theorem the modulus of the last integral
is less than 2a; ^ 2Btt)~1. The left-hand member of A5.5.20) is therefore
greater than x/2 - 2Bt)"'1 > 1.
G) The positivity of all the Cotes numbers X, for 0 5S a =/3 5|3/2 has
been proven in Askey-Fitch 1, and for a,j8^0, a+j8 ^ 1 in Askey 5. Two
inequalities of Vietoris 1 for trigonometric polynomials give the positivity
of another sum which is related to a more general quadrature problem (inter-
(interpolate at the zeros of Pna>p)(x) but integrate with respect to A - x)y(l +x)ldx).
For this and a summary of other positive sums related to quadrature problems
see Askey-Steinig 1.
(8) A new proof of the divergence half of Theorem 15.4 was given by Locher 1.

CHAPTER XVI
POLYNOMIALS ORTHOGONAL ON AN ARBITRARY CURVE
In Chapter XI there were introduced certain systems of polynomials which
play a role with respect to the unit circle similar to that which the orthogonal
polynomials previously discussed do for a real interval. We shall now give a
short survey of a further generalization which uses a rectifiable Jordan curve
in place of either the unit circle, or a real interval.
16.1. Preliminaries; definitions
A) Let T be a simply connected region in the complex a>plane, containing
x — oo as an interior point; let the boundary C of T be a continuum consisting
of a finite number of rectifiable Jordan arcs. When integrating along C, the
parts of C are described in an arbitrarily fixed order; along the parts which
have the character of cuts, we must integrate twice. The integrals considered
will have the form
I
A6.1.1) f(x)\dx\.
Jc
Here f(x) is a Lebesgue-integrable function defined on C, and | dx | is the arc
element on C. We denote the total length of the boundary C by L, counting
the cuts twice.
A particularly important case is that of a rectifiable Jordan curve. Another
remarkable instance is that of a Jordan arc. The case of a finite interval is of
this type.
B) Let
A6.1.2) x = 4>{z) = cz + Co + cxz~l + c2z~2 + ¦ ¦ • , c > 0,
be the analytic function, regular and univalent for | z | > 1, which maps | z | > 1
conformally onto the region T, preserving the point at infinity and the direc-
direction therein. According to a theorem of Osgood and Caratheodory (Cara-
the"odory 1, p. 86), the function 4>{z) is continuous in | z | ^ 1 and furnishes a
one-to-one and continuous correspondence between the unit circle | z | = 1
and the boundary C of T (described in the way indicated above). The function
4>{z) is uniquely determined, and the number c is called the transfinite diameter
(Robin's constant, capacity) of C (cf. §16.2 E)).
Let C be a Jordan curve, and let x0 be a preassigned point in the interior of C.
In this case we may consider also the function
A6.1.3) x = Hz) = xo + dlZ + d2z2 + • • • , ' di > 0,
which is regular and univalent for | z | < 1, and which furnishes the conformal
364

[16.1] PRELIMINARIES; DEFINITIONS 365
mapping of | z \ < 1 onto the interior (/ of C; it carries the origin z = 0 into
x = zn and preserves the direction at the origin; 4>{z) is also uniquely deter-
determined and continuous in | z \ ?= 1.
For the interval -1 ^ x ^ +1 we have <$>(z) = \{z + z~l) (cf. §1.9), while
\{/(z) has no sense in this case. For the circle | x \ = 1, xo = 0, we have 4>{z) =
Hz) = *•
We shall denote the inverse functions of A6.1.2) and A6.1.3) by z = $(x)
and z = y(x), respectively.
A particularly simple case occurs when the boundary C of T (or of U) con-
consists of a finite number of analytic arcs. Then 4>{z) is analytic on | z j = 1,
apart from a finite number of points corresponding to the corners of C. The
length of any subarc of C corresponding to z = a0, di ^ 6 ^ 02 , can be repre-
represented in the form
A6.1.4) ( ' \<t>'(ei0)\d6
(similarly for the mapping A6.1.3)).
C) Let C be a Jordan curve and w(x) a positive, and continuous weight
function denned on C. Then the considerations of §10,2 can be applied to
the functions w{4>(e~l )| and w\^{e'0)\ which are both positive and continuous
on the unit circle z = e , — x ^ 6 ^ +tt. Substituting into the corresponding
analytic functions De(z) and A(z) the functions z = l^ix)}^1 and z = ^?(x),
we obtain certain analytic functions Ae(x) and Ai(x) which have the following
properties:
(a) Ae(x) is regular in T including x = <*>, Ai(x) is regular in V;
(b) Ae(x) ^ 0, Ai{x) ^ 0;
(c) Ae(=o) and A»(a;o) are real and positive.
Furthermore, we have
A6.1.5) Km | Ae(xe) |2 = lim | A<fe) |2 = w(x),
Xe-*X Xi-*x
where xe —» x indicates an exterior approach to the point x of C and Xi —> x
stands for an interior approach to x of C. The convergence is in both cases
uniform with respect to x.
These considerations can be generalized by replacing the continuity of w(x)
by more general conditions, and also the curve C by more general point sets.
If C is an arc (for instance if C is a finite segment), At(x) is meaningless.
D) We define the scalar product of two functions f(x) and g(x), x on C, by
the integral
A6.1.6) (/, g) = j- I f(x)g~(x)w(x) \dx\.
We can then orthogonalize the system
2
A6.1.7) 1, x, x\ ¦ ¦ ¦ , xn, ¦

306 POLYNOMIALS ORTHOGONAL ON AN ARBITRARY CURVE [ XVI ]
and this procedure will lead to a set of polynomials uniquely determined by
the following requirements (cf. §§2.2, 11.1):
(a) pn(x) is a polynomial of precise degree n in which the coefficient of xn
is real and positive;
(b) the system {pn(x)\ is orthonormal, that is,
A6.1.8) T / pn(x)pm{x)w(x) | dx | = <5nm, n, m = 0, 1, 2, ¦ ¦ ¦ .
Li Jc
If C is a finite real segment, or the unit circle, we obtain the polynomials
previously considered. In the next section we take up certain fundamental
properties of the polynomials pn(x) which can be classified as follows: formal
properties (minimum-maximum properties, zeros), asymptotic behavior of
pn(x) if n —» oo and x is in the interior of C, asymptotic behavior of pn(x) if
n —» oo and x is in the exterior of C, asymptotic behavior of pn(x) if n —> °°
and x is on C.
In what follows we give only short indications of the proofs, particularly
when no essential change is necessary in the arguments used for the previous
special cases.
Concerning the definition and principal properties of orthogonal polynomials,
see Szego 5 and Walsh 1, Chapter VI. Most of these properties have analogues
for the polynomials orthogonal on the interior U of C; they are associated with
the following definition of scalar product:
A6.1.9) (/, g) = jj- J J f(x)g~(d da,
where A is the area and da the element of area of U. These polynomials have
been investigated by Carleman A, pp. 20-30). Here a weight function can
also be introduced.
16.2. Formal properties
A) Let Dn > 0 be the determinant of the positive definite quadratic form
(cf. B.2.8) and A1.1.4))
1 I I i i 2 , , n |2
L JC
Mo + UiX + UiX + • • • + unxn | w(x) dx
A6.2.1)
A6.2.2) K = t / x'Sfw(x) \dx\
Li Jc
(cf. B.2.1) and A1.1.2)). The orthogonal polynomials pn(x) can be represented

16.2
FORMAL PROPERTIES
367
as follows (cf. B.2.6), B.2.10), and A1.1.9)):
po(x) = Do*,
Pn(x) = (DB_iD»)
A6.2.3)
kno
ko,n—l
1
x
• /Cn,n
. xn
Lnn\
II I*
(x — xq) (x — Xi) • • • (x — xn-i)
xf f w(xo)w(xi) • • • w(xn-i) \dxo
dXn-
n-l I
We have (cf. B.2.7), B.2.11) and A1.1.5))
A6.2.4)
n
\x, —
• wixo)wixi) • • • wixn) | dxo I I dxi | • • • | dxn I.
BO2 Let fix) be a continuous function defined on C. Then the partial sums
sn(x) of the Fourier expansion
fix) ~/oPo(z) +fiPiix) +,
+
+ fnVnix)
A6.2.5)
1
fn = - fix)pn(x)wix) \dx\,
L Jc
n = 0,1, 2,
minimize the integral
A6.2.6) y I \f& ~ P^ T w& I dx I
if p(x) ranges over the class of all xn . The minimum is
A6.2.7) i- /" \fix) |2 u»(x) | dx | - |/o |2 - l/i |2 - • • • - |A |2.
This also yields Bessel's inequality
(lo.z.oj |/o I + |yi| + |/21 + '*•
Let C be a rectifiable Jordan curve, and let/(x) be an analytic function regular
in the interior of C and continuous on C. Then the equality sign in A6.2.8)
72 Concerning the considerations of B), C), D), cf. §3.1, §11.1, §11.3.

368 POLYNOMIALS ORTHOGONAL ON AN ARBITRARY CURVE [ XVI ]
is to be taken; that is, Parseval's formula holds. This follows from Theorem
1.3.4.
Smirnoff A) investigated the validity of this formula for the more general
class of functions f{x) for which
A6.2.9) /{iK2)}A-C0 {*'(«)}*
is of class H2 in | z \ < 1 (cf. §10.1); here we have used the symbols previously
introduced. Parseval's formula holds for this class if and only if the map
function ^/{z) satisfies the hypothesis
0 ^ r < 1.
(See Smirnoff 1, pp. 164-168.) According to Keldysch and Lavrentieff A),
there exist rectifiable curves for which the hypothesis A6.2.10) is not satisfied.
C) Let p{x) be an arbitrary -irn in which the coefficient of xn equals unity.
Then
A6.2.11) min| / | p{x) I2 w(x) I dx \
L Jc
is attained if and only if p(x) = (Dn/Dn_i) pn(x), and this minimum is Dn/Dn-i.
D) Let xo be an arbitrary but fixed point in the complex plane, and let p(x)
be an arbitrary irn with p(xo) = 1. Then the minimum of A6.2.11) is attained
for p(x) = \Kn(xo, xo))~lKn(xo, x), where
A6.2.12) Kn(x0, x) - po{xo)po(x) + p^XoJp^x) + ¦ • • + pn(xo)pn(x).
The minimum is {Kn(xo, Zo)}- (The same result can be expressed in terms
of a maximum property; see §3.1 C) and §11.3.) The "kernel polynomials"
Kn(xo, x) can be characterized by the condition
A6.2.13) y / Kn(x0, x)p~{xjw(x) | dx \ = pl^j,
L Jc
where p(x) is an arbitrary irn (cf. C.1.12)).
Using Kn(xo, x), we can express the nth partial sum sn(x) of the development
A6.2.5) in the following form (cf. C.1.11)):
A6.2.14) sn(x) =j
f
c
E) The extremum problem of C) can be generalized as in the case of an
interval (§3.11). For instance, the problem of the "Tchebichef deviation"
corresponding to A6.2.11) consists of the determination of the minimum of
max | p(x) |, x on C, when p(x) is an arbitrary 7rn in which the coefficient of
xn is 1. The polynomials which solve this problem have been investigated
by Faber C).

