Bruce C. Berndt
Ramanujan's Notebooks
Part V
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Bruce C. Bcrndi
Department of Mathematics
University of Illinois at Urbana-Champaiun
Urbana, i'l 61801-2975
USA
Mathematics Subject Classification A991): 11-00, 11-03, 01Л60, OIA75, 33E05,
33-00, 33-03, 41-00, 41-03, 41Л58, 41Л60
Library of Congress Caialoging-in-Publication Data
(Revised for vol. 4)
Ramanujan, Alyangar, Srinivasa, 1887-1920.
Ramanujan's notebooks.
Includes bibliocraphies and indexes.
I. Mathematics. 1. Berndt, Bruce С 1939-
II. Title.
QA3.R33 1985 510 84-20201
ISBN 0-387-961 10-0 (\. I)
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Dedicated to the mathematicians who
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To the author's knowledge, only three photographs of Ramanujan are extant. Vari-
Variations of his passport photo appear in our books, Parts I and IV. A group photo
with Ramanujan appearing in cap and gown can be found as the frontispiece of the
publication of Ramanujan's lost notebook [11], and has been excised in several
cropped versions, often with Ramanujan standing alone. The photograph above is
also one of several renditions, the most frequent being one with Ramanujan sitting
alone.

Preface
During the years 1903-1914, Ramanujan recorded most of his mathematical dis-
discoveries without proofs in notebooks. Although many of his results had already
been published by others, most had not. Almost a decade after Ramanujan's death
in 1920, G. N. Watson and В. М. Wilson began to edit Ramanujan's notebooks,
but, despite devoting over ten years to this project, they never completed their task.
An unedited photostat edition of the notebooks was published by the Tata Institute
of Fundamental Research in Bombay in 1957.
This book is the fifth and final volume devoted to the editing of Ramanujan's
notebooks. Parts I—III, published, respectively, in 1985, 1989, and 1991, contain
accounts of Chapters 1-21 in the second notebook, a revised enlarged edition of
the first. Part IV, published in 1994, contains results from the 100 unorganized
pages in the second notebook and the 33 unorganized pages comprising the third
notebook. Also examined in Part IV are the 16 organized chapters in the first
notebook, which contain very little that is not found in the second notebook. In
this fifth volume, we examine the remaining contents from the 133 unorganized
pages in the second and third notebooks, and the claims in the 198 unorganized
pages of the first notebook that cannot be found in the succeeding notebooks. In
contrast to the organized portion of the first notebook, the unorganized material
in the first notebook contains several results, particularly about class invariants,
singular moduli, and values of theta-functions, which are not recorded in the
second and third notebooks.
As in the first four volumes, either proofs are provided for claims not previously
established in the literature, or citations are given for results already proved in the
literature.
Urbana, Illinois Bruce C. Berndt
September, 1997

Contents
Preface ix
Introduction 1
32 Continued Fractions 9
1 The Rogers-Ramanujan Continued Fraction 12
2 Other ^-Continued Fractions 45
3 Continued Fractions Arising from Products of Gamma
Functions 50
4 Other Continued Fractions 66
5 General Theorems 80
33 Ramanujan's Theories of Elliptic Functions to Alternative
Bases 89
1 Introduction 89
2 Ramanujan's Cubic Transformation, the Borweins' Cubic
Theta-Function Identity, and the Inversion Formula 93
3 The Principles of Triplication and Trimidiation 101
4 The Eisenstein Series L, M, and N 105
5 A Hypergeometric Transformation and Associated Transfer
Principle 108
6 More Higher Order Transformations for Hypergeometric
Series 116
7 Modular Equations in the Theory of Signature 3 120
8 The Inversion of an Analogue of К (к) in Signature 3 133
9 The Theory for Signature 4 145
10 Modular Equations in the Theory of Signature 4 153
11 The Theory for Signature 6 161
12 An Identity from the First Notebook and Further
Hypergeometric Transformations 165

xii Contents
13 Some Enigmatic Formulas Near the End of the Third
Notebook
14 Concluding Remarks
34 Class Invariants and Singular Moduli
1 Introduction
2 Table of Class Invariants
3 Computation of Gn and gn when 9|n
4 Kronecker's Limit Formula and General Formulas for Class
Invariants
5 Class Invariants Via Kronecker's Limit Formula
6 Class Invariants Via Modular Equations
7 Class Invariants Via Class Field Theory
8 Miscellaneous Results
9 Singular Moduli
10 A Certain Rational Function of Singular Moduli
11 The Modular y-invariant
35 Values of Theta-Punctions
0 Introduction
1 Elementary Values
2 Nonelementary Values of (p(e~"")
3 A Remarkable Product of Theta-Functions
36 Modular Equations and Theta-Function Identities in
Notebook 1
1 Modular Equations of Degree 3 and Related Theta-Function
Identities
2 Modular Equations of Degree 5 and Related Theta-Function
Identities
3 Other Modular Equations and Related Theta-Function
Identities
4 Identities Involving Lambert Series
5 Identities Involving Eisenstein Series
6 Modular Equations in the Form of Schlafli
7 Modular Equations in the Form of Russell
8 Modular Equations in the Form of Weber
9 Series Transformations Associated with Theta-Functions
10 Miscellaneous Results
37 Infinite Series
38 Approximations and Asymptotic Expansions
175
180
183
183
187
204
216
225
243
257
269
277
306
309
323
323
325
327
337
353
354
363
367
373
376
378
385
391
397
403
409
503
39 Miscellaneous Results in the First Notebook
Contents xiii
565
Location of Entries in the Unorganized Portions of Ramanujan's
First Notebook 579
References 605
Index 619

Introduction
Knowledge comes, but wisdom lingers.
Alfred, Lord Tennyson, Locksley Hall
This book constitutes the fifth and final volume of our attempts to establish all
the results claimed by the great Indian mathematician Srinivasa Ramanujan in his
Notebooks, first published in a photostat edition by the Tata Institute of Fundamen-
Fundamental Research in 1957 [9]. Although each of the five volumes contains many deep
results, perhaps the average depth in this volume is greater than in the first four. As
will be seen in the following paragraphs, several mathematicians made important
contributions to the completion of this volume. However, 1 particularly extend
my deepest gratitude to Heng Huat Chan and Liang-Cheng Zhang without whose
contributions this volume would have been woefully deficient. This volume, how-
however, should not be regarded as the closing chapter on Ramanujan's notebooks.
Instead, it is just the first milestone on our journey to understanding Ramanujan's
ideas. Many of the proofs given here and in other volumes certainly do not reflect
Ramanujan's motivation, insights, proofs, and wisdom. It is our fervent wish that
these volumes will serve as springboards for further investigations by mathemati-
mathematicians intrigued by Ramanujan's remarkable ideas. As in the other four volumes,
for each correct claim, we either provide a proof or cite references in the literature
where proofs can be found. We emphasize that Ramanujan made extremely few
errors, and that most "mistakes" are either minor misprints, or, in fact, they are
errors made by the author arising from misinterpretations of Ramanujan's claims,
which are occasionally fuzzy.
The second notebook is a revised, enlarged edition of the first, and, as with G. N.
Watson and В. М. Wilson, who made the first attempts at editing Ramanujan's
notebooks, the second was our initial focus. It was therefore quite surprising for us
to discover that the unorganized pages of the first notebook contain many beautiful
results, especially in the areas of class invariants, singular moduli, and explicit
values of theta-functions, that Ramanujan failed to record in his second notebook.
The material examined in this volume arises from the unorganized pages in all
three notebooks, and we provide now brief descriptions of the contents of each of
the eight chapters.