[ 16.3 ] ASYMPTOTIC BEHAVIOR OF Kn(x0, x) INTERIOR TO THE CURVE C 369
Let /!„ be the minimum in question. Then (cf. Faber, loc. cit.)
A6.2.15) lim rf/n = c,
where c is the transfinite diameter defined in §16.1 B). We shall see that a
similar formula holds for the minimum of the problem in C) (cf. A6.4.3)).
Fekete A) extended the definition of the transfinite diameter to an arbitrary
closed set in the complex plane by showing that for the corresponding minimum
values Mn (which have meaning if we replace the curve C by an arbitrary closed
set), the limit A6.2.15) still exists.
The general transfinite diameter is a remarkable set-function which can be
calculated in various cases (cf. for instance Szego 6, p. 254).
Concerning the extension of §3.11 E) to an arbitrary curve, see Julia 1.
F) The zeros of pn(x) lie in the least convex region containing the curve C.
See Szego 5, pp. 236-241; Fejer 7. The proof can be based on an argument
similar to that of §3.3 B) (cf. also §16.4 A), (a)).
Concerning the location of the zeros of Kn(xo, x), see Szego 5, pp. 241-244.
See also Theorem 11.4.1 and Problems 5 and 49.
16.3. Asymptotic behavior of Kn(x0, x) in the interior of the curve C
A) In this section we assume that C is an analytic Jordan curve and the
weight function w(x) defined on C is positive and continuous there. Let x0
be an arbitrary point interior to C; denote by z = ^(x) = ^(xo, x) the inverse
function of the map function \p(z) defined by A6.1.3). In the present case
\(/(z) is regular and univalent in a certain circle | z | ^ P where P > 1. Let
Ai(x) have the same meaning as in §16.1 C). We shall prove the following
theorem:
Theorem 16.3. Let \pn(x)} denote the set of orthogonal polynomials defined
by the conditions A6.1.8). Then the series
A6.3.1) K(x0, x) = po(xo)po(x) + pi(xo)pi(x) + ... + pn(x0)pn(x) + • • •
is convergent provided xo and x are arbitrary points in the interior of C. The
convergence is uniform both in x0 and x if x0 and x are limited to a closed set in
the interior of C. Furthermore, we have
A6.3.2) K(x0, x) = ~ {AiMAiWr'WixoWix)}*.
In case w{x) = 1, see Szego 6, pp. 244-251. The assumptions of this theorem
concerning w(x) and C can be generalized (cf. Smirnoff 2, pp. 353-356). For a
Jordan arc (especially for a finite segment) these considerations have no sense.
For the special case where C is the unit circle, see A2.3.17).
As a consequence of the convergence of A6.3.1) we notice that
A6.3.3) lim pn(x) = 0,

370
POLYNOMIALS ORTHOGONAL ON AN ARBITRARY CURVE [ XVI
if x is in the interior of C (uniformly on every closed set in the interior of C).
B) Let p{x) be an arbitrary xn , and let xQ be in the interior of C. Then
according to Cauchy's theorem,
A6.3.4)
Consequently,
A6.3.5) \p(xQ)\2 ^ ~
Zir
x - xo\
2«
ix
X — Xq
dx.
2x<5 L Jc
P(x)
dx
where w{x) ^ m and <5 denotes the least distance of x0 from the points x on the
curve C. Taking into account §16.2 D), we have
A6.3.6)
*n{Xo,Xo) = —.
Hence the series K(x0, Xo), and therefore A6.3.1), are convergent, as is readily
shown by Cauchy's inequality.
C) Now we show that
A6.3.7) K(xQ, xo) = lim Kn(x0, x0) ^ ?- {Ai(xo)}~^'(xo).
To this end we consider the function
A6.3.8)
which is regular in the interior of C, that is, for \z\ < 1; x = ^(z), z = ty{x).
(The last factor is regular even for | z \ ^ 1.) Let r be fixed, 0 < r < 1, and e
an arbitrary positive number. According to Theorem 1.3.4 we can find a
polynomial Q(x) such that
A6.3.9)
F(rz) - Q(x)
x on C.
Writing p{x) = {Q{xo)} lQ{x), we obtain from §16.2 D), for a sufficiently
large n,
A6.3.10) {K{xo, xo)} ^ {Kn{xo, xo)}'1 ^ L^LL2 f \Q(X) \*w(x)\dx |;
whence, on allowing e to approach zero, we obtain
A6.3.11)
Now if r —> 1 — 0, we have
A6.3.12)
F(rz) \2w(x) \dx\.
\F@)
1-2
2x
\*'(x)\\dx\ =
1-2
which is equivalent to A6.3.7).

16.4 ] ASYMPTOTIC BEHAVIOR OF pn(x) EXTERIOR TO THE CURVE C 371
D) Finally, we consider
Jn = lim - .
r-'1-0 L J\z\,
Kn(x0>
Zic
2
\dz
A6.3.13)
= T \Kn(x0, x) \2w(x) \dx\
L Jc
- lim Tr-'riAiixoir'i-Z'ixo)}* f Kn(x0, x)Ai(x) {*'(*) }~* $
r-*l-Q J\*\=r 1Z
The first term equals Kn(x0 , xQ); for the second term we obtain
A6.3.14) -iT1{Ai(xo)r1{?'(xo)}i-2*Kn(xo, xo)A,(xo){^/(xo)}~i = -2Kn(xo, x0);
so
A6.3.15) Jn = ~Kn(x0, xo) + ~ Mxo)}~**'bo)-
Ztr
Therefore, lim,^,, Jn = 0 by A6.3.7). From this it follows, if | z \ < 1, or if
x is in the interior of C, that (cf. G.1.4))
A6.3.16) lim Kn(x0,
which is the same as A6.3.2).
16.4. Asymptotic behavior of pn(x) in the exterior of the curve C
A) We shall introduce the assumptions concerning the curve C and the
weight function w(x) used in §16.3. We denote by z = $(z) the inverse func-
function of the map function x = 4>{z), defined by A6.1.2); in the present case </>(z)
is regular and univalent in the closed exterior of a certain circle | z | = r where
r < 1. Let A«(x) have the same meaning as in §16.1 C). We may then make
the following assertion f
Theokem 16.4. Let {pn(x)\ denote the set of orthogonal polynomials defined
by the conditions A6.1.8). // x is in the exterior of the curve C, we have
A6.4.1) pn{x) ^ (^
The ratio of these expressions as n —> °o tends to unity uniformly in every finite
or infinite closed region exterior to the curve C.
In case w(x) = 1, see Szego 5, pp. 260-263.
We mention the following noteworthy consequences of A6.4.1):
(a) The zeros of pn(x) approach the closed interior of C uniformly as n —> <»
(apply Theorem 1.91.3 (Hurwitz's theorem)).

372 POLYNOMIALS ORTHOGONAL ON AN ARBITRARY CURVE [ XVI ]
(b) If a; is in the exterior of C,
A6.4.2) Km P?+^ = Km {pn(a;)}1/n = *(x).
71"** 00 Jr^\J Tl~~* 00
Cf. A2.2.6).
(c) The convergence domain of the expansion A6.2.5) of an analytic func-
function, regular in the closed interior of C, is the interior of a level curve CR of the
conformal mapping x = <?(z) (cf. §1.3 B)). The determination of R from the
coefficients /„ is analogous to that in the case of a power series (cf. Theorems
1.3.5 and 12.7.3).
(d) Let kn be the coefficient of a;" in pn(x). Then
A6.4.3) kng*(L) {a.Coo)}-^—»,
where c is the transfinite diameter of C (cf. §16.1 B); §16.2 E)). This result
can easily be expressed in terms of the determinants Dn introduced in A6.2.4).
B) The proof of A6.4.1) is based on an argument similar to that in §16.3;
here the minimum property of §16.2 C) is used. We first show that
A6.4.4) Km sup c"^2 ^ ^ {Ae(°o) }2.
n—»oo XJ
To this end we introduce the polynomials of Faber's type fn(x) (Faber 1),
defined as the polynomial part in the Laurent development of the function
A6.4.5) f^-Y {A.(x)}-M*'(x)}M*(x)}» = j/»(x)
around x = °o. The function gn{x) is regular for \z\ ^ r, and fn(x) is clearly
a 7rn . Let Cr be the curve corresponding to | z \ = r. On applying Cauchy's
theorem to the ring-shaped region bounded by Cr and a large circle, we obtain,
if a; is on C,
gn(x) — fnix) = ^—. / d% = —. /
2in Jcr ? — x 2tti Je
d%,
rk- x
where the integration is extended in the negative sense. Hence
A6.4.7) 1 gn(x) - /»(*) 1 < Mrn for x on C,
where M depends only on C and r. The functions gn{x) are uniformly bounded
if a; is on C.
Now let Yi, 72, • • • , Ym be arbitrary constants, n > m. The polynomial
A6.4.8) P(x) = (j?\ cn+i{fn(x)

[ 16.4 ] ASYMPTOTIC BEHAVIOR OF Pn(x) EXTERIOR TO THE CURVE 373
is of degree n, and its highest coefficient is unity. Then, according to §16.2 C)
and A6.4.7),
A6.4.9)
2"+1 ~ I \fn(x)
C2M \ [
+
\*W{x)
+ C
so that
A6.4.10)
or
A6.4.11)
<I
—2
ymz \ w\x) \dz\;
lim sup c-an-V ^~
n-»oo L
y(z) \2w(x) \ dz |,
where 7B) is an arbitrary analytic function, regular for | z \ ^ 1, with 7( 00) = 1.
On putting
A6.4.12)
_
and allowing R to approach 1 + 0, we obtain the inequality A6.4.4).
C) Now we consider
'n = lim T
B-»l+0 L
2tt
\dz
A6.4.13)
1
L ic
dx
iz
The second term is
J'n = 2 - 2 (~J Ae(oo)cn+ikn.
Therefore,
A6.4.14)
In view of A6.4.4), this implies limn_«, j'n = 0, which establishes the statement
A6.4.1).

374
POLYNOMIALS ORTHOGONAL ON AN ARBITRARY CURVE [ XVI
16.5. Asymptotic behavior of pn(x) on the curve C
Finally we have the following result:
Theorem 16.5. The asymptotic formula A6.4.1) holds uniformly outside and on
the curve C provided the function Ae(x) is regular in the closed exterior of C; more
precisely we have
A6.5.1) Pn(x) =
where 0 < h < 1; this constant h depends on C and w(x).
The same formula holds in a sufficiently small neighborhood of C in the interior
ofC.
A) For the proof we use the polynomials Fn (x) of Faber's type associated with
A6.5.2)
2ir
(which is the "principal part" of the right-hand side of A6.5.1)) in the same way
as the polynomials fn(x) denned in §16.4 B) are associated with A6.4.5). If
0 < r < 1 and r is sufficiently near to 1, the function Gn{x) is regular for \z\ =? r.
We have again
A6.5.3)
(?„(*) - Fn(x) | < Mrn
for x on C,
where M depends only on C, r and w(x). The functions Gn(x) are uni-
uniformly bounded on C.
We write (cf. §16.4 B))
A6.5.4) P(x) = (J^j \e(«>)cn+iFn(x).
This is a irn with the highest coefficient unity. Hence
| Fn(x) |2 w(x) I dx
A6.5.5) = {^j l {Ae(oo)\2c2n+1 i j | Gn(x) |2 w{x) \dx
c2nO{rn
O{rn)
(The simpler nature of this argument is due to the fact that Ae(x) is regular on C.)
On the other hand,
A6.5.6)
1 = t / I Vn(x) |2 w(x) | dx | = —
L Jc Zir
Gn(x)
dz
^ lim
Gn{x)

[ 16.5 ] ASYMPTOTIC BEHAVIOR OF Vn(x) ON THE CURVE C 375
so that
A6.5.7) *? ^ {*.(») jV1*1;
consequently
A6.5.8) K2 = ^ {Ae(°°)}Yn+1 + cin0{rn).
B) Now let
A6.5.9) pB(x) = XoFo(x) + XxFi(x) 4- X2F2(x) + • • ¦ + X»F»(x),
where Xo, Xi, X2, • • • , Xn are proper constants, X, = X(i/, n); we have
A6.5.10) XB = (^
so that Xn is real and, by A6.5.8),
A6.5.11) Xn = 1 + O(rn).
From the definition of Fn(x) we conclude that
*)}-»{?,.(*) - \nFn(x)}
xn—
< 2tt J^
1
1 / i / \ i2
=? J pn{x)Fn{x)w{x) \ dx \ + —-
A6.5.12) = Xo + Xx*(x) + X2{$(a;)}2 + • • • + Xn_i{«l>(a;)}n~1 + y
+ Y2Z
where y,, y'y are certain constants. Therefore,
|2
Li \ fn v"v/ I \XJ | OjX I
i" (a;) \ w {x) dx
c

376 POLYNOMIALS ORTHOGONAL ON AN ARBITRARY CURVE [ XVI ]
In the second term Fn(x) can be replaced by X^pnCx); in the third term we use
A6.5.11) and A6.5.3) and obtain
A6.5.14) |Xo|2+ |XX|2 + ¦•• + |Xn_!|2^ -1 + y [\Gn(x)\2w(x)\dx\ + O(rn),
L Jc
or
A6.5.15) | Xo | 2 + | Xx | 2 + • • • + | Xn_x | 2 = O(rn).
Now Fn(x) = 0A), uniformly if x is on C. Whence, according to Schwarz's
inequality,
A6.5.16) \0F0(x) + XxFi(x) + • • • + XB-!FB_i(x) = O(nVn/2);
and from A6.5.9), A6.5.11), and A6.5.3),
p»(x) = KFn(x) + OinV2) = Fn(x) + OinV12)
A6.5.17)
= Gn(x) + O(nVn/2).
This establishes the statement. The extension of A6.5.1) to the interior of C
also follows immediately, since A6.5.3) holds if x is in the interior of C and
sufficiently near to C.