2 Introduction
Ramanujan loved continued fractions, and many of his most beautiful results
involve continued fractions. Chapter 32 contains about 70 results on continued
fractions scattered among the unorganized pages in his second and third note-
notebooks, and four evaluations of the Rogers-Ramanujan continued fraction from his
first notebook. Several modular equations for the Rogers-Ramanujan continued
fraction R(q) can be found; in less technical language these are functional equa-
equations relating R{q) at two different arguments. Other ^-continued fractions were
also examined by Ramanujan in these unorganized pages. Several results arise
from Ramanujan's beautiful continued fractions for quotients of gamma functions
found in Chapter 12 of his second notebook. The present chapter primarily con-
constitutes a reorganized and partially rewritten version of the memoir published by
G. E. Andrews, the author, L. Jacobsen, and R. L. Lamphere [2]; a preview and
discussion of some of the results was published by the four of us in [ 1 ]. The four
evaluations of the Rogers-Ramanujan continued fraction from the first notebook
first appeared in a paper with H. H. Chan [1], and in Chan's doctoral thesis [2].
In the classical theory of elliptic functions, the ordinary hypergeometric func-
function 2Fd{, 5; 1; jc) plays a very important role. In his famous paper [3], in the
course of stating without proofs some remarkable series representations for \/ж,
Ramanujan remarked that several of his series arose from alternative theories of
elliptic functions wherein the aforementioned hypergeometric function is replaced
by either 2Fi E, |; 1;л), гР\(\, \; 1; *), or2fi(g, \\ 1; x). These theories were
never developed, but the first six pages in the unorganized section at the end of his
second notebook are devoted to these theories. This is the content of Chapter 33,
most of which was first published in a paper with S. Bhargava and F. G. Garvan
[ 1 ]. A few results from the first notebook have been added to this presentation. The
first of the three alternative theories is the most interesting and the most important,
and we feel that a large body of work remains to be discovered here.
Like Chapter 33, it took several years for us to satisfactorily examine all of the
material in Chapter 34, which is devoted to class invariants and singular moduli.
Most of this work has appeared in papers with Chan and L.-C. Zhang [ 1 ]-[3], [5]
and with Chan [3], [4]. A summary, written with Chan and Zhang appears in F],
and several results were established in Chan's Ph.D. thesis 12]. Ramanujan did a
prodigious amount of work in calculating over 100 class invariants. For reasons
that are unclear to us, he failed to record many of these values in his second
notebook. To establish most of Ramanujan's hitherto unproved class invariants,
we had to develop methods that were completely unknown to Ramanujan. Thus,
Ramanujan's methods and insights into class invariants remain largely a mystery
to us. It is also puzzling to us that, except for four values, Ramanujan did not record
in his second notebook the more than 30 representations for singular moduli found
in his first notebook. For even n, Ramanujan left us a beautiful formula to aid in
the calculation of singular moduli, although we are uncertain how he found it, but
for odd n, Ramanujan's methods are unknown to us.
The values of the classical theta-function Z)*l-ooe 2"*^ are beautiful al-
algebraic numbers when и is a positive rational number. Chapter 35 is devoted
Introduction 3
to Ramanujan's explicit values of theta-functions found in his first notebook.
Most of this chapter previously appeared in papers with Chan [2] and Chan and
Zhang [4].
Chapters 19-21 in Ramanujan's second notebook contain several hundred mod-
modular equations. Surprisingly, some of his deepest results on modular equations in
the forms of Russell, Schlafli, or Weber appear only in the first notebook. In Chap-
Chapter 36 we establish all the results on modular equations found in the first notebook
but not in the second. Some are easy variations of results in the second notebook
that we proved in our third volume [3].
We return in Chapter 37 to the second notebook. All the results in this chapter
pertain to infinite series. Many were very difficult for us to prove, and we owe
our thanks to R. A. Askey, G. Bachman, P. Bialek, D. Bradley, R. J. Evans, J. L.
Hafner, and A. Hildebrand for important contributions. Although it is impossible
to summarize in one brief paragraph the contents of this long and varied chapter,
we mention just a few highlights. The first several sections of the chapter are de-
devoted to interesting variants of the Abel-Plana summation formula and examples
thereof. In Section 21, we examine Ramanujan's surprising transformation for-
formulas for two certain doubly exponential cousins of two classical theta-functions.
In Section 22, an intriguing formula for the logarithmic derivative of the gamma
function is derived. In Section 42, Ramanujan offers some very remarkable the-
theorems on the explicit behavior of partial sums of certain divergent alternating
series.
In previous volumes we marveled about Ramanujan's insights into asymptotic
analysis; see, in particular, Chapter 13 in our second book [2]. In Chapter 38, we
gather all of Ramanujan's approximations and asymptotic expansions found in
the unorganized pages of his second and third notebooks. We are very grateful
to Askey, R. P. Brent, Evans, and M. L. Glasser for their valuable contributions
to this chapter. Again it is difficult to succinctly summarize this work. The first
several sections are devoted to the asymptotic analysis of series which are hybrids
of the Riemann zeta-function and hypergeometric series.
Last, in Chapter 39 we collect together results from the unorganized pages of
the first notebook which do not fall under the purviews of Chapters 32-36. Most
are from analysis, but some are elementary.
In Part IV we provided a chapter documenting each entry in the 16 organized
chapters of the first notebook. The vast majority of these results can be found in
the second and third notebooks, but for those that are not we gave proofs. At the
end of this volume, we provide a similar account for all the claims made in the 198
unorganized pages of the first notebook. Thus, for each entry we indicate where a
proof can be found in Parts I-V.
Except for the massive amount of material in Chapters 33 and 34 related to
Ramanujan's paper on modular equations and approximations to л [3], [10, pp.
23-39], in contrast to the first four volumes, very few claims in this volume pertain
to Ramanujan's published papers and problems. The following table summarizes
these connections:

Introduction
Paper
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
Related Material in this Volume
See final chapter on location of entries, in particular, pp. 347-349
Entry 18 in Chapter 37
Most of Chapters 33 and 34
Entry 30 in Chapter 37
Entries 27-30 in Chapter 37
Entries 24-26 in Chapter 37
Entry 23 in Chapter 37
First section in Chapter 32
The following table gives the number of results in each of the eight chapters in
this volume:
Chapter
32
33
34
35
36
37
38
39
Total
Number of Results
73
62
196
24
87
53
46
24
565
The next table summarizes our reckonings of the results examined in each of
the five volumes.
Volume
I
II
III
IV
V
Total
Number of Results
759
605
834
491
565
3254
This total is in agreement with Hardy's original estimate that Ramanujan's
notebooks contain the statements of approximately 3000—4000 theorems.
In the sequel, equation numbers always refer to equalities in the same chapter,
unless otherwise indicated. Several references will be made to our first four vol-
volumes [l]-[4], and we almost always use the abbreviations Part I, Part II, Part III,
and Part IV, respectively. In chapters where claims from both the first and second
notebooks are discussed, we append the abbreviations NB I or NB 2 to a page
Introduction 5
number to indicate that the result is in the first or second notebook, respectively.
Throughout the text, the residue of a meromorphic function F(z) at a pole zo is
denoted by RZo.
To prove Ramanujan's claims in Chapters 33 and 34, it was necessary to derive
many ancillary results. Thus, in contrast to the remaining chapters, the formats in
these two chapters are different.
The road through Ramanujan's notebooks has been a long one, and many people
on this journey deserve my gratitude.
After taking a course in modular forms from my subsequent thesis advisor,
J. R. Smart, my first introduction to the work of Ramanujan came in a course on
modular forms with applications to number theory taught by Marvin Knopp at
the University of Wisconsin in the Fall of 1964. Here I learned about the Hardy-
Ramanujan asymptotic formula for the partition function p(n) and Ramanujan's
congruences for p(n). Knopp's book [1] arose from this course.
I learned of the existence of Ramanujan's notebooks one day in early 1967
in Robert Rankin's office at the University of Glasgow. However, when Rankin
asked me if I would be interested in examining his copy of the Tata Institute's
publication of the notebooks [9], I declined.
I owe a huge debt to the late Emil Grosswald for my initial interest in the
notebooks. It was on a cold winter day in early February, 1974, while I was on leave
at the Institute for Advanced Study, that I was reading two papers by Grosswald [ 1 ],
[2] in which he proved some formulas from Ramanujan's notebooks. I suddenly
realized that I could also prove these formulas by using some transformation
formulas for Eisenstein series that I had proved about two years earlier. I was
naturally curious to determine if there were other formulas in the notebooks that
I could prove with my methods. Fortunately, the library at Princeton University
possesses a copy of the Tata Institute's publication [9]. I found a few more formulas
which I could prove, but I also found a few thousand others which I could not
prove. My papers [6] and [7] contain proofs of my initial findings and several
other formulas in the same genre.
At the close of the spring semester in May, 1977, at the University of Illinois, I
decided to attempt to find proofs for all of the formulas (a total of 87) in Chapter
14 of Ramanujan's second notebook, where the formulas which Grosswald proved
can be found. After working on this project for nearly a year, George Andrews
visited the University of Illinois and informed me that the attempts of G. N. Watson
and В. М. Wilson to edit Ramanujan's notebooks in the late 1920s and 1930s had
been preserved in the library at Trinity College, Cambridge. Thinking that with a
copy of their notes, I could edit further chapters, I wrote Trinity College. Indeed,
Watson and Wilson's notes were very useful, and, in particular, Watson's work
on modular equations in Chapters 19-21 of the second notebook was invaluable.
We are pleased to record here our thanks to the Master and Fellows of Trinity
College, Cambridge, for providing us with a copy of these notes. Thus, to bring
us to the end of the story, since May, 1977,1 have devoted all of my research time
to proving the claims made by Ramanujan in his notebooks.

6 Introduction
This work could not have been completed without the help of several people
First, С Adiga, G. E. Andrews, S. Bhargava, A. J. Biagioli, P. Bialek, H. H. Chan
R. J. Evans, F. G. Garvan, J. L. Hafner, P. T. Joshi, R. L. Lamphere, L. Lorentzen
J. M. Purtilo, and L.-C. Zhang collaborated with me in writing papers on chapters
or sets of formulas from the notebooks, and I extend to them my sincere gratitude
for their collaborations.
Others have made contributions in papers that they have individually written,
or in work that appeared only in our accounts. Thus, I wish to thank the following
mathematicians without whose proofs these volumes could not have been com-
completed: G. E. Andrews, R. A. Askey, G. Bachman, J. M. Borwein, P. B. Borwein,
D. Bradley, H. Cohen, M. L. Glasser, A. Hildebrand, L. Lorentzen, R. Mclntosh,
K. S. Williams, and D. Zagier. Some names appear in each of the past two para-
paragraphs, because these mathematicians also made contributions independent of any
collaboration with me.
Finally, I emphasize the enormous help given by three people. For several years,
when I became stymied for months or perhaps years over one of Ramanujan's
enigmatic formulas, I turned to Ron Evans. On each occasion, he was able to
supply a proof, and some of the most difficult proofs in these five volumes are due
to Evans. Preliminary versions of many of the chapters were read by Dick Askey,
who found mistakes, supplied references, gave insights, and sometimes provided
a proof. Lastly, in recent years, Heng Huat Chan not only served as a valuable
collaborator, but he offered many additional comments and insights, supplied
further proofs, and critically read preliminary versions of several chapters.
Jaebum Sohn also read in detail several chapters in this volume, and I thank
him for the several errors he uncovered.
I have given hundreds of lectures on Ramanujan's work to graduate students
and colleagues at the University of Illinois during the past two decades, and I am
grateful for the many meaningful comments they have provided.
Others who offered important comments and insights are cited in the Introduc-
Introductions to the first four volumes [l]-[4].
It is important that mathematicians are cited for the relevant contributions they
have made to a subject. At times, it is difficult to unearth their work, and 1 thank
Nancy Anderson, Mathematics Librarian at the University of Illinois, for helping
me locate many obscure papers.
One day in the early 1980s Heini Halberstam called me to his office here at
Illinois to meet Springer-Verlag's Mathematics Editor, Walter Kaufmann-Buhler,
who suggested that I compile my work on Ramanujan's notebooks into volumes
for Springer-Verlag. Thus, I express my sincere thanks to the late Kaufmann-
Buhler and the current Mathematics Editor, Ina Lindemann, for their support of
my work.
In the early years devoted to Ramanujan's notebooks, I received support from
the Vaughn Foundation, and I express my deep gratitude to James Vaughn for
his financial support. In more recent years, the National Science Foundation, the
Sloan Foundation, and the National Security Agency have provided grants, and
I thank these agencies for their support. I also am pleased to thank The Center
Introduction 7
for Advanced Sludy at The University of Illinois for three appointments which
provided me with time that I could exclusively devote to Ramanujan's notebooks
The author bears the responsibility for all errors and would appreciate being
notified of such, whether they be minor or serious.