PROBLEMS AND EXERCISES
1. We denote by i\ < in < • ¦ • the positive zeros of Airy's function A{x)
(§1.81); then i, ~ vl if j»-> ». (Use A.81.4), A.81.1), and A.71.7).)
2. Let i4.(x) denote Airy's function (§1.81). We have, for real values of x,
2irt73
exp (-p - Pe2*tl3x)dP} = A(x).
(Develop both sides in a power series in x; see A.81.4), A.81.1), and A.7.3).)
3. Let pn(x) denote the Poisson-Charlier polynomial B.81.2); then
W*) =P.(x)
satisfies, for x = 0, 1, 2, • • • , the relation
4. By use of the notation B.2.1), B.2.7), the "kernel" polynomials Kn(x0, x)
(cf. C.1.9)) can be represented as follows:
—i
Kn(x0, x) = —Dn
Co cx
C\ C2
c2
c3
Cn 1
Cn+1 ^0
Cn
1 X
Cn+2
2
X
xn 0
(Use C.1.12) with p(x) = x", j» = 0, 1, • • • , n.)
5. Location of the zeros of the "kernel" polynomials Kn(xo, or) (cf. C.1.9)).
Let a and b be finite, and let xo be an arbitrary non-real number. Every zero
? of Kn(xo, x) lies in the area bounded by the interval [a, b] and by the circular
arc through a and b whose continuation passes through xq . (Cf. Szego 5, p. 244.
By C.1.12)
{x _. xo)
X —
x-S
X —
- da{x) = 0.
In the conformal mapping (a; — ?)/(x — x0) = x', the image of a 5S x ^ b is a
circular arc; the segment bounded by this arc and its chord contains x' — 0.)
6. Derive from C.2.1) the representation, n — 1, 2, 3, ¦ • ¦ ,
n(x) = po{x)
o
C\
A2x
A3x
0
0
0
Anx + Bn
377

378 PROBLEMS AND EXERCISES
7. Prove the reality of the zeros of pn(x) using B.2.9), or using Problem 6.
8. Let Xi,xs, •• • ,xnbe distinct values in [a, b], and let/(z) have a derivative
of order 2n in [a, b]. If H(x) is the 7r2n_i satisfying the conditions
H(x,) = fix,), H\xv) = /'(*,), „ = 1, 2, .. ¦ , nf
there exists a value ? == ?(x) in [a, b] such that
/(*) - H(x) = J1^f (x - xtf{x - x2J ...(*- xn)\
(See A. Markoff 5, pp. 6-8. Let ? be a fixed value, x ?± x, ; and to the function
of z given by
apply Rolle's theorem.)
9. Let f(x) have a continuous derivative of order 2n in [a, b]. By using the
notation of Theorem 3.4.1 obtain
f(x) daix) = J^L
Here ? is a proper value in [a, b], and kn is the highest coefficient of the ortho-
normal polynomial pn(x) associated with the distribution daix) (cf. B.2.15)).
(See A. Markoff 5, p. 81; use the preceding problem.)
10. Let Xi, Xii, • • • , xn be the zeros of the orthogonal polynomial pnix) asso-
associated with a given distribution daix) on the interval [a, b], and let Xi, X2, • • • ,
Xn denote the Christoffel numbers C.4.3). Define the scalar product of two
functions fix), gix) by
n
(f, q) = 23 Kfix,)gix,).
y-l
The functions 1, x, x2, ¦ ¦ ¦ ,xn x are linearly independent; by orthogonalization
we obtain the polynomials poix), piix), • ¦ • , Pn-iix) which are the same as
those associated with daix).
11. If daix) = wix)dx, wi—x) = wix) and a + b = 0, we have for the Chris-
Christoffel numbers C.4.3)
X, = XB+1_,, v = 1, 2, • • • , n.
12. For the Jacobi polynomial P^'^ix) the Christoffel numbers are
X' = 2^T1 A + Xy)> » = l,2,>",n.
(Cf. D.1.8) and A5.3.1).)
13. In the special case P?~il~i)(a;)) the numbers y, = cos <f>,, 0 < <f>, < t, of
the separation theorem of §3.41 are

PROBLEMS AND EXERCISES 379
$, = (n — v)r/n, v = 1, 2, • • • , n — 1.
Verify the separation theorem for PB~* ~4)(a;), PBU)(:c), P^A\x).
14. Let xi, x2, ¦ ¦ ¦ , xn be the zeros of Pnal/3)(x). Then
xi + x, + • • • + zB = 03 - a) - , w , a.
2n + a + /3
(Cf. D.21.2).)
15. New proof of D.7.31). Combining the second part of D.1.5) (polynomials
in 1 - 2x2) with the first formula in D.22.1), we find
P?\x) = 2" (n + I ~ ^ xnF(-n/2, A - n)/2, -n - X + 1; rT2).
16. The generating functions D.7.16) and D.7.23) of the ultraspherical poly-
polynomials P(n\x) are identical if and only if X = |, that is, in the Legendre case.
17. The functional equation
A - *)/'(*) = X/(-z),
where X is a parameter, has a polynomial solution f{x) ^ 0 if and only if X =
(-l)"(n + 1) and/(z) = const. \Pn(x) + PB+i(:c)}, n = -1, 0, 1, • • • ; P_i(z)
= 0. (Write
AT
V — — 1
where c_i, Co, • ¦ ¦ , cN are constants, and use
which follows from D.7.27).)
18. Show that
QB@) = (-1) —^ j- TV, n even,
/(*) = E cv\Pv(x) + Pr+I{x)\,
1
1-3
22-
2-
•
4
4
• ¦ ¦ n
(n-
.(n-
1)'
-1)
Q/r\\ ( i\(»+l)/2 ^ * V'* J-^ jj
»@) = (-1) —3»5 ... n—"' n
Here Qo"(O) = 1, Qi@) = -1. (Use the recurrence formula D.62.13), D.62.14),
D.62.3).)
19. The "Laplace transform"
f(s) = / e~'lF(t) dt,
Jo
of Laguerre's function F(t) = taLl"\t) is
(Cf. Sonin 1, p. 42; use the method of the "generating function".)

380 PROBLEMS AND EXERCISES
20. Writing
xaL(na\x)(T(n + a + I)) = f(n, a; x),
we have, f or a > /3 > — 1,
1 f'
f(n, a; x) = -, / (* - u)a"'3-1/(n, j8; u) du.
T(a — /3) Jo
(Kogbetliantz 22, p. 156. Take the generating function of both sides and use
E.1.16); the resulting formula for Bessel functions can be verified by means of
A.71.1) and A.7.5).)
21. Writing e-*'ix"iLln"\x) = /»(*), we have
(Hardy 1, p. 139; use E.1.9) and A.71.1).)
22. Assume x ^ 0, y ^ 0, max (x, y) > 0. Then
iax(at.v)
(E. R. Neumann 1, Watson 6. Apply Theorem 9.1.5 and use the formula
(n + 1) [' Ln(t)dt = x{Ln{x) - L'n(x)\
Jo
(cf. E.1.2)).)
23. From E.1.15) derive Mehler's formula
n-0
nl
» = (i - w^ exp !2xyw ~
[ l -
(See Watson 5, Erd&yi 2.)
24. From E.1.9) and E.6.1) derive the following generating function for
Hermite polynomials:
5 ^r •"" -A
5
where m = [n/2]. (Cf. Doetsch 1, p. 590, G).)
25. Assume k > — \ and let H^\x) denote the orthogonal polynomials cor-
corresponding to the weight function e~** | x |2* in [— <», + »]. Then the follow-
following differential equations are satisfied:
, f0> w even,
- ex"^ = .0, e =
Bfc dd;y = H(nk\x);
z" + hn + 2k+l-x2+ (l)^fc% = 0; « = e~Mn x*
(Generalizations of E.5.2).)

PROBLEMS AND EXERCISES 381
26. For fixed 0, n, and x we have
\ 2
(Use D.21.2).)
27. Let {xy\ be the zeros of Pia'?)(x) in decreasing order, a > -1, /3 > -1.
Then for v = 1, 2, ... , n
lim x, = — 1, lim a;, = +1.
a-*+oo j3-*+oo
In the first case /3 and n, and in the second case a and n, are fixed. (Use Prob-
Problem 26 and D.1.3).)
28. Let [x,\ be the zeros of Pf\x), X > 0. We have for fixed n
lim x, = 0.
\-»+00
(From D.7.6) lim^ BX)"nP™'(*) = *7n!; see also E.6.3).)
29. Assume a > -1, /3 > -1; if {*,} and {^} denote the zeros of Piai/3)(:c)
and Lia)(x) in decreasing and increasing order, respectively, we have (a, n, and
v fixed)
lim /3A - a;,) = 2^.
|3—+oo
(Use E.3.4).)
30. For fixed n and x we have
—n r(a)/ \ \i- X)
L\ '(ax) =
l* —n r(a)/ \
lim a L\ '(ax) =
Tt I
(Use E.1.6).)
31. Let jo = 0 < ji < jz < ¦ ¦ ¦ be the positive zeros of Ja(x). Then {j,}
is a convex sequence if — \ < a < +|, that is, jy+1 — jv is increasing. Further-
Furthermore v~ljv is increasing. (Apply Theorem 1.82.2 to A.8.9).)
32. With the same notation and assumptions as in Problem 31, we have
j, > (v + a/2 - 1/4)*, v = 1, 2, 3, • • • .
(Put n = 2v - 1 in F.3.13).)
33. We have for the factor C,n , defined by F.31.13),
GV4J < Crn < 4.
Here ji is the least positive root of Jo(x). These bounds are the best possible.
(Use the increasing property of v~xjv ; see Problem 31.)
34. Assume a > -1; denote by Xi < x2 < ¦ ¦ ¦ < xn the zeros of Lia)(x). If
a 5= |, the sequence x\ — x\-i, v = 2, 3, • • • , n, is increasing; if a > \, this is
true for x,^ > (a - ?)*. (Apply Theorem 1.82.2 to the fourth equation in
E.1.2).)

382 PROBLEMS AND EXERCISES
35. With the notation of the previous problem,
min (** - z,*_i) ~ n, max (a;* - z*_i) = x{ - s*_i ^ 2~i3~ife - ii)n~l,
n —¦ oo;
where ii and i2 are the least positive zeros of Airy's function A (x). (Use (8.1.8),
(8.22.1), and (8.9.15).)
36. Let Xi > x2 > • • • > ?[(B+i)/2] be the non-negative zeros of the Hermite
polynomial Hn(x). Write
xv = xrn = hn - Fk,r*k,, K = Bn + 1)*.
Then for fixed y and increasing n, the numbers <„„ are decreasing. Further,
show that for 1 ^ v ^ (n + l)/2,
where P < p,n < Q; here P and Q are two absolute positive constants. (On
writing $ = Fhn)~H in F.32.10), we have
+ \t/Z - {<ShnTH*)z = 0.
The monotonic character of t,n follows by means of Theorem 1.82.1. Further-
Furthermore, see F.32.3),
Now apply Problem 1.)
37. Consider n unit "masses", n ^ 2, at the variable points X\, X2, • • • , xn
in the interval [ — 1, +1]. For what position of these points does the expression
n i Xv - x? i
become a maximum? (Stieltjes 4, p. 441; the maximum position is the same
as in Theorem 6.7.1, obtained by replacing n by n — 2 and writing p = q = 1.
We have A - z2)P»-V(*) = const. \Pn(x) - PB_2(z)} = const. A - x*)P'n-i(x);
see D.7.27).)
38. Consider n unit "masses", n ^ 2, at the variable points x\, X2, • • • , ?„
in the interval [0, + °° ], such that
rT'ix! + x2 + • • ¦ + xn) ^ K
where if is a fixed positive number. For what position of these points does the
expression
become a maximum? (Cf. Problem 37 and Theorem 6.7.2. We have xL{n-i(x)
= const. {Ln(x) — LB_i(:c)} = const. xL'n{x); see E.1.14).)