32
Continued Fractions
Chapter 12 in Ramanujan's second notebook is devoted almost entirely to contin-
continued fractions. Further continued fractions can be found in other chapters, espe-
especially in Chapter 16. See Parts II [2] and HI [3] for accounts of Chapters 12 and
16, respectively. The 100 pages of unorganized material at the end of the second
notebook and the 33 unorganized pages in the third notebook contain about 60
further results on continued fractions. These and four evaluations of the Rogers-
Ramanujan continued fraction from the first notebook will be examined in this
chapter.
We have divided the entries into five categories. Section 1 is devoted to the
famous Rogers-Ramanujan continued fraction, the only continued fraction ap-
appearing in Ramanujan's published papers [8], [10, pp. 214-215], and certain gen-
generalizations. Other ^-continued fractions are examined in Section 2. In Section
3 we establish several continued fractions arising from Ramanujan's beautiful
continued fractions for quotients of Г-functions in Chapter 12 of his second note-
notebook. Most of the continued fractions in Section 4 arise from special functions, in
particular, hypergeometric functions. General theorems are the focus of Section
5.
We next describe a few highlights in this chapter.
Let, for \q\ < 1,
R( ,. 4US Я Я2 Я3
and
S(q) := -R(-q)
denote the famous Rogers-Ramanujan continued fractions. Entries 1-6 provide
beautiful equations relating R(q) with each of R(-q), R(q2), /?(?3), R(q4), and
R(q5). In both his first and second letters to Hardy, Ramanujan [10, pp. xxvii,
xxviii] communicated theorems about R(q) and S(q). In particular, in his first
letter, he asserted that
<„.„

10 Ramanujал s Notebooks, Part V
and
V5-1
@.2)
The evaluation @.1) follows easily from a reciprocity theorem for R(q), which
Ramanujan gave in his second letter, and which was first proved by G. N. Watson
[2]. The evaluation @.2) follows from a similar reciprocity theorem for S(q),
which apparently Ramanujan did not communicate to Hardy, but which is found
in his notebooks [9, p. 204]; see also Part III [3, p. 83]. The latter theorem was
first proved by K. G. Ramanathan [2], but @.2) was first established by Watson
[1] in a different manner. In his second letter, Ramanujan also claimed that
¦Л+1
1/5
@.3)
which was also first proved in print by Watson [2]; Ramanathan [2] also estab-
established @.3). Entries 7-10 offer four particular evaluations of R(q) from page
311 of Ramanujan s first notebook. Several further evaluations of R(q) and S(q)
were recorded by Ramanujan in his "lost notebook" [11], and these have been
proved by the author, H. H. Chan, and L.-C. Zhang [3]. Entry 11 is a fascinating
theorem concerning the oscillating behavior of the divergent Rogers-Ramanujan
continued fraction for q > 1. Ramanujan offers some "approximating" continued
fractions, with the most interesting results being Entries 13 and 14 involving mod-
modest generalizations of the Rogers-Ramanujan continued fraction. In each case,
one continued fraction is approximated by another continued fraction. D. Zagier
[1] has discovered the proper interpretations for these fascinating results, and we
briefly describe his work. Some elegant continued fractions, for example, Entry
15, are instances of more general continued fractions found later by Ramanujan
and recorded in his "lost notebook" [11].
Some ^-continued fractions give representations for certain ^-products. A few
of these results were established by A. Selberg [1], [2, pp. 1-23] in 1936. We
particularly call attention to Entry 19. To prove this, we employ a continued fraction
found by G. E. Andrews [1] which may be the unidentified continued fraction to
which Ramanujan alludes in his first letter to Hardy [9, p. xxviii].
Ramanujan had a tremendous facility for extracting interesting, important, and
elegant special cases from his theorems. The unorganized material contains numer-
numerous corollaries arising from his amazing work on continued fractions for products
of gamma functions in Chapter 12. Two of the most curious results in this vein are
Entries 41 and 42.
Before commencing our examination of Ramanujan's continued fractions, we
must offer several remarks about notation. All "chapter" references refer to chap-
chapters in Ramanujan's second notebook [9].
We employ the notation
for the continued fraction
°1
ь2
32. Continued Fractions 11
@.4)
Occasionally, for brevity, we shall use the notation K(an/bn) instead of @.4). We
let An and Bn denote the nth numerator and denominator, respectively, for @.4).
Thus, for n > 1,
O\ d2 <*з On _ А„
where
and
А„ — ЬпАп-\
В„ —Ь„Вп-\
@.5)
@.6)
where Д-, = 1 = Во and Ао = 0 = В_, (Н. S. Wall [I, p. 15]).
The set of natural numbers is denoted by N, while the set of complex numbers
is designated by С Furthermore, set С = С U {ею). The set of real numbers is
denoted by ffi., and we set ffi. = R U {oo). The set of integers is denoted by Z.
If a,v = 0 for some N e N, we say that the continued fraction @.4) terminates,
and we assign to il the value
f — — —
J '~ b~x+ b2 +
В
N-\
if an Ф 0 for n <
@.4) converges if l
Jim,,-»» А
N. If an Ф 0, 1 < n < oo, then the continued fraction
m,,-^ An/Bn exists in С Its value is then given by / =
» А„/В„, and we write
b2 + b3
@.7)
^Note that @.7) includes the case "oo = oo.") If lim,,-^ А„/В„ does not exist in
С (and an Ф 0,1 < n < oo), we say that @.4) diverges.
Many of Ramanujan's continued fractions arise from equivalence transforma-
transformations or from the "even parts" of continued fractions. In contrast to many contem
porary authors, we employ equality signs when invoking these ideas.
Several results depend upon continued fractions from Chapter 12, and so we
frequently make reference to our account [2] of this chapter. Our convergence
statements for many of the continued fractions of this chapter are based upon a

12 Ramanujan 's Notebooks, Part V
theorem of L. Jacobsen [4, Theorem 2.3], which is a consequence of the parabola
theorems (W. B. Jones and W. J. Thron [1]).
Several entries below concern theta-functions, and we employ the notation
Ramanujan introduced in Chapter 16 [9] (Part III [3]). For \ab\ < 1, put
f(a,b):=
@8)
Also set
:= f(q,93), and f(-q):=f(-q,-q2).
@.9)
For each nonnegative integer n, let
(a;q),, := (l-a)(l - aq) ¦ ¦¦ (\ -aq"~]),
@.10)
and, if \q\ < 1, set
(a,q)oo ¦= Mm (a\q)n.
We employ the contemporary convention for Bernoulli numbers and not that
used by Ramanujan in his notebooks. Thus, the Bernoulli numbers В„, п > 0, are
here defined by
ex - 1
n=0
We always employ the principal branch of each multivalued relation such as
z~l/1, log z, and tan z.
Lastly, as customary, set
Г(а+к)
1. The Rogers-Ramanujan Continued Fraction
Entry 1 (p.326). Let\q\<\,
and
Then
(i)
v -\-uz
v :=
= uv2
=: Vl/5.
and
(ii) UV2(U2 + V) + U2 - V + IOUV(UV - U + V + 1) = 0.
32. Continued Fractions
13
(iii) If U = n (—-) , then V = *2I±A
\l +nj 1 - n
Proof of (i). In Part III [3, p. 80, eq. C9.1)], we showed that
In order to derive a relation involving и and v, we need to rely on Ramanujan's
work on modular equations.
Let д/а and V? be the moduli associated with the variables ql/5 and q5, respec-
respectively. (Ordinarily, of course, these variables would be designated by q and q2i,
respectively.) Let m denote the multiplier associated with a and /3. Furthermore,
set
P_
R_
o_ md R_
qx/5f(-q5)' ^ g2/5/(-910)' qx/sf(q5)'
It will be easier to first derive a relation involving Q and R. It will then be an easy
matter to deduce a formula connecting P and Q.
From Entries 12(i), (iii) of Chapter 17 (Part III [3, p. 124]), we easily deduce
that
It follows that
fi(]-d)
R
Q
A.2)
Now rewrite Entries 15(i), (ii) of Chapter 19 (Part III [3, p. 291]) in the respective
forms
(SI
\-a
/jg(t-/3)V/8 _
\a(l-a)j
a(l-a)
and
These two equalities are, in fact, modular equations of degree 25. Multiplying the
former equality by (a(l - a)I/8 and the latter by (/J(l - /3))l/8, we see that we
can combine the resulting two modular equations to obtain the single equation

14 Ramanujan's Notebooks, Part V
Using A.2), we write this equality in the form
G3 ' " Q2 ' Q ~ Ql I «3 + 2 л2 + Rl
Clearing denominators and rearranging, we find that
R3 +2R2Q-2RQ2 - Qy =5QR- Q2R2.
We now replace ql/S by -ql/5. Then R is replaced by -P, and Q remains unaf-
unaffected. So,
- 2PQ2 - 2P2Q + P3 - P2Q2 +5PQ.
A.3)
Next, set Pi = 1 + P and Q\ — \ + Q. After some elementary tedious algebra,
we deduce, from A.3), that
(P2 + 2)(Q2 +2)- P,3 - Q\ + P, 0i - 4P, - 4Bi = 0.
By A.1), this last equality may be rewritten in the form
which can be written in the shape
мV(m2 + v)(v2 + u) + (иV + и V) + (ubv + ui - «V - иV)
- (м V + и V) + (и2 - v)(v2 - u) = 0.
Upon factorization, we arrive at
(uv2(u2 + v) + u2 - v)(u2v(v2 + u) + v2 - u) = 0.
From the definitions of и and v, it is clear that и — 0(<7l/;)andi/ = O(q2/S) asq
tends to 0. Hence, the first factor (and not the second) vaiishes for q sufficiently
small. By the identity theorem, the first factor vanishes fir \q\ < I. This proves
(i).
A proof of (i) was also given by L. J. Rogers [4, eq. E.4)].
It will be convenient to prove (iii) before (ii). We shal: alter Ramanujan's for-
formulation of (iii) by defining n by uv2. We shall then estiblish the two proffered
formulas for U and V.
Proof of (iii). From part (i),
V — U2
n =
Solving for u2, we find that
V +U2'
32. Continued Fractions 15
But since u2i>4 — n2, we deduce that
\ — n
which proves the formula for V. Now,
which completes the proof.
Proof of (ii). From part (iii), we obtain the two cubic equations in n,
л3+л2 + Vn - V -0 A.4)
and
n3 - (I/ -(- 2)n2 - B1/ - \)n - U = 0.
By subtraction,
(I/+ 3)n2 + B1/+ V- \)n + (U- V) = 0. A.5)
We now obtain three equations, in addition to A.4) and A.5), by multiplying A.4)
by n and A.5) by n and n2. Thus, we obtain five homogeneous equations in the
"unknowns," 1, n, n2, л3, and n4. Since there exists a solution (n = uv2), the
determinant of the coefficients is equal to 0, i.e..
1 1
0 1
U +3 2U + V - 1
0
0
1 1
0 1
U + 3
0
V
1
V -v 0
I V -V
и -v о о
2U +V -I U -V 0
U + 3 2V + V - 1 U -V
-V 0
V -V
0 0 4-5V-UV IV-V2 UV+V2-4V
0 0 U + V-4 U-UV-4V UV+3V
00 U + 3 2U + V - I U -V
= 4A0t/2V2 - 10U2V + lOi/V2 + UV + U2 + UZV2 + t/V3 - V).
This completes the proof of (ii).

16 Ramanujan's Notebooks, Part V
Entry 2 (p. 321). Let \q\ < 1. If
and
then
v : =
1 1 <L i!
1 +1+1 + 1 +
и - v)A - и V(« - vJ + 2и V = (м - v)(\ + и V). B.1)
Proof. Let
u> :=
1 + 1 + 1 + 1 +
By a direct application of Entry 1 (i), we find that
w — v2 ,
w + v2
Replacing q by — q in Entry l(i), we also find that
w — u2 ,
— —UW.
w + u2
These two equalities yield, respectively,
vw* + ti3ui2 — w + v2 = 0
and
uw3 +
+ w — u2 = 0.
B.2)
B.3)
Multiply the first of the last two equalities by u2 and the second by v2 and then
add the resulting two equalities. Second, multiply the first equality by и and the
second by v and then subtract them. Upon cancelling the nonvanishing factor u + v
in each case, we rind that, respectively,
uvw2 + u2v2w - (u - v) = 0
B.4)
and
Kt;(tt — v)w2 + w — uv = 0. B.5)
Next, divide B.2) by B.3) and cancel the nonvanishing factor и + v to obtain the
quadratic equation
w2 + (u-vJw-u2v2 = 0. B.6)
The three quadratic equations B.4)-B.6) may be written in the more succinct form
„2,
uv
uv(u —
1
v)
и
(и
V
1
- V
v -и \ /w1\ /0\
-uv )[w )= 0
-л2/\ i / W
32. Continued Fractions 17
The determinant of the 3 x 3 matrix on the left side must therefore equal 0, and
this is equivalent to B.1). This completes the proof.
Entry 3 (p. 321). Let \q\ < 1. //
and
+
then
Entry 3 was established by Rogers [4, p. 392, eq. F.2)]. (In fact, F.2) should
be designated by F.3), as apparently there is a misprint.)
Entry 4 (p. 326). Let, for \q\ < I,
1 + 1 + 1 + 1 +
2/5 f(-q\-qu)
m :=
f(-q\ -q«) Vl
and
Then
m — n = mn =
1 -n
Proof. Simple algebra shows that mn = m2/(l + m) if and only if m - n = mn
and that mn = и2/A -и) if and only if m - n = mn.
By Entry 38(iii) of Chapter 16 (Part III [3, p. 79]), mn = ми3 if and only if
/(-V. -<?")/(-<?, -qU)f\~q\ -q12) f(-q, -q*)p{-q\ -q»)
fi-q1, -q*)f(-q\ -qn)f2(-q6, ~q9) f(-q2, -q3)f4-q6, -q*)'
or
= f(-q, ~qA)f(-q\ ~qU)f(-q\ -q*)f(-q2, -qn).
D.1)

18 Ramanujan 's Notebooks, Part V
By applying the Jacobi triple product identity, Entry 19 of Chapter 16 (Part HI [3,
p. 35]), to each of the eight theta-functions in D.1), we, indeed, verify that D.1)
is valid. Thus, we have shown that mn = uv3.
It remains to prove that m — n — mn. Using Entry 38(iii) of Chapter 16 as
before, we find that the proposed identity is equivalent to the identity
/(-g\ ~<ls)f(-q2, -
f(-q7,-qH) f(-q2,
which is equivalent to
f{-q\ -V3) f(-g\ -V) f(-q\ -qn
To prove D.2), we shall apply the quintuple product identity three times. Referring
to Part HI [3, p. 80, eq. C8.2)], we replace^ by ql5/2 and set, in turn, В = -qiy2,
-q1/2, and -qvl. Accordingly, we find that
f(-q2\ V) - q7f(-q~\ -g«) = /(-g") i^'/l'
and
f(-qU, -q33) - q3f(-q\ -q42) = /(-<j
Thus, D.2) is equivalent to the identity
= qf(-q12, -q*) - q4f(-q\ -q42)-
D.3)
However, from a basic property of theta-functions (Part III [3, p. 34, Entry 18(iv)]),
and
Substituting these facts into D.3), we see that D.3) reduces to a tautology. This
then completes the proof of the equality m — n — mn.
Entry 5 (p. 326). lf\q\ < 1,
и : =
i si sL
and
v : =
1 + 1 + 1 + 1 +
32. Continued Fractions
19
then
Proof. Let
--IJ.
w :=
1 +1 + 1 + 1 + ••¦
From Entry l(i), it follows that
мю3 + mV -w+u2-0
and
v2w3 + w2 + v3w - v - 0.
Eliminating u>3 from this pair of equations and then cancelling the nonvanishing
factor uv + 1, we find that
u(uv — \)w — v w + uv = 0.
E.1)
We now take this pair of cubic equations in w and eliminate the "constant" terms
from the pair. After dividing out the nonvanishing factor w(uv + 1), we deduce
that
uvw + и w + v(uv — 1) = 0.
E.2)
Next, we take E.1) and E.2) and eliminate the "constant" terms. Second, we
eliminate the terms quadratic in iw from E.1) and E.2). We then obtain the pair
of equations
w
1
tt'v + u3(«u — 1) uv(uv — IJ — u2v2 —uv3 — u*(uv — 1)
We thus can derive two formulas for w2, namely,
u*v + v*(uv — 1) 2 Iuviuv -
—uv3 — u^{uv — 1) \ uv2 + и
- IJ — h2v2
Therefore,
(и3 + v2(uv - \))(v3 + u2(uv - 1)) + uv(u2v2 - 3uv + IJ = 0.
The desired result now follows after some elementary algebra.
Entry 6 (Formula C), p. 289). For \q\ < 1, let
Я Ч Я Я
and
1 + 1 + 1 + 1 +
q4i я q2 q*

20 Ramanujan's Notebooks, Part V
Then
,5 .1-2*-
This result was communicated by Ramanujan [10, p. xxvii] in his first letter
to Hardy. The first proof of Entry 6 is due to Rogers [4, p. 392, eq. G.1)]. A
second proof has been given by Watson [ 1 ], and another proof has been found by
Ramanathan [2]. Entry 6 is connected with modular equations of degree 5.
Rogers [4] also derived a modular equation relating R(q) and R(q ").
We now establish the four values for R(q) stated on page 311 in Ramanujan's
first notebook. Each of our proofs employs an eta-function identity from the unor-
unorganized portions of the second notebook [9] (Part IV [4, Chap. 25]). Ramanathan
[3] gave a different proof of Entry 7, but proofs of Theorems 8, 9, and 10 were
first given by the author and Chan [ 1]. In order to state the first four theorems, we
set
lc :=
a - b
G.1)
where a and b are certain real numbers to be specified below.
Entry 7. Lela = 5]/4,b = 1, and с be given by G.1). Then
R(e-4") = s/c2+ 1 - с
G.2)
Entry 8. Let a - 3 + ^2 - V5, b = B0)l/4, and с be given by G.1). Then
R(e-*") = Ус2 + 1-с. (8.1)
-л/10),
(9.1)
Entry9. Leta = 5]/\4-V2),b= 1 + -Д + V5-21/4C -
and с be given by G.1). Then
= Vc2 + l-c.
Entry 10. Let a = F0)'/*, b = 2 - л/3 + V3, and с be given by G.1). Then
Л(г~6я)= Ус2 + 1 -с. A0.1)
Before proving Entries 7-10, we offer some needed notation and preliminary
results. Recall that f(—q) is defined in @.9). We shall need two related trans-
transformation formulas for / (Part III [3, p. 43, Entry 27(iii), (iv)]). If a, fi > 0 and
afl = л2, then
e-'Wfi-e-2") = e-Vu/$l<4f(-e-2») G.3)
and
G.4)
32. Continued Fractions 21
Following Ramanujan [3], [10, p. 23], we define the two class invariants
G _ 2~1/4o~l/24(-<r q2)x and #„ = 2~1/49~1/24(^; tf2)oo. G-5)
where n > 0 and <? = e~"^". At the beginning of Section 2 of Chapter 34, we
establish two simple relations for these invariants,
*4.=2l/4A.Gn G.6)
and
First Proof of Entry 7. Recall that (Part III, [3, p. 84, eq. C9.1)])
— R(e-a) - 1 = g-"/s/A '
/?(«)
G.7)
G-8)
f(-e-5a) '
where a > 0 and / is defined by @.9). After some elementary algebraic manipu-
manipulation, we find that G.2) is equivalent to the identity
1
with a = 4n. Thus, from G.8), G.9), and G.1), we must prove that
G.9)
a _
-Vs,
G.Ю)
where a and b are as stated in Entry 7.
We shall employ Entry 58 of Chapter 25 of Part IV [4, pp. 212-213]. Let
0=
q2/Sf(_qWy
Then
(PCJ + 5PB = P' - 1P2Q - 2PQ2 + Q\
Lett? = <T2\ Then, by G.3),
P = V5.
Using G.12) in G.11), we find that
G.11)
G.12)
G.13)
It will be convenient to set Q = V5T, so that G.13) takes the form
¦/ST2 + V5T = 1 - 2T - 2T2 + Ty = (T2 - ЪТ + \)(T + 1).
Since clearly T Ф -1,
-/5T = T2 -3T+\.