PROBLEMS AND EXERCISES 383
39. Assume a > -1, /3 > -1. With the notation of G.32.2) we have, for
n —* <x>,
i-i
max
{rfo+DpV ifg^ -i,
r n |a + $|a | /3 + f | |a + /3 -f 1 | "
if -1 < q ^ -J.
(Cf. G.32.2).)
40. From G.33.10) derive
PB(cos 9) - PB+i(cos 9) = 0}0Gr*), 0 < 9 ^ tt/2.
(Use
A - x){PB(x) + P'n+1(x)} = (n-f
(see Problem 17) and G.33.9).)
41. We have for 0 < 9 < t
(sin 0)* | Qn (cos 0) | <
The constant Gr/2)* cannot be replaced by a smaller one. (See Hobson 1,
p. 309, where the bound (ir/n) is obtained; compare Theorem 7.3.3, and Prob-
Problem 18.)
42. Let f(x) be an arbitrary vn , non-negative for all real values of x, and let
"+0O
e~xif(x) dx = 1.
f
J—o
s/n\ -}3-5 ••• Bm + 1) _i j r ,.,
max/@) = iT* ¦-— —^— ^ t n\ m = [n/4j, n
Then
(Use Theorem 1.21.2, G.71.2), E.5.9), and E.5.5).)
43. .A mean-value theorem for polynomials. hetf(x) be a 7T2B • Then
fib) - f{a) - F - oiTtt),
where $ is a proper point in the interval
\{fl + b) -
Here Xi denotes the greatest zero of the Legendre polynomial PB(:c). (Tchaka-
lofif 1; use C.4.1).)
44. Derive the formula of Lipschitz
Jo
e~atJ0(bt) dt = (a2 + fc2)"*, a > 0, b > 0,
from the generating function of the Legendre polynomials, that is, from D.7.23)
for X = \. (Write w = e~alN, x = cos (b/N), a and b fixed, N—> <», separate

384
PROBLEMS AND EXERCISES
23"-o Pm(x)wm into m ^ wN and m > uN (« a fixed positive number), and
use (8.1.1) and G.3.8).)
45. Derive E.4.2) from E.1.9) by writing x = a/N, w = e~blN, a and b fixed,
b > 0, JV -» + oo. (See Problem 44; use (8.1.8) and G.6.8).)
46. JVeu> p^oo/ o/ the asymptotic formula (8.22.4) of Hilb's type for Laguerre
polynomials. From the generating function E.1.9) we obtain
-*/2r(«)
= n~, exp<--
L±I?
@+)
i0+)
xl+e
A -
2 1 -
- A -
where $(x, z) is regular in | z \ < 2ir. If we develop $(x, z) in a power series
in 2, the resulting integrals can be reduced to Bessel functions (Watson 3, p. 176,
A)). (This argument furnishes not only (8.22.4) for an arbitrary real a, but
also an asymptotic expansion of Hilb's type whose terms are Bessel functions;
see Wright 1. This is the analogue of Szego's argument used in 15 for Legendre
polynomials.)
47. Let a be real, but different from zero. The infinite series
-X tanl/2
n e
n-l
is convergent if X > \, and divergent if X ^ \. (For X > 0 the convergence of
the series is equivalent to that of the integral Jf aTV111 dx.)
48. The polynomial sn(a, z) (see A1.3.3)) admits the following representation,
in which the notation of §11.1 is used:-
sn(a, z) = -D
—i
Co C_i
C-n+l
1
Co • • • C_B+2 C_n+i &
CB CB_i
1 Z
n—1
Co a"
2B 0
(Cf. Problem 4.)
49. Second proof of Theorem 11.4.1 on the zeros of sn(a, z). Use the argument
of Problem 5. (If z0 is the zero in question, and (z — zo)/(z — a) = z', the image
of | z | = 1 contains z' = 0.)
50. Theorem 11.3.3 on the convergence of 2Z"=o \<f>n(z) \2,\z\ < 1, is not true
if the weight function f(9) is such that log fF) is not integrable in Lebesgue's
sense. (Let/@) = 0 for -e < 9 < +e, and/@) = 1 for e ^ 8 ^ 2tt - e, and
apply A6.4.2). In this case | <f>n@) |1/n -» R, R > 1.)

PROBLEMS AND EXERCISES 385
51. Let/@) be a weight function on the unit circle, -ir ^ 0 ^ +v, for which
tl log/@) dd exists. Assume p > 0, and let ixn = jin(J; p) be the minimum of
where p{z) = zn + • • • is an arbitrary irn with the highest term zn. Then
lim /xn(/;p) = expj^ / log/(fl)dfl) = ©(/). '
n—oo [ZiT J-tc )
(This is a generalization of A2.3.3); see the argument used there.)
52. Let knQ be the highest coefficient of the orthonormal polynomial pn{x)
associated with the distribution du{x) on the finite interval [a, b]. Then
 ~\-h
knQ> 22n~1F - a)~n i f" da(x)\
(Cf. Theorem 12.7.1; Shohat 2, p. 575, B4). Use the extremum property of
Theorem 3.1.2, choosing P{x) = 21~2n(b - a)nTn{2(x - a)/(b - a) - 1).)
53. Use the notation of Problem 52. Let [a', b'] be a subinterval of [a, b]
such that a(x) is constant in [a', b']. Then two positive constants A, B exist,
B > 4/F - a), such that /cn0 > ABn. (Cf. Shohat 2, p. 577. Use the extremum
property of Theorem 3.1.2, and choose for p{x) the "Tchebichef polynomial"
corresponding to two disjoint segments in the sense of §16.2 E)..)
54. By use of the notation of §12.7 A) we have, under the assumption of
Theorem 12.7.1,
v even
2" "n" d0 di, v odd.
Here v is fixed, n —¦ ». If di = 0, the second formula is to be read as follows:
Urn,,-,,* 2~nnA~')/2fcn, = 0. (See the special cases in Shohat 2, p. 577. Use
Theorem 12.1.2 and observe that on putting
we have ynv = B2'i/!)~1( — n)", v fixed, n —»<».)
55. In case of the Jacobi polynomials
we have, v fixed, n —¦ °°,
/n'= (v/2)! ' { P) [(v-l)/2}\ '
according as v is even or odd. In the second case we assume a ^ /3. (See

386 PROBLEMS AND EXERCISES
Geronimus 1, p. 380; use the result of Problem 54; in this case we have d0 =
o-(a+/3+l)/2 J fa \ n-(<*+P+l)l2 \
* , «i = (P - a) 2 .)
56. With the notation and assumption of Problem 54 we have
tt = ~n/4 + c + (n, lim en = 0,
where c is a constant. (Cf. Shohat 2, p. 577. Use the same argument as in
Problem 54.)
57. Let ly(x), v = 1, 2, • • • , n, be the fundamental polynomials of the La-
grange interpolation with the zeros of Tn{x) as abscissas. Then if k is even,
r+i
/ ly.(x)lySx) • • • lyAx)(l ~ X^) "^ dX = 0.
Here v\, v2, • • ¦ , vk are distinct integers between 1 and n. (E. Feldheim 1;
\Tn{x)}k~]', x = cos 9, is a cosine polynomial not containing terms with cos v9,
v < n.)
58. In the ultraspherical case a = /3, —1 <a = /3<0, we have for — 1 S x
^ +1 (notation as in Problem 57)
l}2+ ••• + \ln(x)}2 ^\a\-\
(For a = — | see Fejer 13, p. 5. Use Problem 59 and the first identity in
A4.1.11).)
59. In the ultraspherical case a = P > —1 we have (cf. A4.5.2))
/ v _ 1 - 2(a + Dxx, + Ba + I)*2
Vy\X) — g ' ¦ ^ (Xj 1 — X —
1 — Xy
60. In the Legendre case we have
3tt . ^
(Cf. Fejft 13, p. 23. Use F.6.5).)

FURTHER PROBLEMS AND EXERCISES
61. Prove the following identities:
P'2n(x) = Bn + l)xP^\Bx*-l),
xP'n{x) = nPn(x) + Bn - 3)Pn_2(z) + Bn -
A - x*)P'n'{x) = -n(n - l)Pn(x) + 2Bn - 3)Pn_2(x)
+ 2Bn - 7)Pn_4(x)
62. Prove:
Pi»(x) = Un(x);
{—x if n = 1,
/ Pn-i(t)dt if n ^ 2.
63. Prove the formula:
1P (x)x"^dx - 1 {n + 2v)! r(y + ^
BV~' " 2" Bv)l T(n + v
64. Prove the identity
/ Pn(x)e~iXxdx = i~
= 0, 1, 2, ¦ ¦ ¦ .
(Use Problem 63 and A.71.1).)
65. Prove that
{ 0 if x > 1 or x < -1.
(Use Problem 64 applying Fourier's inversion formula.)
66. Prove the identity
in particular the identity
{\ + 2xy
(Use D.7.23).)
67. Prove the identity
387

388 FURTHER PROBLEMS AND EXERCISES
(Use E.1.9).)
68. Prove the identity
H,(x)ByY-' = Hn(x + y).
(Use E.5.7).)
69. Let x be a parameter, — 1 < x < +1. The zeros of the polynomial in z:
Po(x) + (;) Pdx)z + r;) p2(xJ2 + • • • + Pn(z)*«
are all real.
(Use Problem 66.)
70. Prove Turan's inequality
IP (r)J — P ,(x)P Jr\ > 0 — 1 < -r <
(Turan 1, see also Szego 22, Karlin-Szego 1 and Csordas-Williamson 1.) If
a,\ and a2 denote the first and second elementary-symmetric function of n
real numbers, we have:
(Use Problem 69.')
71. Derive from Problem 66 the generating series D.10.6) and, in particular,
D.10.7). (Put y = n/z, n -> °o; use (8.1.1).)
72. Derive from Problem 67 the generating series E.1.16). (Put y = n/z,
n-> °o; use (8.1.8).)
73. Prove the following formula of the "Rodrigues type":
x) = ^<n+1(
x) n<( J
(Use Taylor's formula and E.1.9).—In the special case a = 0 this is due to
G. P61ya, 1941.)
74. Let {un| and {j;n| be two sequences; n = 0, 1, 2, • • • . One of the relations
^ (n + a\ , ^ (n + a\ . ,,
implies the other.
75. Using Problem 74 and E.1.6) prove
n\ fz
76. Let un and vn be two sequences; n — 0, 1, 2, • • • . One of the relations
implies the other.