22 Ramanujan's Notebooks, Part V
Solving this quadratic equation, we find that
T =
3 + ч/5 ± v/lO
If we took the minus sign above, we would find that Q = sfbT < 1. But clearly
Q > 1, and so we deduce that
e =
)¦
By G.10) and G.14), it remains to show that
51/4+l
5'/4-Г
G.14)
G.15)
However,
= 3 + V5 +
C + 5'/4 +
- 1)
2E1/4
and thus G.15) has been shown to complete the proof.
Ramanathan [3] gave a more difficult proof of Entry 7 in which class invariants
were employed. We have also discovered a proof of Entry 7 that utilizes class
invariants. Since our proof is simpler than that of Ramanathan and much different
from our proof above, we give it below. Like Ramanathan's proof, our proof
requires the value of G25, and so we give a simple derivation of this evaluation
next.
Lemma 7.1.
1+V5
Proof. We employ a modular equation of degree 5 found in Entry 13(xiv) of
Chapter 19 of Ramanujan's second notebook (Part HI [3, p. 282)). Let
P =
and
where 0 has degree 5 over a. Then
32. Continued Fractions 23
G.16)
Recalling the definition of Gn in G.5), recalling that (Part HI [3, p. 37]) *(?):=
(—q; *?2)oo, and using Entry 12(v) of Chapter 17 ofRamanujan's second notebook
(Part III [3, p. 124]), we find that
G25 = 2-1/4^/2421/6<Г"/24{0A - /3)Г1/24 = 2-"|2{0A - /3)Г1/24.
Since it is well known and easy to prove that G\ = 1, we have, by the same
reasoning as above,
С1=2-'/12{«A-а)Г1/24=1.
Hence, it follows that
Gh
G\b
Therefore, by G.16), if jc = G25,
= 0.
Since x + 1/x ф 0 and x > 0, we conclude that
1 + V
G25 =x =
-—,
and so the proof is complete.
We remark that the value of G25 is given without proof in Ramanujan's paper
[3], [10, p. 26].
Second Proof of Entry 7. Recall from our first proof that it suffices to prove
G.10).
Set a = 27Г/5 in G.3), so that j9 = 5тг/2. After some simplification, we find
that

24 Ramanujan's Notebooks, Part V
Thus,
( е
( e
")
Since
we deduce from G.17) that
Q = v5g25glOO.
G.18)
where gn is defined in G.5). We thus must determine #25 and gioo. Since G25 was
computed in Lemma 7.1, we see from G.6) and G.7) that Q can be calculated.
For brevity, set x = g\b and a = G*5. Thus, from G.7),
ax2-a2x+{=0, G.19)
Since x > 0, from G.19), we deduce that
2
x = -
la
By Lemma 7.1, Gf5 = B + V5J = 9 + Ф/5, and so
by G.20) and Lemma 7.1,
G.20)
= 9 - 4V5. Hence,
2-51/4J-
G.21)
Thus, from G.18), G.21), and Lemma 7.1,
Q = yf52'!Ag\bG2b
3/2
2-51/4)l/2
-ft
5'/" - 1'
32. Continued Fractions
Thus, G.10) is established, and the proof is complete.
25
Proof of Entry 8. We will again employ Entry 58 of Chapter 25 in Part IV [4, p.
212-213], but now with
By the same reasoning as that used in the first proof of G.2), in order to prove
(8.1), it suffices to prove that
* fl-fr '
where a and b are given in the statement of Entry 8.
Write G.11) in the form
_ P2 Q2
~~Q~2 +~P'
(8.2)
(8.3)
From Entry 7, P - EI/4 + 1)V5/E1/4 - 1). Putting this in (8.3) and setting
Q = л/57", we find that
51/4 _
t~2\
51/4+1" -¦'
By an elementary verification, it is easily checked that T = 1 is a root of (8.4).
Since clearly Q > V5, this root is not the one that we seek. Writing (8.4) in the
form Я3Г3 + a2T2 + u| T + a0 = 0, and dividing by T - 1, we find that
E1/4-lJ7-2-2(H-5l/4+N/5+53/4)r-(9+65l/4+3V5+2-5V4) = 0. (8.5)
Now set
T =
a+b
a-b
in (8.5) to deduce, with the help of Mathematica, that
-E + 3V5)a2 + F 5I/4 + 2 ¦ 5V4)ab + C - 3V5)fc2 = 0.
Solving for a, we find that
a =
- I5)b2
We now set b = 5l/4-s/2 and choose the plus sign above, because if we had chosen
the minus sign, we would find that T < 0, which is impossible. Hence,
- 30л/5
= -Jl 4- V 14 - 6^5 = V2 + УC - ч/5J = л/2 + 3 - V5.

26 Ramanujan's Notebooks, Part V
Hence,
Q =
and so (8.2) has been shown to complete the proof.
Proof of Entry 9. We again employ Entry 58 of Chapter 25 in Part IV, but now
we set
By the same argument that we used in the proofs of Entries 7 and 8, to prove (9.1),
it suffices to prove that
a + b r-
Q =
a-b
(9.2)
where a and b are prescribed in the statement of Entry 9.
Set A = 3 + л/2 - л/5 and В - B0)l/4. As in the last proof, let Q = -J5T.
Thus, by Entry 8 and (8.3), we know that
A +
A-
Let
T =
a + b
a-b
in (9.3). Clearing fractions and simplifying with the help of Mathematica, we find
that
(-10 - 7л/2 + 4л/5 + 2л/Ш)а3 + 51/4(8 + 9^2 - 2л/5 -
+ (-20 - 15л/2 + 6л/5 + 4л/Ш)аЬ2
+ 51/4A0 + 15\/2 - 4л/5 - 6л/Ш)&3 = 0.
Let a = 51/4d, cancel 5i/4Vl, and simplify to deduce t»at
(_7 _ 5л/2 + 2ч/5 + 2Vi0)d3 + (9 + 4л/2 - 2ч/5 -
+ D + 3n^ - Зл/5 - 2VT0)*>2</
+ (-6 - 2л/2 + Зл/5 + л/Ш)Ь3 = 0. (9.4)
Observe that d = b is a root of (9.4). If this were the rooi that we are seeking, then
Q would equal EI/4 + 1)л/5/E|/4 - 1). Thus, with Pand Q interchanged, we
have the same solutions to (8.3) that we had in the procf of Entry 8. Clearly, this
is not the solution that we want. Hence, dividing (9.4) by (d - b), we find that
D + 6л/2 - 2л/5 - ЗУШ)Ь2 + 2(-l + V2 + y/l)bd
+ (-10 - 7л/2 + 4л/5 + 2л/К))</2 = 0.
32. Continued Fractions 27
Solving for b, we find that
2A - л/5 - л/5)</±:
b=
Since
16 + 83л/2 -
2D + 6л/2 - 2л/5 -
2A-л/5-л/5) A — л/2—
'
2D + 6>/2 - 2л/5 - Зл/ТО) B + Зл/5)B - л/5)
14
14
we are motivated to set d — 4 — s/2. Therefore, a = 5l/4D — л/2) in agreement
with what Ramanujan claimed. Thus, by (9.5),
6 - 5л/5 - 4л/5 + л/Т0 ± 25'4л/283 + 19Ол/5 - 125л/5 - 86л/Т0
4 + 6л/2 - 2л/5 - Зл/ТО
(9.6)
Observe that
4 + 6л/2 - 2л/5 - Зл/ТО
We next wish to write
4B83 + 190л/2 - 125л/5 - 86л/Ш) = (ш + * л/2 + ул/5 + гл/ШJ,
for certain rational integers w,x, у, and z- Thus,
to2 + 2jt2 + 5y2 + 10z2 = 1132,
wx + 5yz = 380,
wy + 2xz = -250,
= -172.
(9.7)
(9.8)
David Bradley kindly wrote a program to determine the 24 positive solutions
to (9.8). We then found the unique solution of the system of four diophantine
equations to be
w = 20, x = 9, у = -8, and г = -5.
Thus,
2у/283 + 190л/2 - 125л/5 - 86л/10 _ 20 + 9л/2 - 8л/5 - 5л/Ш
4 + 6л/2 - 2л/5 - Зл/Ю 4 + 6л/2 - 2л/5 -
= 3 - л/5 + л/5 - л/Ш. (9.9)

28 Ramanujan's Notebooks, Part V
Putting (9.7) and (9.9) in (9.6), we find that
b = 1 + л/2 + V5 ± 2l/4C - л/2 + S - %/lO).
If we choose the plus sign above, we would find that a - b < 0 and Г < 0, which
is impossible. Thus, we conclude that
b = 1 + л/г + л/5 - 21/4C - л/г + л/5 - л/10),
which is what Ramanujan asserted. Thus, (9.2) is proved, and the proof of Entry
9 is complete.
Our proof of Entry 9 heavily relies on computation in the later stages. Although
Ramanujan possibly used Entry 58 of Chapter 25, he undoubtedly found a less
computational proof.
Proof of Entry 10. By the same reasoning in the proofs of Entries 7-9, to prove
A0.1), it suffices to prove that
eu;i
A0.2)
where a and b are specified in the statement of Entry 10.
Apply the transformation formula G.3) with a = Зтг/5 and /3 = 5луЗ. After
some simplification.
Thus,
Because 30 = 9 • у, we are led to Ramanujan's cubic continued fraction
< 1.
+1 + 1
From Part III [3, p. 345, Entry l(i)],
С(-„ = -,^<"
where
In particular,
A0.4)
225
32. Continued Fractions 29
by G.5). Recalling the value G2s = A + \/5)/2 from Lemma 7.1 and the value
G225 = id + >/5)B + V3)l/3 jv/TWB + A5I/4J
from Ramanujan's paper [3], [ 10, p. 28], or from the table of Section 2 of Chapter
34, but apparently first proved in print by Watson [7], we find that, from A0.4),
B - ч/3)C - V5)
Now
24л/Т5 = \/5У^Fл/3 + 4л/5J
1
Thus,
B -
B - лД)C - л/5)Fл/Ъ + 4л/5 - 3F0)|/4 - n/15F0)i/4)
8
A0.5)
Now Chan [1, Theorem 1] has shown that G(q) satisfies the modular equation
G\q) + 2G2(q2)G(q) - G{q2) - 0.
Replacing q by -q and solving for G{q2), we find that
G(q2) = ' - ^ - 8СЗ<-«>
A0.6)
4G(-q)
Set q = e~5", as above, and v = G(e~10"). Recall the definition of A from
A0.3). Then Entry l(iv) of Chapter 20 in Ramanujan's second notebook (Part III
[3, p. 345]) can be written in the form
= - + 4v2.
v
(Ю.7)

30 Ramanujan's Notebooks, Part V
Thus, by A0.7) and A0.6), with w := G(-<r5") given by A0.5),
(w + lJBw - 1)
by a somewhat lengthy, but straightforward, calculation.
Hence, by A0.2), A0.3), and A0.8), it remains to show that
/ e-
+ b
4
b
A0.8)
A0.9)
where a and b are specified in the statement of Entry 10. We used Mathematica
to verify A0.9) and complete the proof.
Another proof of Entry 10 has been given by the author, Chan, and Zhang in
[3].
The next entry is actually recorded twice by Ramanujan in his notebooks. We
quote Ramanujan.
Entry 11 (pp.374,382). lfq>\.
oscillates between
and
l~l tf~4 0~8 tf~12
T + T" + "T + ПГ
A1.1)
A1.2)
A1.3)
From the general theory of continued fractions, if all the elements of a divergent
continued fraction are positive, then the even and odd approximants approach
distinct limits. Thus, since A1.1) diverges for g > 1, Ramanujan is indicating that
its odd approximants tend to A1.2) while its even approximants approach A1.3).
In fact, we shall prove Entry 11 for \q\ > 1.
First Proof. Recall the definition of the Gaussian binomial coefficient
where \q\ < 1 and n and к are integers with 0 < к < п. Define
\ aq aq1
()
aq aq1
aq"
32. Continued Fractions 31
Then by a result of Ramanujan (Part III C, p. 31, Entry 16]),
cn(a;q)= " \Ч (Ц.4)
where
Dn(a;q) =
0<2j<n L J
We will need two further results of Ramanujan (Part III [3, p. 77, Entries 38(i),
(»)]), viz.,
G(q) :=
and
00 „n '+n
A1.5)
(П.6)
These are the famous Rogers-Ramanujan identities first established by Rogers
[ 1 ]. Lastly, we require the following identities due to Rogers [ 1 ] and Watson [11],
[ 14] independently:
^ i.q\q)b,+\
A1-7)
(П.8)
(П9)
and
A1.10)
It will be convenient to replace q by \/q in Entry 11. Letting c(q) denote
the infinite continued fraction lim^» cn(\; q), we may rephrase Entry 11 in the
following way. For 0 < q < 1,
the odd approximants of c(q~l) tend to Y/c(-q), (П.П)
while
the even approximants of c(q ') tend lo qc(q4). A1.12)

32 Ramanujan s Notebooks, Part V
We first examine the odd approximants. Using A1.4), replacing jbyn-j, and
utilizing the fact
we find that
L4-
A1.13)
h
2i
j=0
Hence, by A1.8) and A1.9),
;=u
f qj4i
lim c2n(l;q' ) =
n->oo
G(-q)
?:
where in the last step we employed A1.5), A1.6), and the Rogers-Ramanujan
continued fraction (Part III [3, p. 79, Entry 38(iii)]). This establishes A1.11).
We next examine the even approximants. Employing A1.4), reversing the order
of summation in both the numerator and denominator, and applying A1.13), we
find that
n-\\i\q )=
2-.
;=0
n-l
;=o
Thus, by A1.7) and A1.10),
lim с
Л-ЮО
= 4-
oo ^
32. Continued Fractions
33
where we used A1.5), A1.6), and the Rogers-Ramanujan continued fraction.
Thus, A1.12) is proved.
Entry 11 is a truly amazing result. It is very remarkable that the Rogers-
Ramanujan continued fraction reappears in determining the limits of both the
even and odd approximants of the divergent Rogers-Ramanujan continued frac-
fraction. It also should be noted that the continued fractions A1.2) and A1.3) are
"near" each other in the sense of Entry 13 below. More precisely, if we set x = 1
in A3.1) and A3.2), we obtain A1.2) and A1.3), respectively, but with q replaced
by \/q.
We will now give a second proof of Entry 11. This proof shows that Entry 11
arises from infinitely many Bauer-Muir transformations.
Second Proof. The even part of A1.1) is given by (see F4.1))
1 fl3 q1 «»
1 + q ~ 1 + q2 + ql ~ 1 + q4 + q5 ~ 1 + q6 + q1 ~
q~[ q~l q~x
1+<Г' - 1 +<?-' +q-i ~ l+<?-' +<TS ~ 1 +<Г' +<?~7 '
A1.14)
The latter continued fraction is a limit 1-periodic continued fraction for |<71 > 1.
Since the linear fractional transformation
t(u>) = —
]~l + w
has the two fixed points-g~' and-l,itfollowsthatA1.14)convergesfor|<i'| > 1.
Moreover, the modified approximants
,-з — ... _
+wn
also converge to the same value if {wn} does not have a limit point at -1. (For in-
instance, see Jacobsen's paper [1].) Applying the Bauer-Muir transformation A3.7)
to the second continued fraction in A1.14), with «;0 = 0 and ш„ = -1 /q, n > 1,
we obtain the continued fraction
,-3
,-3
A1.15)
which converges to the same value for |^| > 1. Again, this is a limit periodic
continued fraction. The attractive fixed point of
t,(w) = --.

34 Ramanujan's Notebooks, Part V
is -<T3. Hence, with w0 = ш, = 0 and wn = -q~y for n > 2, the Bauer-Muir
transformation of A1.15) is given by
q ' q "
1 + 1 +
,-5
-5
9g-5 - 14-<г5+?-7 - 1 + q-s+q-g '
which converges to the same value. Repeating this process к times with
w0 = w\ = • • • = tu*_i = 0 and wn = -q~lk+\ n > k, A1.16)
we obtain the limit periodic continued fraction
.-4
„-4*
Я'
1 + 1
+ 1 +
~-2к-1
+ i+<
r2*-' -
,-2*-3
A1.17)
,-2*-1
„-2А-5
which converges to the same value as A1.14). Repeating this process infinitely
many times, we obtain A1.3). Since A1.17) converges uniformly with respect to
к by the uniform parabola theorem (Jones and Thron [1, p. 99]), and since A1.3)
converges, we conclude that A1.3) converges to the same value as A1.14). We
have thus proved that the even part of A1.1) converges to the value of A1.3) for
\q\> I-
To prove that the odd part of A1.1) converges to the value of A1.2) for |^| > 1,
we can use the same idea, and even the same choices A1.16) for wn. We then find
that the odd part (Jones and Thron [1, p. 43, eq. B.4.29)], where the first minus
sign is misplaced) of A1.1) equals
1 -
= 1 -
= 1 -
= 1 -
1 + q + q2 ~
q6 - •••
+ <r2-
_-5
-Ч-<7-4-
9
+ g-6_...
<Г3
+ 9
~2
л-8 - ...
-5
-5
q-7 + i _ q-i
„-5
-* + 1 + q-6 - 1 + q
-5
,-8
,-9
„-13
-* + l-qS
-6 + l-q-'
+
- ! \+q-i + l-q-i q q
The last continued fraction is the even part of A1.2). Since A1.2) converges for
\q\ > 1, the second proof of Entry 11 is complete by the same argument as above.
32. Continued Fractions 35
K. Alladi A ] has given another proof of Entry 11 that is similar to our first proof.
However, he related his proof to the continued fraction
11
To be more precise, let Pn(q) and Qn(q) denote the nth numerator and denomi-
denominator, respectively, of the nth convergent of r(q). Let T(q) :— q~l/5R(q). Then
Alladi [1, pp. 225-229] proved that, for \q\ < 1,
and
lim
lim
G(q16) T(qt(>)
P2n(q)
Moreover, in the sense of modified convergence, Alladi proved that r(q) tends to
For the definition, importance, and historical background of modified conver-
convergence, see Jacobsen's paper [2].
For the last entry on continued fractions found in the third notebook, we quote
Ramanujan.
Entry 12 (p. 383). //
и : —
<71/5 Я Я2
then и2 + и — 1 — 0 when q" = 1, where n is any positive integer except multiples
of 5 in which case и is not definite.
This statement is not quite correct. However, I. Schur [1], [2, pp. 117-136] has
established the following theorem.
Theorem 12.1. Let
q q2 q2
where q is a primitive nth root of unity. If n is a multiple of 5, K(q) diverges.
When n is not a multiple of 5, let X = (j), the Legendre symbol. Furthermore, let
p denote the least positive residue ofn modulo 5. Then, for n^O (mod 5),

36 Ramanujan's Notebooks, Part V
Note that it is elementary that K(\) = (л/S + l)/2 and K(-l) = (V5 - l)/2.
We provide a short table of further values of К (q).
n
3
4
6
7
K(q)
^ 2
-5^/5+1
9 2
Ramanujan's lost notebook [11, p. 57] contains some claims on finite, general-
generalized Rogers-Ramanujan continued fractions, and these results have recently been
proved by S.-S. Huang [1]. Perhaps the main result gives a formula for evaluat-
evaluating certain finite generalized Rogers-Ramanujan continued fractions at primitive
roots of unity x. At the bottom of page 57 is a table of general formulas arranged
according to residue classes of n modulo 5, when x is a primitive nth root of unity.
However, the table contains some errors. When this table is used in Ramanujan's
primary formula, specialized to the ordinary Rogers-Ramanujan continued frac-
fraction, we obtain Entry 12 as Ramanujan recorded it. This is evidence that these
results in the lost notebook were derived before Entry 12. For more details, see
Huang's paper [1].
The next result is stated exactly as Ramanujan recorded it.
Entry 13 (Formula D), p. 289).
qx q1 fx_ <f_ i!i
'~T + T- l+i-i +•¦•
*+* + * + * + "
conventional only
Both continued fractions converge to meromorphic functions of x in С — @)
for |^| < 1. We shall indeed prove that
oo (_x)y
2-- Г^4. ,
9дг 92 ill 1_
1 + 1-1+1 -•••
-^-fe-. M<1. (".1)
and
x + x + x + x +
E
\q\ < 1. A3.2)
32. Continued Fractions 37
Let /(x; g) and g(x; q), respectively, denote the values of the continued fractions
in A3.1) and A3.2). We shall then prove that both f{x\ q) and g{x; q) satisfy the
same functional equation
F(xq2;q)= -L -, A3.3)
xq + F(x;q)
for 0 < \q\ < I. Furthermore, we shall prove that f(q; q) = g(q; q). It then
follows from A3.3) that
/(92"+I;e) = «(«2"+l;«) A3.4)
for every nonnegative integer n. Lastly, we shall describe some work of Zagier A ]
which indicates in what sense f(x; q) and g(x; q) are "nearly" equal.
The continued fraction on the left side of A3.1) may be identified by employing
a continued fraction found in Ramanujan's "lost notebook" [11] and proved by
Andrews [5], M. D. Hirschhorn [2], S. Bhargava and С Adiga [1], [2], Bhargava,
Adiga, and D. D. Somashekara [1], and others. Let
F(a b X ) =
and
Then
bq
(?; я)п(-Ья\ q)n
в(а,Ь,к,я)
A3.5)
. у„,^,..,ч/ G(aq b Xq qy
Setting b = 1 and X = 0, replacing q by q2, and setting a = —x/q, we establish
A3.1).
The continued fraction on the right side of Entry 13 is equivalent to
?A q*jJ_ qW_ ql2/x2
1 + 1 + 1 + 1 + •••¦
Thus, A3.2) follows from the corollary to Entry 15 of Chapter 16 (Part III [3, p.
30]). This corollary also appears in Ramanujan's [10, p. xxviii] second letter to
Hardy.
To prove A3.3) for F(x; q) = f(x; q), we shall use the Bauer-Muir transfor-
transformation found in Perron's book [ 1, p. 47]. Briefly, a Bauer-Muir transformation of
a continued fraction by + K(a,,/bn) is a (new) continued fraction whose approxi-
mants have the values
P[ -r t>2 -r ¦ ¦ ¦ + bk + Wk
Such a transformation exists if
A* := ak - wk-x{bk + wk) ф 0,
, fe = 0,1,2,....
A3.6)