FURTHER PROBLEMS AND EXERCISES 389
77. Prove the identity:
n\ fa v\ (n - 2v)\'
(Use Problem 76 and E.5.4).)
78. Prove the following identities:
y" + 2xy' + 2(n + l)y = 0, y = e^Hn(x);
J0 v=0
79. Prove that
iim M" ^n (il\ , ^.
n^= \n/ \2yJ
(Use E.5.4). Cf. P. Turan, Matematikai Lapok, vol. 5 A954), pp. 134-137.)
80. Prove that
Km a-nllL^(ah + a) = (- l)-«/2(n!)"'//„B~>x).
81. Let \pm(x)\ be the orthonormal polynomials associated with the distribu-
distribution da{x) in 0 ? x < + x . Denoting by ?i, $2, • • • , ?x- any zeros of pm(.r) we
have
f(t) = e-"[(^-x) '¦¦ (^-x)}-1{pm(x)}ida(x)>0, t > 0.
(See Karlin-McGregor 1, pp. 507-509. Since/@) = 0, we have
/@ = e-{*( / ei*T<p(T) dr. Induction.)
J
82. Notation and assumption as in Problem 81, pm(Q) > 0 for all m. Prove
that
/ z) rfa(x) > 0, / > 0.
(See Karlin-McGregor 1, loo. cit. Letm > n. We represent pn(x) by Lagrange's
interpolation formula corresponding to the abscissas $Ml, • • • , ?Mn+1 chosen as
in 3.3 F); use Problem 81.)
83. Let \pn(x)} be the Poisson-Charlier polynomials B.81, sgn pn@) =
(-1)"). We have, X > 0, m ^ n, j(x) as in B.81.1),

390 FURTHER PROBLEMS AND EXERCISES
]? e-'Kzj{x)pn{x)pm{x)
.r=0
_ ovri . . _. ... ..__ ." .'n
- l)]a*<"+m)(?n!/n!)* V (
e i)
{m — v)\
84. Let X > — 5, X ^ 0. Using the notation D.9.21) we have
+i
A - x2)*-lP^(x)P%\x)Pp(x) dx
provided that ? + m + n = 2s is even and a triangle with sides I, m, n exists.
The integral on the left is zero in every other case. (Cf. D.7.15); cf. Hsu 1.)
85. Let ln(x) have the same meaning as in B.8.1); we have then
tn(x) = (-inn(N-i-x).
Here n and x run over the range 0,1, • • •, N — 1. Prove also that
86. Let pn(x) have the same meaning as in B.81.2); we have then
pB@) = ( - l)n(a7"!I/2 = ( - 1)V/2OX")I/2.
Prove also that, writing pn{x)/pn@) = cn{x,a), we have
cn(x;a) =cx(n;a).
87. Writing Hn(x) = Hn we have
f" -*2
I a i t
provided that, a + 13 + y = 2s is an even integer and s ^ a, s ^ 13, s ^ 7.
In all other cases the integral is zero.
88. Prove that
y
89. Prove the following identities:
L(na){x) = y, (e-xxa + ly'Y + nT*xay = 0;
e~x/2Ln(x) = h, xh" + h'

FURTHER PROBLEMS AND EXERCISES 391
90. Prove the identity
*=0
91. Obtain the estimates
2/x < «p - 2+p/2,
2ft = up -2+P/2,
2ft > up - 2 +p/2,
!^<*p-2M-2
n-p/2logn,
where a,/3,/x> —1 and p>0 are fixed. (See G.34.1).)
92. Prove that
1
0
3 =(n + l)sin0 = 5' n = = 2'
and show that all the bounds are obtained for some 0 and n. (Use G.8.1) for
93. Prove that
d2"
(
and
dx
2n
x"V>0, X>0, a^O.
(Use Theorems 6.72 and 6.73. These results were conjectured by I. Joo.)
94. Prove that
(_!)*+»+» f Lla)(x)L%)(x)L?)(x)xae-xdx^O, a > - 1, k,m, n -0, 1, • • •.
Jo
(Use E.1.9).)
95. Prove that
n+m
pn{x)pm(x) = J2 a(k,m,n)pk(x)
k=\n—m\
with a{k,m,n) ^0 if pn(x) satisfies
Pi(x)pn(x) =pn+i(x) +unpn(x)
where an ^ 0, ft, > 0, an+1 ^ «„, Pn+1^:0n, n = 1,2, • • -,po(*) = 1, PiU) =
x +a. (Askey 4. Use induction. This contains Problem 94.)

392 FURTHER PROBLEMS AND EXERCISES
96. Prove that
Pk(x)+Pm(x) gl+Pn(x), O^x^l
when n(n + l) =k(k + l) +m(m +1). Also show that
J0(x) +J0(y) ^l+J0(z)
when x2 +y2 =z2. Griinbaum 1, 2, Askey 7.
97. If Xln=oOn is finite, a> — 1, and if
n=0 r(n+a
prove that
an=( x^mUMmU) ^ 0. (Sarmonov 1.)
Jo
98. If o*,m,n are defined by
1
A -r)(l -s) +A -r)(l -t) +A -s)(l -f)
show that
k,m,n= I Lk(x)Lm(x)Ln(x)e 3xdx.
Jo
Then use Problem 84 for X = | and D.5.4) to show that dk,m,n ^ 0. Generalize
to fo™Lia)(x)Li°)(x)Lina)(x)xae-3xdx> a^ - ?. (Szego 26, Askey-Gasper 3.)
99. Prove that
J - ¦» J - ¦» ,_i
for all choices of plus and minus signs and any choice of integers n,-. (Use
E.5.11) and Problem 87. See Ginibre 1.)
100. Show that
2nnl?(-l)kLkBx2 + 2y2) =? (?) [Hk(x)]2[Hn_k(y)]2
and in particular that
[n/2]
-J\n —6j) I
>=o J -J-\n
Extensions of this sum to Pn"'0)(x) are given in Askey-Gasper 4.
101. Show that
This formula of Howell is given in Bailey 2.

APPENDIX
ON A SINGULAR CASE OF ORTHOGONAL POLYNOMIALS
In recent publications F. Pollaczek A-4) has introduced certain remarkable
generalizations of the Legendre and other classical polynomials which should be
treated in this Appendix in a brief way. The polynomials of F. Pollaczek show
in many respects a singular behavior. For a short treatment of this topic we
refer to Bateman Manuscript Project 1, vol. 2, pp. 218-221. Cf. also Szego 24.
1. Definitions and formal properties
Let a and b be real parameters, a > \b\. We write
and define the polynomials Pn(x; a, b) = knxn + • • • by the generating function
00
f(x, w) = /(cos 6, w) = ]T Pn(X; a, b)wn
A.^) n=0
or, in another form:
A.3) f(x, w) = A - 2xw + w>)~) exp { (ax + b) ? — Um-i(x)
{ m=l m
i m
where Um-i has the same meaning as in A.12.3). The polynomials Pn(x; a, b)
reduce to the Legendre polynomials in the limiting case a = b = 0.
The following identities are easy to establish:
A.4) Pn(x;a,b) = (-l)»PB(-x;a, -6),
A.5) P»(l;a, b) = Ln{-a-b), P»(-l; a, b) = (-l)«Ln(-a +b)
where Ln{x) = Z4,0)(x) is Laguerre's polynomial (Chapter V). The highest
coefficient kn of Pn(x; a, b) can be obtained by replacing w by w/x in A.3) and
taking x —> <x>. We find
The following recurrence formula holds (cf. Bateman Manuscript Project,
loc. cit.):
nPn(x; a, b) = [Bn - 1 + 2a)x + 2b]Pn-i{x; a, b)
(L? - (n - 1 )Pn_2^; a, b), n = 2, 3, 4, • • • .
393

394 APPENDIX
Here Po = 1, Pi = Ba + l)x + 26.
We have the important relation of orthogonality (cf. Szego 24):
r+i
A8) / Pn(x;a,b)Pm(x;a,b)w(x;a,b)dx=(n + %(a + l))-lSnm,
n, m = 0, 1, 2, • • • ,
where the weight function is defined by
A.9) w(cos 0; a, b) = e<"—>*<«> [cosh (tt/i^))]-1.
We note that
A.10) w(cos6;a, b) = 2 exp {(a + 6)A - t/6)\ as 0->+0.
The behavior of w is similar as 6 —> tt — 0.
The following representation in terms of the hypergeometric function holds
(Bateman Manuscript Project, loc. cit.):
A.11) Pn(cos6;a,b) = eMF(-n,\ +
2. Generalization
Let X be real, X > — \. We define the polynomials P^(x; a,b) by the gener-
generating function
B.1) ?P<,X)(z;a,fc)w« = A - 2xw + w*)~* exp / (ax + 6) ? — Um-i(x)\.
n=0 I m=l m )
For a = b = 0 we obtain the ultraspherical polynomials Pj,X)(z). The case
dealt with in 1. corresponds to X = \. The polynomials P^ix; a, b) are orthog-
orthogonal in —1 ^ x ^ 1, x = cos 6, with the weight function
B.2) w<X)(z; a, 6) = x-i2**-V"-»>*<»> (sin 0JX-1|r(X + t/i@))|2.
Concerning a recurrence formula and a representation in terms of the hyper-
hypergeometric function, see Bateman Manuscript Project, loc. cit.
3. Integral representations
The following generalizations of the Laplace integral D.8.11) and of the Mehler
integrals D.8.6) and D.8.7) hold (Novikoff 1):
Pn(cos 6; a, b) = ir-ie-™hW cosh (rh(p))
= e~6hW cosh (irhF))
2 f
C.2) • - I cos
1 exp{
• (cos
2tfc@)
0 + t
t n\
B cos
logctg^y
cos t sin 0)
sin
sin
t — 2 cos
3 1
/ —f—
2
0 —
2
0)H
J
dt

APPENDIX 395
= e<*-«*<« cosh (rh@))
. t + e
sin —=—
. t - e
sin
C.3) • - / sini {n + \)t - hF) log
T Je
{ Olil " 2
• B cos 6 - cos <)~J dt.
In these formulas 0 < 6 < tt.
4. Infinite interval
Pollaczek C) defines another remarkable class of polynomials P^(x; a) by the
following generating function:
D.1) X) P^ix; a)wn = A -
n=0
Here 0 < a < ir and X > 0. These polynomials are orthogonal in the interval
— oo < x < oo with the weight function
D.2) t-1B sin aJX-1e-c-2«)^|r(X + ix)\\
Laguerre polynomials appear as a limiting case. Indeed, replacing x by x/a
and assuming a —> 0 we obtain
I IX 1
lim A — w;eia)-x(l — w;e~ia)-x exp < — log
«-*o \a & 1 -
we~xa\
= A -
so that, cf. E.1.9),
D.3) limP^(x/a;a) = Lf)(-2x), /3 = 2X - 1.
It is also clear that the polynomials in 2. arise from P^X)(x; a) as follows:
D.4)
5. Asymptotic properties
(a) By means of the generating function A.1) it is not difficult to obtain an
asymptotic expression for Pn(x;a,b) when n —> oo. We may use Darboux's
method (§8.4).
First let x be outside of the closed interval [— 1, +1]. Writing x = \(z + z~l),
z =-¦ eiS, $6 > 0, we find
Pn(x;a, b) = {r(| +
E.1)
It is not difficult to extend this to an asymptotic expansion. Now let
— 1 < x < +1; forming the real part of the right-hand expression in E.1),

396 APPENDIX
the asymptotic formula for \Pn(x;a, b) arises. Finally, in view of A.5) we
find from (8.22.3)
E.2) P»(l;a, b) ^7r-*e-*<o+fc>(a + 6)-*n"* • exp {2(a + 6)*n*}.
(b) Novikoff A) investigated the asymptotic behavior of Pn(cos (*n-*);a, 6)
where t > 0 is fixed. His principal results are as follows:
Pn(cos (tn-*);a,b) = |x-*(a + 6 - *2)~s exp (-|(a + b))
E.3)
X(i) = aj, — arc tg ai,
ai = t-\a + b - i2)*, 0 < t < (a + 6)*;
Pn(cos (in-i); a, 6) = ^(t2 - a - 6)-' exp (-|(a + 6))
a, = r1^2 - a - 6)*, t > (a + b)K
From E.3) and E.4) interesting conclusions can be drawn about the "extreme"
zeros of Pn(x; a,b). Let us denote the zeros of these polynomials, as in Chap-
Chapter VI, by cos 0, where 0 < 0i < 02 < • • • < dn < tt; 0, = 0,n. Then, for any
fixed value of v, .
E.5) lim n*0,n = (a + 6)*.
n—> "o
6. Associated orthogonal polynomials
(a) We consider the system \<f>n(z)} of polynomials which are orthogonal
on the unit circle \z\ = 1 relative to the weight function
F.1) /@) = w(cos0)|sin0|;
here w(x) has the same meaning as in A.9). These polynomials show also in
many respects a singular behavior.
The relation to the Pollaczek polynomials can be established by means of
the formulas A1.5.2):
F.2)