38 Ramanujan's Notebooks, Part V
and it is given by
0 °
b\ +
+ w-i -
+ by + u>3 -
+
A3.7)
Suppose that Ьр + К(ап/Ь„) converges to a value / e С Let t0 Ф f be arbitrarily
chosen from C, and define tk = S^ Co), k > I. Then Sk(wk) tends to / as well,
unless the chordal distance d(wk, tk) has a limit point at 0. Hence, the Bauer-
Muir transformation A3.7) converges to / as well, if Ит*-^ inf d(wk, tk) > 0
(Jacobsen 15]).
We shall apply this to the odd part
1_ <?3* q7x qux
ЯХ 1 + q2{\ -qx) + \+q4{l - qx) + 1 + qb(\ - qx) + • ••
= /(*;?) A3.8)
of A3.1). Let
а„ = q*"-xx (n > 1) and bn = \ + q2n(l - qx) (n > 0),
so that A3.8) can be written as -\+bo + K(an/bn) = /(x,g).Letu4 = -q2k for
eachfc e N. Then the modified approximants—l + S*(ii>t) of A3.8) also converge
to f(x; q) by the following argument. The continued fraction -1 + bo + К(а„ /bn)
is limit periodic since limn_oo а„ = 0 and lim,,-»» bn — 1. It is then well known
(see, for instance, Jacobsen's paper [3]) that -1 + ft0 + K(an/bn) converges to a
value <p and that tk tends to -1 for every tail sequence of -1+ bo + К(а„ jbn) with
Го ф <P- In particular, this implies that lim*-,.,» inf d(wk, tk) — d@, -1) Ф 0, so
that lim*-3o Sk(wk) = f(x; q). Since A3.1) obviously holds for q = 0, we may
assume that q Ф 0.
A simple calculation shows that, for k > 0,
kk=ak- wk-\
0,
so that the Bauer-Muir transformation exists, converges to f(x\ q), and is given
by
1
fix; q) = -qx + -~TX + 1 + ЧЦ\
Comparing A3.8) and A3.9), we find that
+
+ ¦
A3.9)
f(q2x,q)'
which proves A3.3) for F(x; q) = /(*; q)-
32. Continued Fractions 39
To prove A3.3) for F(x; q) — g(x; q), we just observe that replacing x by q2x
in A3.2) yields
which is what we wanted to show.
For дс = q, the continued fractions in Entry 13 reduce to, respectively,
and
t0
A3.10)
A3.11)
Thus, A3.11) is the even part of A3.10), and so f(q; q) = g(q; q).
In order to examine how "close" the continued fractions f(x; q) and g(x; q)
are to each other, by A3.1) and A3.2), we are led to examine
_
r-2nn4n2 oo
Quite remarkably, Ramanujan stated an identity for F(x; q) in his lost notebook
[11], namely
F(x; q) =
E
n=— oo
(-*)"<?"'
A3.12)
where the last equality is obtained from an application of the Jacobi triple product
identity. The identity A3.12) was proved by Andrews in [6, pp. 25-32] and is
mentioned by him in his Introduction to the lost notebook [11, p. xxi, eq. A0.6)].
Zagier [1] independently also established A3.12). From A3.12), it is obvious that
F(q2n+ l;q) = 0 for each nonnegative integer n. We thus have obtained a second
proof of A3.4). Of course, the second proof is shorter than the first, but the first
proof is more elementary than the second, because A3.12) is somewhat difficult
to prove.
If x is not an odd power of q, in what sense are f{x; q) and g(x; q) near each
other? H. Cohen performed extensive calculations to answer this question, and
Zagier [1] established Cohen's conjectures as well as much more. We give a brief
summary of some of Zagier's results. We always assume here that 0 < q < 1 and
x > 0.

40 Ramanujan's Notebooks, Part V
Let Q = exp A~— )¦ Then, for x = 1,
Q3-
and
-nW-
In particular, as q tends to 1-,
/a;«) - *o
However, as x tends to 0,
Note that the asymptotic behaviors for x — 1 and x near 0 are different.
In general, Zagier [1] has shown that
A3.13)
as q tends to 1 -, where в — Or logjc)/B logq) and
^ log2 (ymV4 + x/2j - logJt log
where Lijd) denotes the dilogarithm
+ x
/2)
Since Li2(l) = тт2/6 and Li2 (C - V5)/2) = тг2/15 - log2 (A +
(Lewin [1, p. 7]), we find that c@) = я2/4 and c(l) = я2/5, in agreement with
A3.13) and A3.U).
Zagier's analysis shows that, for instance, /(jc; 0.99) and g(:t; 0.99) agree to
about 85 decimal places if v is near 1, about 96 places if jc is near |, and about
107 places if x is close to 0. The function c(x) becomes negative for x larger than
about 6.177. Thus, for л larger than this, f(x; q)-g{x; q) becomes exponentially
large as<? tends to 1-.
We do not know the meaning of the words "conventional only" in Entry 13.
For Entry 14, we again quote Ramanujan.
32. Continued Fractions 41
Entry 14 (Formula B), p. 290).
qx яъх
1 -
Я2*
1 + q + 1 + q2 ~ 1 +
X+X+X+X+---
+ l+q4 ~
nearly.
The analysis for Entry 14 is very similar to that for Entry 13. Both continued
fractions converge to meromorphic functions of x in С - {0} for \q \ < 1. We shall
prove that
qx
q2x
l+q + l+q2 - l+q* +
(q2;q2)n
q6x
'
I + q
6 _ ...
A4.1)
and
(Я2; Я2),,
X+X+X+X +
„to (q2;q2)n
Let /(*; q) and g(x; q), respectively, denote the continued fractions on the left
sides of A4.1) and A4.2). We shall prove that /(<?"; q) = g(q";q), for every
nonnegative integer и, by invoking the same theorem from Ramanujan's lost note-
notebook that we used in Section 13. Thus, we shall easily see that the "closeness" of
f(x; q) and g(x; q) can be determined merely by changing the variables in the
analysis of Section 13.
Proof. For \q\ < 1 and each nonnegative integer m define
By straightforward calculations, we then find that
S
qm+tx
and
qimx
+q2m)
Ръп+[(Х).

42 Ramanujan's Notebooks, Part V
Since Pm@) = 1 for each m e N, we thus deduce that
qx дгх
Pp(x) _ A + 1)A
PiW ~ 1
q'x
+ <?2) A
1
3)d
_!!_*?_ q*x q2x i6* 1
2\\+q + \+ q2-\+qi + l+q* Г
Hence, the first continued fraction in Entry 14 converges for \q\ < 1 to the value
„to (?; <?)„(-?;
= -1 +
? (<?2;?2)«
which immediately proves A4.1).
By the corollary of Entry 15 in Chapter 16 (Part III [3, p. 30]),
I i
i I
x+ x+ x + x +¦•¦ x\ 1 + 1 + 1 +
and so the proof of A4.2) is completed.
From A4.1) and A4.2),
f(x\q)-g(x;q)
h (я2;я2)*
со (_
?
32. Continued Fractions 43
(я;
у llf
x-2nqtf-n ¦
where we have employed A3.12) with q2 replaced by q and дс replaced by x^fq.
It follows that f(q"; q) = g(q"; <?), for each nonnegative integer и. We also see
that Zagier's analysis can be applied mutatis mutandi, after the aforementioned
changes of variables are made.
It is interesting to note that the continued fractions in A3.1) and A4.1) are
connected to the basic hypergeometric functions
)
^ (c; q)n(q; q)n
For example, we shall show that the continued fraction in A4.1) can be derived
from E. Heine's [1] continued fraction expansion
2(p\(a,b;c\q\ z)
- \ -'hi ^i? ^1
2tp\(a,bq;cq;q;z) 1 - 1 - 1 -
A4.4)
where
qk(\-aqk)(b-cqk)
- cq2k+])
and Д2* =
- bqk)(a - cqk)
- cq2k~l)(l - cq2k) '
Let a — 0, с = -1, and z = xq/b, and let b approach oo. We then find that
and —
A +q2k)(\+q2k+])
Thus, the continued fraction in A4.4) reduces to the one in A4.3). Likewise, since
lim (fc; q)nb-n = (-l)"q{n2-n)/2 and lim (bq; q)nb~n = (-l)"^(+'1)/2
the left side of A4.4) reduces to the left side of A4.3). The identity A4.3) follows
then, since the continued fraction A4.4) converges uniformly with respect to b in
a neighborhood of b = oo. A rigorous proof of this statement can be given along
the same lines as that given for B4.5) below.
Entry 15 (p. 373). Let«, b, and q be complex numbers with \q\ < 1. Define
<p(a) =
q" an
Then
<p{aq) ~ 1 +
аЯ2 ~
aqi aqA - bq2
+ ~T~ + I
Л
Proof. In A3.5), let a = 0, replace b by -b, and set X = a. Upon observing that
(-X/«; q)nan = x."qn("-]>n.
(
a-»0