APPENDIX
397
We havB pB(x) = [n + \{a + l)}*PB(x;a, b) [cf. A.8)]; the system \qn(x)} is
orthonormal with the weight function A — x2)w{x). Moreover, mn denotes
the (positive) highest coefficient of <t>in(z). Between x and z the relation
x = \{z + z~l) holds.
(b) Let \z\ < 1. We investigate the asymptotic behavior of the polynomials
<t>n(z) as n —> oo. Let 2 ^ 0, 2 = eifl, 30 > 0. We rewrite E.1) in the form
F.3)
Vn(x) =
A(x) =
Now the polynomials \qn(x)\ can be represented in terms of \pn(x)} by using
ChristoffePs formula 2.5, taking A.5) into account. Thus
Vi(x;a,6) Pn{x;a, b) Pn+l(x;a,b)
A - x2)qn~i(x) = const. Ln_! Ln
Ln = Ln(-a -b), L'n = Ln(-a
Denoting the latter determinant by An(x) we find by an easy calculation:
A — x2)qn-i(x) = (Ln_iL^ + LnL'n_l)~i(LnL'n+l + Ln+iL'n)~^
|n -
OMMx).
Using A.5), (8.22.3), and A.6) we obtain
e = (a + 6)*n~*, «' = (a — 6)*n~'.
(c) From F.3) and F.4) we conclude, since 1 - x2 = -A - 22JB2)~2,
F.5)
Denoting by k'n ?= jn + ?(a + l)}*A;n the coefficient of x" in pn(x) and by
Zn_i the coefficient of xn~l in ^n_i(x), we find from F.5) for z —> 0:

APPENDIX
Now the first formula in F.2) yields:
so that
K2n = tt*! (knJ + Zj|_i}* • 2~" = 7rM/cn • 2~"+i ( 1 —
F.6)
F.7)
r(i(a + l))n ^ 2 +U\n))-
In particular,
F.8) — =
Kin 2 \n
(d) We obtain from F.2), \z\ < 1, in view of F.3), F.5), F.8),
F.9) • (l + ^ + O^+^O'-d-^ + O)^! + o(l
F.10)
where we set for abbreviation:
, v h(a ~ 3) + bz + \(a
a{z) = ¦ ——
F.11) g(|)J(a + l) + h
1
(z) = H« i) + 6z +
(l))
= (i) ^rmirn^{ia + mi + ') -{a~ mi~z)] A + 0(n))

APPENDIX 399
The argument of the T-function can be written as follows:
ia ,<ii at \ a 1 + 22 , l z ,1
F.12) /3B)=_r_ + 6___ + _
and the real part of this expression is >\ for \z\ < 1. Hence the function 1/r
is never zero in \z\ < 1. The only point of accumulation of the zeros of the poly-
polynomials \4>m(z)\ in \z\ < 1 is the point
(q + 6)* - (q - 6)* = (a2 - 62)* - a
(q + 6)* + (q-6)* 6
appearing only for m = 2n — I.
The exponent 7B) of n has a real part > —\. Hence the series
is divergent for all \z\ < 1. In particular,
F.13)
(^j &)* db (q - 6)*)
Taking A1.3.6) into account, this yields again the main term of F.6).
(e) Recapitulating, we may point out certain properties of the Pollaczek
polynomials which indicate a rather singular behavior compared with the clas-
classical polynomials.
The weight functions w{x) or/@) in Theorems 12.1.1 and 12.1.2, respectively,
are such that log w(cos 0) and log/@) are integrable. The weight function w(x)
of the Pollaczek polynomials vanishes at the endpoints x = ±1 so strongly
that logw(cos0) is not integrable [cf. A.10)].
The normalized Jacobi polynomials are at x = ±1 of the order na+i and
n,P+i, respectively. The orthonormal Pollaczek polynomials at x = ±1 are of
order n* exp {2(a + 6)»n»} [cf. E.2)].
The Toeplitz minima ixn(f) A2.3) associated with the weight function fF) tend
to a positive limit under the assumption of Theorem 12.1.1. In case of the poly-
polynomials discussed in 6. this limit is zero; the weight function defines a "determin-
"deterministic" process. We have in this case: fj.n(f) = n~2 ~ n~". Let fF) = 0 in a
certain interval, say — e < 8 < +«, 0 < e < tt and /@) = 1 otherwise. We
have then again m*(/) ~~* 0, and, more precisely, ixn{f) ~ rn, r < 1; cf. A6.4.3)
and Problem 50.
The orthonormal polynomials defined in Theorem 12.1.2 are asymptotically
of the order (x + (x2 - 1)»)" if z is not on the cut [-1, +1]). The Pollaczek
polynomials are under the same condition of the order nK(x + (x2 — 1)*)"
where K is a function of x. A similar discrepancy arises if x is on the segment
[-1, +!]•
For the "largest" zeros cos 0,, 0 < 0, < ir, 0, = 0,(n), v fixed, n —> *>, of the
Jacobi polynomials we have 0,(n) ~ n~ljy where -j, is the corresponding zero of

400 APPENDIX
an appropriate Bessel function; see F.3.15). The similar zeros of the Pollaczek
polynomials satisfy the relation 0,(n) ^ n~^(a + 6I, see E.5); the order of
magnitude is different and the constant does not depend on p.

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73 The asterisks indicate items not dealing with orthogonal polynomials.
401

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404 LIST OF REFERENCES
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LIST OF REFERENCES 405
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406 LIST OF REFERENCES
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LIST OF REFERENCES 407
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408 LIST OF REFERENCES
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LIST OF REFERENCES 409
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410 LIST OF REFERENCES
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LIST OF RKFHRENCKS 411
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LIST OF REFERENCES 413
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414 LIST OF REFERENCES
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416 FURTHER REFERENCES
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INDEX
The numbers refer to pages
Abel, v, 100, 401.
— 's continuity theorem, 92.
— 's transformation, 2, 91,135.
— summability, 245, 273.
See Inequality.
Achieser, 5, 37, 42, 401, 416.
Adamoff, 203, 248, 249, 250, 401.
Addition theorems, 58, 97 ff.
Additive operation, 11,12.
Airy's function, 18, 19, 131, 201, 234, 239,
243, 377, 382.
See Zeros.
Antipole condition, 246, 249, 265.
Approximation, 5 ff.
Askey, xi, 98, 99, 110, 273, 348, 363, 391,
392, 416.
Askey-Fitch, 96, 363, 416.
Askey-Gasper, 97,99,110,190, 392, 417.
Askey-Pollard, 273, 417.
Askey-Steinig, 97,158, 363, 417.
Askey-Wainger, 99,190, 273, 417.
Associated function, Legendre, 84.
Asymptotic formula of Bessel functions,
15,16.
— — —classical polynomials, 191 ff.
— — — general orthogonal polynomials, v,
vi, 296 ff.
— - -Hermite polynomials, 132, 191, 194,
198 ff., 218 ff., 235, 242.
— — — Jacobi functions of second kind,
201, 225.
polynomials, 58, 167, 168, 191 ff.,
201,202,212,225.
of Hilb's type, 169, 197, 202,
214.
— — — — —of Mehler-Heine type, 167,
192 ff., 202.
See Darboux's formula.
kernel polynomials, 369 ff.
Laguerre polynomials, 132, 191, 193,
194,198 ff., 236, 237, 240 ff.
of Hilb's type, 177, 199, 203,
216 ff., 219 ff., 384.
of Mehler-Heine type, 193,194,
217.
of Plancherel-Rotach type, 200,
201, 227 ff.
SeeFejer's formula, Perron's formula.
Legendre functions of second kind,
198, 212, 222 ff.
polynomials, 191, 193, 194 ff., 243.
Asymptotic formula of Legendre polynomials
of Hilb's type, 195, 202, 212 ff.
See Laplace formula, Laplace-Heine
formula.
— — — polynomials orthogonal on a curve,
371 ff.
the unit circle, 297 ff.
ultraspherical polynomials, 196. 197,
206, 208, 209, 212.
Bailey, 273, 392, 417.
Balazs-Turan, 347, 417.
Banach, 13, 401.
Bateman, 96,98,99, 273, 417.
Bateman Project, ix, 95, 110, 243, 393, 394,
417.
Bernstein, S., vi, 9, 31, 42, 159, 165, 168 ff.,
296, 299, 300, 303, 314, 315, 330, 401.
— 's theorem on trigonometric polynomials,
5, 280, 304.
See Polynomials.
Bessel functions, v, 14 ff., 95, 96, 102 ff.,
126 ff., 140, 166, 167, 191, 193, 202, 203,
212 ff., 225, 243, 272, 353, 362, 363, 380,
384, 387, 392, 400.
See Asymptotic formula, Zeros.
- 's inequality, 25, 38, 289, 316, 322, 333, 367.
Blumenthal, 268, 310, 401.
Bochner, 108, 273, 401, 417.
Bonami-Clerc, 273, 417.
Bottema, 132, 401.
Bounded variation, functions of, 12.
Brauer, A., 131, 401.
Bruns, 122, 125,136,138, 401.
425

426
INDEX
Buell, 124 ff., 401.
Butlewski, 166, 417.
Capacity, see Transfinite diameter.
Caratheodory, 364, 402.
See Osgood-Caratheodory.
Carleman, 366, 402.
Carleson, 273, 417.
Cauchy-Hadamard formula, 248, 253, 312.
Cauchy's principle value, 278.
— theorem, 69, 106, 221, 275, 289, 324, 370,
372.
Centroid, 189.
Cesaro summability, vi, 244, 246 ff., 258.
— — (means) of Fourier series, 12, 14, 246,
250.
Hermite series, 250, 251.
Jacobi series, vi, 246, 248, 249, 256 ff.
Cesaro summability of Laguerre series, 247,
248, 250, 251, 271 ff.
— — — Legendre series, 14, 249.
power series, 325.
— — — ultraspherical series, 248.
Characteristic values, 49,187.
Charlier, see Polynomials.
Christoffel, iii, 42, 43, 47, 402.
— (-Darboux) formula, 42 ff1., 318, 326.
— formula of, 29 ff., 397.
-numbers, 46 ff., 114, 115, 187, 351 ff., 378.
Clerc, see Bonomi-Clerc.
Closure, see Orthogonal polynomials.
Coifman-G. Weiss, 273, 417,
Completely monotonic sequence, 136 ff., 155.
Confluent hypergeometric function, 104.
Conformal mapping, 21, 160, 364, 365, 372.
Conjugate function, 279.
— points (interpolation), 332, 347.
Continued fractions, v, 54 ff.
Continuous operation, 12.
Convergence in mean, 330.
— — — , generalized, 330.
— of interpolation, 330.
— , quadrature, 330, 333, 350 ff.
Convergent (continued fraction), 55 ff.
Cotes numbers, 12, 349 ff., 363.
Courant-Hilbert, see Hilbert-Courant.
Cramer, 251, 402.
Csordas-Williamson, 388, 418.
Darboux, 42, 43,195,196, 211, 402.
Darboux's formula for Jacobi polynomials,
168, 196, 225, 236, 237, 248, 253, 337, 353.
— method, 202, 206 ff., 265, 269, 395.
Davis-Hirschman, 99, 418.
Davis-Rabinowitz, 158, 418.
DeBruijn, 110, 418.
de la Vallee-Poisson summability, 273.
Deviation, quadratic, 38, 41, 288.
-, Tchebichef, 41,368.
Differences, 34, 35,136.
Differential equation, v, 16 ff., 37, 152, 159,
164,166, 210.
- — of Bessel functions, 15.
Hermite polynomials, 106, 176, 380.
- - -Jacobi(hypergeometric) polynomials,
iv, 60 ff., 117,141.
Laguerre polynomials, 100, 117, 176.
—ultraspherical polynomials, 80, 81.
Dirichlet, 85, 402.
Dirichlet-Mehler, see Integral representation.
Dirichlet's integral, 12,14.
Discriminant of classical polynomials, 143 ff.
Distribution, 8, 9, 26, 29, 38, 40, 57, 181, 274,
330,351,378.
- of Stieltjes type, 9, 33 ff., 38.
Doetsch, 34, 35, 380, 402, 418.
Du Bois, Reymond, 14, 402.
Eagleson, 37, 418.
Egervary-Turan, 348, 418.
Ellipse, 8, 21, 251 ff.
-of convergence, 245, 248, 252, 311, 312.
Elliptic functions, v, 60.
Equiconvergence, 31, 244, 246, 247, 249,
313 ff.
Erdelyi, ix, 104, 243, 273, 380, 402, 418.
Erdelyi-Swanson, 243, 418.
Erdos, vii, 347, 348, 418.
Erdos-Feldheim, 334, 402.
Erdos-Grunwald, 347, 418.
Erdos-Lengyel, 347, 418.
Erdos-Turan, 113, 114, 115, 332, 333, 335,
347, 348, 402, 418.
Euler, 72, 402.
— 's constant, 15.
— 's integral of the first kind, 14.
second kind, see Gamma function.
Expansion, v, vi, 58.
—, finite cosine, of Legendre polynomials,
90 ff.
—, infinite sine, of Legendre polynomials,
90 ff.
— in power series, 313, 314.
— in series of classical polynomials, 244 ff.,
251 ff.

INDEX
427
— — — — general orthogonal polynomials,
313 ff.
— — — - Hermite polynomials, 244, 245,
247, 251,253, 269 ff.
— — — — Jacobi polynomials, 244 ff.
— — — — Laguerre polynomials, 244 ff.,
249 ff., 253,266 ff., 313.
Legendre polynomials, 248, 249.
— — — — Tchebichef polynomials, 248, 314.
ultraspherical polynomials, 248,
249.
See Fourier series, Orthogonal poly-
polynomials.
Faber, 299, 330,368, 369, 372, 402.
See Polynomials.
Fatou, 275, 403.
— 's theorem, 274.
Favard, 43, 403.
Fejer, vii, 14, 45, 58, 89, 91, 96, 134, 136 ff.,
157, 165, 172, 174, 175, 179, 198, 202, 249,
264, 330 ff., 336, 348, 350, 386, 403.
SeeStekloff-Fejer.
—'s asymptotic formula for Laguerre poly-
polynomials, 198, 202, 203, 237, 240, 269.
Fejer's generalization of Legendre poly-
polynomials, 135 ff., 174 ff., 206.
— 's second generalization of Legendre poly-
polynomials, 137,138,155.
— 's integral, 12.
— 's representation of positive trigonometric
polynomials, 3, 4, 274.
— — — — — —, generalization of, 275 ff.
Fejer-Szego, 175, 404.
Fekete, 369, 404.
Feldheim, 95, 96, 97, 334, 335, 386, 404, 418.
See Erdos-Feldheim.
Fitch, see Askey-Fitch.
Fourier, v.
— coefficient (constant), 24, 287.
— 's inversion formula, 387.
— series, 12, 14, 24, 25, 28, 38, 39, 244, 246,
253, 274, 289, 311, 314, 323, 347, 367.
See Ces&ro means.
Freud, 309, 418.
Friedrichs, 404.
Fujiwara, 145, 404.
Functions of second kind, Jacobi's, 73 ff., 251.
, Legendre's, 74, 78, 88, 89, 92,
379, 383.
See Asymptotic formula, Integral
representation, Zeros.
fundamental polynomials of Hermite inter-
interpolation, 330 ff.
Lagrange interpolation, 12, 47, 329 ff.
Galbrun, 251,404.
Gamma-function, 14, 75.
Gasper, 37,98, 99,190, 273, 419.
See Askey -Gasper.
Gatteschi, 242, 419.
Gauss, iii, 33, 47, 49, 62, 404.
See Mean-value theorem, Mechanical
quadrature.
Gauss-Weierstrass summability, 273.
Gegenbauer, 80,94, 98,99, 404.
"Gegenbeispiel", 250, 272.
Generating function, 35, 36.
of Hermite polynomials, 106,380.
Jacobi polynomials, 69 ff., 82, 83, 95.
Laguerre polynomials, 101, 202, 379.
Legendre polynomials, 90, 206, 207,
383.
Pollaczek polynomials, 393.
ultraspherical polynomials, 82, 83.
Geometric mean, 275, 296, 385.
Geronimus, 42,189, 404.
Gibbs' phenomenon, 251.
Ginibre, 392, 419.
Gottlieb, 37, 405.
Grenander-Szego, 274, 287, 419.
Gronwall, 165, 249, 405.
Griinbaum, 392, 419.
Griinwald, vii, 330, 347, 405, 419.
See Erdos-Griinwald.
Griinwald-Turan, 346, 405.
Haar, 14, 248, 405.
Hahn, W., 33, 107, 111, 112, 129, 132, 151,
405, 419.
Hamburger, 57, 110, 405.
Hankel, see Quadratic form.
Hardy, 102, 380, 405.
Hardy-Littlewood-Polya, 2, 405.
Hartman-Wintner, 20, 419.
Hausdorff, 136,155, 405.
Heine, iii, 27, 37, 47, 54, 80, 90 ff., 151 ff.,
192,194, 251, 405.
Heine-Stieltjes theorem, iv, 151 ff.
Helly, 13, 405.
-'s theorem, 13, 14, 258, 272, 330, 339, 350.
Hermite, v, 156, 405, 406.
- interpolation, 330 ff., 340 ff., 347, 348.
Hermite-Stieltjes, 89,156,157,172, 406.

428
INDEX
Hermitian form, 287.
Hilb, 195,249, 406.
Hilbert, 142,145, 406.
Hilbert-Courant, 58, 59, 100, 106, 109, 406.
Hildebrandt, 8,10, 406.
Hille, 102,105,126,131, 242, 251, 406.
See Shohat-Hille-Walsh.
Hirschman, 99, 273, 419.
See Davis-Hirschman.
Hobson, iv, 58, 84, 92, 382, 406.
Hollo, 335, 406.
Horton, 99, 273, 419.
l'Hospital, rule of, 307.
Howell, 392.
Hsu, 390,419.
Hua, 99, 419.
Hunt, 273, 420.
Hurwitz, theorem of, 21, 150, 193, 239, 371.
Hypergeometric function, iii, 62 ff., 83, 84,
96.
See Differential equation.
Inequalities, 159 ff.
Inequality, Abel's, 2,174, 205.
-, Cauchy's, 2, 39, 120, 160, 183, 291, 302,
306,321,370.
— for the arithmetic and geometric mean, 2,
300.
-, Schwarz's, 2, 9, 110, 136, 162, 268, 276,
317, 376.
-, Turan's, 190, 388.
See Bessel's inequality.
Integral, Lebesgue, vii, 9, 26, 159, 246 ff.,
274, 275, 287, 291, 364, 384.
-, Riemann, 9, 10, 280, 281, 282, 298, 310,
333, 335, 351, 361.
-, Riemann-Stieltjes, 8, 9, 11, 50, 333, 350,
351.
-, Stieltjes-Lebesgue, v, 1, 8,10, 38,187.
— equations, v, 218, 251.
— representation of Legendre functions of
second kind, 88 ff.
polynomials, 85 ff.
, Dirichlet-Mehler, 85 ff., 96.
.Laplace (first), 86,176.
, Laplace (second), 86 ff.
, Stieltjes, 87 fl".
— — — ultraspherical polynomials, Dirichlet-
Mehler, 89.
, Stieltjes, 89 fl'.
Interpolation, v, ix, 12, 14, 58, 329 ff., 347.
-, Lagrange, 12, 47, 180, 329 ff., 348, 349,
389.
— on Hermite abscissas, 340.
— — Jacobi abscissas, 335 ff.
Laguerre abscissas, 340, 344 ff.
Legendre abscissas, 386.
Tchebichef abscissas, 330, 334, 335, 386.
ultraspherical abscissas, 386.
See Conjugate points, Fundamental
polynomials, Hermite interpolation,
Lagrange polynomials.
Jackson, vii, 6, 7, 42, 331, 406.
Jacob, 251, 406.
Jacobi, iii, 49, 58, 69, 86, 406, 407.
See Functions of second kind, Me-
Mechanical quadrature, Polynomials,
Series.
Jensen's theorem, 301.
Joo, 391.
Jordan, C, 58, 407.
- arc, 8, 364.
- curve, 8, 21,364,365.
-, Ch., 34, 407.
Jordan-Pochhammer integral, 75.
Julia, 369, 407.
Kaczmarz-Steinhaus, 1,10, 407.
Kanter, vii.
Karlin-McGregor, 37, 273, 389, 420.
Karlin-Studden, 5, 420.
Karlin-Szego, 190, 388, 420.
Keldysch-Lawrentieff, 368, 407.
Kernel polynomials, 39, 40, 44, 71, 101, 180,
183, 249, 290, 322, 323, 333, 351, 368, 377.
See Asymptotic formula, Zeros.
Klein, 145, 407.
Kogbetliantz, 102, 168, 172, 203, 240, 248,
249, 251, 256, 273, 380, 407.
Koornwinder, 98,99, 420.
Korous, 131,132,162,167, 212, 251, 303, 408.
Koschmieder, 60, 408.
Kowalewski, 24, 408.
Kowallik, 251, 408.
Krall, 107, 408.
Krawtchouk, 113, 408.
See Polynomials.
Kronecker, 408.
Lagrange, 100, 408.
- polynomials, 329 ff., 347.
— series, 70.
See Interpolation.
Laguerre, v. 100,117,131, 408,409.
See Polynomials.

INDEX
429
Lame function, 151.
Langer, 204, 210, 409.
Laplace, iv, 86.
Laplace's formula for Legendre jpolynomials,
194,198, 201, 202, 204 ff., 211 ff., 224.
— , Darboux's generalization, 195,
201, 206 ff., 211.
, Stieltjes' generalization, 195,
202,209 ff.
Laplace-Heine formula, 194, 204 ff., 208, 243.
, generalization, 194.
Laplace series, 96, 249.
- transform, 379.
See Integral representation.
Laurent expansion, 252.
Lawton, 151, 409.
Lebesgue, 14,15, 409.
See Integral.
- constant, 13, 258, 330, 336, 338, 339, 350.
Legendre, v, 70,409.
See Associated function, Polynomials.
Lengyel, 113.
See Erdos-Lengyel.
LeRoy, 103, 409.
Level curve, 8.
Leibniz, rule of, 67, 68,101.
Limited operation, 12.
Linear operation, see Operation.
Liouville-Stekloff, method of, 202 ff., 210 ff.,
299.
Lipschitz, 383.
- condition, 6,162,163,186.
Lipschitz-Dini condition, 279, 297, 324.
Littlewood, see Hardy-Littlewood-Polya.
Locher, 363, 420.
Lorch, xi, 190, 249, 420.
Lorch-Muldoon-P. Szego, 158, 420.
Lorch-P. Szego, 158, 420.
Lukacs, 4,178 ff., 249, 409.
Magnus-Oberhettinger, 420.
Makai, 20,148,157,190, 420, 421.
Makai-Turan, 158, 421.
Marcinkiewicz, 273, 330, 347, 409.
Markoff, A., v, 33, 37, 50, 57, 115, 116, 121,
122,139, 259,378, 409.
McGregor, see Karlin-McGregor.
Mean approximation, 10,11.
Mean-Value theorem, 383.
of Gauss, 275, 310,311.
, second, 2, 202,357, 363.
Mechanical quadrature, v, ix, 12, 14, 58, 187,
329,348 ff.
for classical abscissas, 352 ff.
Jacobi abscissas, 355 ff., 378, 379.
, Gauss-Jacobi, 47 ff., Ill, 348 ff.
Mehler, iii, 47, 49, 85,192, 380, 394, 409, 410.
Mehler-Heine, see Asymptotic formula.
Meixner, 34, 35, 410.
Method of Liouville-Stekloff, see Liouville-
Stekloff.
steepest descent, 202, 203, 221 ff.
Modulus of continuity, 6, 7, 335, 336, 340,
346.
Moecklin, 194, 203, 410.
Moment problem of Stieltjes, iii, iv, v, 40.
Muckenhoupt, 190, 243, 273, 421.
Muckenhoupt-Stein, 273, 421.
Muldoon, see Lorch-Muldoon-P. Szego.
Mtintz, 251, 410.
Myller-Lebedeff, 251,-410.
Neumann, E. R., 130, 251, 380, 410.
Neumann, F., 248.
Neumann, J. von, 108.
Newman-Rudin, 273, 421.
Newton's formula, 259.
Normalization, 28, 58,160.
Norm function, see Weight function.
Norm of operation, 12.
Novikoff,394, 396, 421.
Obrechkoff, 198, 248, 249, 410.
Olver, 243, 421.
Operation, linear functional, 11 ff.
Orthogonal polynomials and continued
fractions, 54 ff.
, asymptotic formula of, v, ix, 4, 296 ff.
, Christoffel-Darboux formula for, 42 ff.
, classical, 29, 49.
, closure, 40,108 ff.
, definition, ix, 23 ff.
, expansion in series of, v, ix, 28, 38, 39,
41, 311,312, 313 ff.
— —, extremum properties of, 28, 38 ff.
, general properties of, 38 ff.
— — , highest coefficient of, 28.
, recurrence formula for, 42 ff., 55, 391.
, representation of, 27.
, zeros of, v, ix, 44 ff., 121 ff., 188, 189.
Orthogonality, Orthogonalization, v, vi, 8,
23 ff., 44.
Orthonormal set, 23, 25, 68.
Osgood-Caratheodory's theorem, 364.
Parabola of convergence, 253.

430
INDEX
Parseval's formula, 40, 289,368.
Peano, 1.
Peetre, 110, 421.
Pencil, 49.
Perron, 45, 54,144, 202, 410.
— 's formula for Laguerre polynomials, 198,
199, 202, 203, 220, 221, 225 ff.
P-function, contiguous Riemann, 71.
Planch erel, 250.
Plancherel-Rotach, 132, 201, 203, 410.
Pochhammer-Barnes, notation of, 103.
Poincare, 310.
Poisson's integral, 276, 292.
See Polynomials.
Pollaczek, ix, 37, 393, 421.
Pollard, 273, 421.
See Askey-Pollard.
Polya, vii, 42, 53, 117, 134, 154, 166, 350,
388,410,411.
See Hardy-Littlewood-Polya.
Polya-Szego, vii, 5, 11, 21, 24, 34, 40, 70, 87,
105, 117, 134, 135, 175, 178, 179, 184, 212,
310,311,411.
Polynomials associated with a curve
(Tehebichef polynomials), 368,385.
— , classical, v, 29,159.
See Asymptotic formula, Expansion,
Mechanical quadrature.
-,Faber,372,374.
—, Fejer's generalization of Legendre, see
Fejer.
— , Gegenbauer, see Polynomials, ultra-
spherical.
-, Hermite, vi. ix, 29, 36, 37, 105 ff., 110,
111, 176 ff., 190, 331, 388 ff., 392.
See Asymptotic formula, Differential
equation, Expansion, Generating
function, Interpolation, Recurrence
formula, Rodriques' formula, Zeros.
— , Jacobi, vi, ix, 3, 29, 58 ff., 94 ff., 103, 105,
107, 161, 167 ff., 172 ff, 179 ff., 243, 249,
295,348,383, 385, 399.
See Asymptotic formula, Differential
equation, Expansion, Generating
function, Interpolation, Mechanical
quadrature, Recurrence formula,
Rodrigues' formula, Zeros.
-, Laguerre, vi, ix, 29, 35, 100 ff., 110, 111,
164, 176 ff., 184, 185, 190, 243, 379, 380,
382,387 ff., 391,393,395.
See Asymptotic formula, Differential
equation, Expansion, Generating
function, Interpolation, Recurrence
formula, Rodrigues' formula.
-, Legendre, vi, 29,30, 33, 34, 48, 58, 63, 70,
85 ff., 95, 96, 136 ff., 162,164 ff., 167, 172,
189,346, 348, 379, 382 ff., 392,393.
See Asymptotic formula, Expansion,
Generating function, Integral repre-
representation, Interpolation, Zeros.
-, Krawtchouk, 35 ff.
—, of S. Bernstein and Szego, 31 ff.
-, Poisson-Charlier, 34, 35,37, 377, 389, 390.
— , Pollaczek, 37, 393 ff.
— orthogonal on a curve, vii, 364 ff.
See Asymptotic formula, Zeros.
segment, see Orthogonal poly-
polynomials.
Polynomials orthogonal on the unit circle, ix,
287 ff., 384.
-, Stieltjes-Wigert, 33.
—, Tchebichef (of the first and second kind),
3, 26, 29, 30, 60, 63, 112, 136 ff., 162, 347,
348,387,391.
See Expansion, Interpolation, Zeros.
-, ultraspherical, vi, 29, 58 ff., 80 ff., 93 ff.,
107,135 ff., 167, 170 ff., 379, 387, 390, 394.
See Asymptotic formula, Differential
equation, Expansion, Generating
function, Integral representation,
Interpolation, Recurrence formula,
Rodrigues' formula, Zeros.
Popoviciu, 46,140,142, 411.
Principle of argument, 21,157.
Probability, calculus of, 35.
Quadratic form, 24,123,187,366.
of Hankel (recurrent) type, 27, 309.
Quantum mechanics, iii.
Rabinowitz, see Davis-Rabinowitz.
Rau, 197, 214, 249, 411.
Recurrence formula, general, 42 ff.
— —of Hermite polynomials, 106.
Jacobi functions of second kind, 78 ff.,
379.
polynomials, 71 ff.
Laguerre polynomials, 101.
polynomials orthogonal on the unit
circle, 293.
ultraspherical polynomials, 81, 82.
Riemann, see Integral, P-function.
-'s lemma, 254, 267,319.
— 's theory of trigonometric series, 248.
Riesz, F., 12, 275, 411.
Riesz, M., 5, 57,304, 411.
Robin's constant, see Transfinite diameter.

432
INDEX
Turan, vii, ix, 158,190, 388, 389, 422.
See Balazs-Turan, Egervary-Turan,
Erdos-Turan, Grunwald-Turan,
Makai-Turan, Suranyi-Turan,
Szego-Turan.
Uniqueness theorems, 248
Uspensky, 53, 107, 203, 219, 220, 251, 415.
Van Veen, 132, 204, 415.
Vector, 9.
— space, 10.
Vietoris, 363, 422.
Vitali-Sansone, 415, 423.
Vitali's theorem, 57.
Volterra equation, 211.
Wainger, see Askey-Wainger.
Walsh, 7, 366, 415.
SeeSholat-Hille-Walsh.
Wangerin, 84, 415.
Watson, 18, 20, 96, 102, 104, 107, 159, 162,
192, 193, 202, 203, 222, 251, 253, 363, 380,
384, 415.
See Whittaker-Watson.
Webster, 347, 423.
Weierstrass, theorem of, 5 ff., 10,110.
Weight function, 9, 37, 68, 159, 160, 162 ff.,
185, 188, 287, 294, 296, 297, 299, 309, 312,
314, 316, 347,380, 385.
Weiss, G., see Coifman-G. Weiss.
Weyl, 110, 251,310, 415.
Whittaker-Watson, 14 ff., 58, 63, 65, 66, 71,
75, 84,86, 88, 93,103, 248, 415.
Widom-Wilf, 243, 423.
Wigert, 33,102, 251, 413.
See Polynomials.
Wilf, see Widom-Wilf.
Williamson, see Csordas-Williamson.
Wiman, 131,416.
Winston, 131,351,416.
Wintner, see Hartman-Wintner.
Wright, 203, 384, 416.
Young, W. H., 10, 248, 414.
Zernike, 132, 416.
Zero-function, 9, 40, 45.
Zeros, distribution of, v, 310.
— , electrostatical interpretation of, 140, 382.
— of Airy's function, 18,19,377, 382.
— — analytic functions, 21.
Bessel functions, 126 ff., 140, 192, 193,
381.
Hermite polynomials, 117 ff., 123,
127 ff., 141 ff., 240, 353.
— — Jacobi polynomials, vi, 116 ff., 140 ff.,
144 ff., 192, 193, 237 ff., 250 ff., 379, 381.
kernel polynomials, 369, 377.
Laguerre polynomials, 116 ff., 122 ff.,
127 ff., 141 ff., 150 ff., 237 ff., 353, 381, 382.
— — Legendre functions of second kind,
155 ff.
polynomials, 111, 122,125, 353.
numerators of continued fractions, 57.
— — polynomials orthogonal on a curve, 369.
the unit circle, 292, 384.
— — solutions of differential equations, see
Sturm's theorem.
Tchebichef polynomials, 330,351.
— — ultraspherical polynomials, 119, 121 ff.,
138,352,381.
See Orthogonal polynomials.
Zygmund, 248, 253, 254, 274, 276, 279, 281,
359, 416, 423.

INDEX
431
Rodrigues' formula for Hermite polynomials,
106.
Jacobi polynomials, 67 ff., 73, 74, 94,
99,117.
— Laguerre polynomials, 101, 117, 388.
— — — ultraspherical polynomials, 81.
Roever, vii.
Rogosinski, 347.
Rolle's theorem, 51,53,117, 378.
Roosenrad, 110,421.
Ross, vii.
Rotach, 203, 249, 250, 411.
See Plancherel-Rotach.
Rouche's theorem, 21,149.
Rudin, see Newman-Rudin.
Runge, 7.
Runge-Walsh, theorem of, 7.
Saddle-point, 219, 231.
Sapiro, 98, 421.
Sarmonov, 392, 421.
Scalar product, 8, 9,10, 25,365,367, 378.
Schmeiser, 158, 422.
Schmidt, E., 34, 35, 411.
Schoenberg, 273.
Schoenberg-Szego, 190, 422.
Schur, I., vi, 140,142, 411.
Schwid, 204, 416.
Seidel-Szasz, 94,422.
Sen-Rangachariar, 145, 411.
Series, Jacobi, vi, 273.
—, Laguerre, 240.
—, Legendre, vi, 42.
See Expansion, Orthogonal poly-
polynomials.
Sherman, 57, 411.
Shibata, 145, 412.
Shohat, v, vii, 42, 48, 159, 163, 189, 309, 333,
336, 340, 345,347, 350,385, 386, 412.
Shohat-Hille-Walsh, 422.
Singular integral, 13.
Smirnoff, 275, 276, 289, 368, 369, 412.
Smith, 412.
Sonin, 96, 100, 102, 104, 159, 166, 169, 176,
189,379,412.
Spencer, 132, 412.
Statistics, mathematical, v.
Stein, see Muckenhoupt-Stein.
Steinig, see Askey-Steinig.
Stekloff, 9, 210,350, 412.
See Liouville-Stekloff.
Stekloff-Fejer, theorem of, 350 ff., 360.
Step function, 34, 35, 37,164.
- polynomial, 329, 339 ff.
, generalized, 331.
Stieltjes, v, 33, 37, 46, 50, 51, 53, 54, 58, 87,
89, 90, 92, 93, 121 ff., 136, 139, 140, 142,
145, 151 ff., 156, 165, 172, 174, 175, 193,
195, 211, 382, 412, 413.
See Heine-Stieltjes, Hermite-Stieltjes,
Integral, Laplace, Moments, Poly-
Polynomials.
Stirling's formula, 227.
— series, 212.
Stone, 10, 23, 251, 413.
Strip of convergence, 253.
Studden, see Karlin-Studden.
Sturm-Liouville type, 210.
Sturm's theorem on differential equations,
19 ff.
(method) — zeros, vi, 45, 111, 121,
124 ff., 139.
Summability, see Abel, Cesaro.
Suranyi-Turan, 347, 422.
Szasz, 5,190, 413, 422.
See Seidel-Szasz.
Szego, G., 19, 27, 31, 33,37, 45,59, 87, 93, 94,
96,110, 125, 127, 128, 135, 137, 140, 158,
163, 164, 167 ff., 189, 190, 197, 198, 203,
206, 214, 243, 248, 249, 251, 274 ff., 287,
289, 294 ff., 299, 300, 309, 310, 312, 313,
340, 346, 355, 360, 361, 366, 369, 371, 377,
384, 388,392, 393,394, 413, 414, 422.
See Fejer-Szego, Grenander-Szego,
Karlin-Szego, Polya-Szego, Poly-
Polynomials, Schoenberg-Szego.
Szego, P., xi.
See Lorch-P. Szego, Lorch-Muldoon-
P. Szego.
Szego-Turan, 157,158, 422.
Tamarkin, vii, ix, 42, 414.
Tchakaloff, 189, 383, 414.
Tchebichef, v, 33, 47, 50, 54, 70, 100, 186,
188,189, 414.
See Deviation, Polynomials.
Thome, 243, 422.
Titchmarsh, 37, 57, 301, 310, 311, 415.
Toeplitz matrix, 287, 399.
—, operator, 99.
Total variation, 12, 34, 35.
Transfinite diameter, 364, 369.
Tricomi, ix, 242, 243, 415, 422.
Trigonometric polynomials, 3, 5 ff., 11.
— representation, 90 ff.