44 Ramanujan's Notebooks, Part V
we see immediately that Entry 15 follows, since the continued fraction expansion
of F(a, b, k,q) in the proof of Entry 13 converges locally uniformly in our domain.
Observe that another continued fraction for tp(a)/<p(aq) is given in Entry 15 of
Chapter 16 (Part III [3, p. 30]). Furthermore, another representation for <p(a) can
be found in Entry 9 of Chapter 16 (Part III [3, p. 18]). With the help of these two
observations, Ramanathan [4] has found another proof of Entry 15.
Entry 16 (p. 373). For\q\<\,
f(-q,-q4)
4;<74)„
and
where, as before, x(q) = (—q\ q2)oo- Moreover,
qf(q2
qf(q\q7) I /(<?,<?9) =Я
(-q\-q12)/ П-Я*.-Я16) 1
+
A6.1)
A6.2)
A6.3)
Proof. First, A6.3) follows immediately from A6.1), A6.2), and a standard rep-
representation for the Rogers-Ramanujan continued fraction given in Entry 38(iii) of
Chapter 16 (Part III [3, p. 79]).
We next demonstrate the first equalities of A6.1) and A6.2). By Entry 19 (the
Jacobi triple product identity), B2.2) (Euler's identity), and Entry 22, all in Chapter
16 (Part III [3, pp. 35, 37, 36]), we find that
f(q,q9) __ (-<?;<?l0)oo(-<?9;gl0W°;<7l0)oo
(-98;«?10)ос(?й;?|0)оо(-<?10;^|0)о0
(-q*\ q$)o
A6.4)
32. Continued Fractions 45
which establishes the first equality of A6.1). The proof of the first equality in
A6.2) is completely analogous.
By A6.4) and its analogue, in order to prove the second equalities of A6.1) and
A6.2), it suffices to show that, respectively,
and
tSa <9*: «')- <-<?2; 42U(q2; ^Ы*?3; qs)oo'
These last two identities have been proved by L. J. Rogers [1, pp. 330,331]. Hence,
the proof of Entry 16 is complete.
Entry 17 (p. 374). Let a, b, and q be complex numbers with \q\ < 1. Define
o^ q(n2+n)/2an
(p(a) = > — .
tZ(q;q)n(-bq;q)n
Then
4>(a) aq bq aq2 bq2 aq* bq2
Proof. The result follows immediately from setting X = 0 in A3.5).
2. Other ^-Continued Fractions
Entry 18 (p. 373). For \q\ < 1,
f(-q, -qs) ^ 1 q+q2 q2 + q4
fi-qK-q*) 1 + 1 + 1 +
Beneath this continued fraction, Ramanujan writes
Num? =
fhll and Den? = , .
fi-я) f(-q2)
In fact, he has incorrectly inverted the identifications of the "numerator" and
"denominator" on the left side of Entry 18.
By Entry 22 and Example (u). Section 31 of Chapter 16 (Part III [3, pp. 36,
5Ц),
f(-q, -q5)
П-qK-q3)
(q\
(q;

46 Ramanujan's Notebooks, Part V
where х(я) = (~Я> Я2)°о- On the other hand, by Entry 22 of Chapter 16 (Part III
[3, p. 36]),
q2) (qb; qb)oo(q; q)<x> (Я'< 92)°о
Hence, we have shown that Ramanujan has mistakenly confused the roles of the
"numerator" and "denominator." Moreover, we now see that Entry 18 can be
written in the more transparent form
C(q) : =
(<?; <?2)oo
to3;*6K*,
1 g + g
1 + 1 +
A8.1)
The first proofs of A8.1) in print are due to Watson [2] in 1929 and Selberg [1,
p. 19] in 1936. B. Gordon [1] and Andrews [1] found proofs in 1965 and 1968,
respectively, while Hirschhorn [3] has shown that A8.1) can be deduced from
Ramanujan's continued fraction A3.5). Ramanathan [2], [3] has briefly discussed
A8.1). L.-C. Zhang [1] has examined A8.1) when q is a root of unity.
A detailed study of G(q) has been made by H. H. Chan [1]. He has derived
modular equations relating G(q) with each of G(-q), G(q2), and G(<?3). Using
these and other modular equations, he has determined values for G(±e""v*) for
several positive rational numbers n. The author, Chan, and L.-C. Zhang [1] have
found general formulas that enable one to evaluate G(±e~''"/") in terms of class
invariants.
Entry 19 (Formula C), p. 290). For \q\ < 1,
(<?2;<?3)oo l я яг
Proof. We apply Theorem 6 in Andrews' paper [1] with at = wxq, a2 = ш xxq,
a = -1/(jtV). and * = !/(*?).where ш = ехрBтг|/3). Thus, l/(a,a2) = -a
and I/a, + \/a2 - -b, as required in Theorem 6. Accordingly, by the same
argument as in the justification of the limiting procedures in Entry 24, using the
uniform parabola theorem (Jones and Thron [1, p. 99]), we find that
lim
H2,\(.(joxq,o) 'xq;x;q)
,5
= 2-
j,w-\xq;xq;q) " 1 + <? ~ 1 +Я2 ~ 1 + Яъ ~ ' ^ ^
where the identification of Н2Л will be made shortly. Comparing Entry 19 and
A9.1), we see that it remains to show that
1
Нг.\((охд,ш [xq;x;q)
o~xxq; xq; q)
A9.2)
32. Continued Fractions 47
By Andrews' paper [1, eq. A.1)],
и / ч (xqlauq)oc(xq/a2\q)von . . ,.„..
H2.i(al,a2;x;q)= C2,{(ai,a2;x;q). A9.3)
; q)o
Furthermore, by [1, eq. A.1)],
lim C2,\(,o)xq,(t)~ixq;x; q)
fr?(q,q)Aw x\4)n(<o;q)n
1 ~ (i _ or VH - wqn)(.-l)nq3ni"-l)/2
A -co~l)(l - со) *—i (я*>Яг)п
1 / '-^/
- E
i-D/2
A9.4)
я=о
„=0
>;я3)п
я=0
where we have employed an identity of Euler (Andrews [4, p. 19]). By a similar
argument,
A9.5)
ton C,., («*«.«-*,; xq;q) = 2l(q,q)n{0)-iq,q)n{(oq,q)n
, 3
уя 9
, 3)
- уя >9 too,
я=0
by another application of Euler's identity.
Putting each of A9.4) and A9.5) in A9.3) and using the results on the left side
of A9.2), we find that
1
]xq;x;q)
, w~lxq; xq; q)
1
- 1
_
This completes the proof of A9.2) and hence also of Entry 19.
The functions Q,, studied by Andrews in [1] also appear prominently in his
paper [3, Sect. 2]. These functions play a crucial role in the full three parameter
general Rogers-Ramanujan theorem (Andrews [2]).

48 Ramanujan's Notebooks, Part V
In Ramanujan's [10, p. xxviii] first letter to Hardy, he states the Rogers-
Ramanujan continued fraction and some identities involving it. Ramanujan con-
continues by claiming "The above theorem is a particular case of a theorem on the
continued fraction
1
ax
ax2
ахъ
ах4
ax5
which is a particular case of the continued fraction
1
ax
ax*
axJ
1 + l+bx + \+bx2 + l+bx3 +¦'
which is a particular case of a general theorem on continued fractions." It seems
possible that Andrews' Theorem 6 of [1] giving an evaluation of
- axg2)
1 + bxq2 +
1 + bxq +
1 + bxq3 +¦¦
is the "general theorem" about which Ramanujan writes. However, Hirschhorn [ 1 ],
[3] has also put forth a very good candidate for this "general theorem." The most
general continued fraction containing the Rogers-Ramanujan continued fraction
as a special case is undoubtedly that of Andrews and D. Bowman [1].
Entry 20 (Formula D), p. 290). For\q\ < 1,
\-l+q2-\+q'-\+q
6 _ ... •
This result is simply the case a = 1, b - 0 of Entry 12 of Chapter 16 (Part III
[3, p. 24]). Entry 20 can also be found in the "lost notebook" [11]. Ramanathan
[4] has also given a proof of Entry 20. Another continued fraction for the left side
of Entry 20 is found in the "lost notebook" and has been proved by Andrews [7]
as well as by Ramanathan [4].
It is interesting to note that the continued fraction in Entry 20 also converges
for|o| > 1.In fact, seta = I/a, so that \a\ < l.Then
1 л „3 „5
+q2 -
I/a
1_
T - 1 + I/a2 -
1 a
~ l+q6
I/a3
I/a5
1 + 1/a4 - 1 + 1/a6
= (д3^4)^ ^ (I/a3; I/a4),*,
~ (a;a4)oo (I/a; 1/a4)^'
This is, indeed, a beautiful example of symmetry.
B0.1)
32. Continued Fractions 49
It follows more generally from Entry 12 of Chapter 16 that
(N3;<?4)°o _ 1 bq bq3 bq5
(bq-^^oo 1 - 1+a2 - 1+a* - 1+a* -••
Now let \q\ > 1 and seta = I/a, so that \a\ < l.Then, as in B0.1),
1 bq bq1 Ъаъ (ba3;a4)^
\q\ < 1.
1 _ = W А?)оо
1 - 1 + q2 - 1 + a4 - 1+ a6 " (ba; a*)*, (b/q; I/a4),» '
Although the continued fraction above is symmetric in q and I/a, the product
(bq3; q*)<xl(bq; q*)oo does not share this invariance. However, if b = -1, then
and the latter quotient is invariant when q is replaced by I/a. These observations
are due to K. Alladi and B. Gordon [1, p. 298].
The convergence of B0.1) when о is a primitive root of unity has been examined
by Zhang [1].
Entry 21 (Formula E), p. 290). For \q\ < 1,
I 4
1 + 1 +
1
+ 1 +
1
si
+ 1 +
B1.1)
Entry 21 was first proved in print by Selberg [1, eq. E4)]. Another proof has
been given by Ramanathan [4]. We provide yet another proof based on Entry 15.
Proof. Applying Entry 15 with a = 1 and b = — 1, we find that
q+"
1 + 1 +
So
(Alternatively, this can also be proved by using A3.5) in Chapter 16 of Part III
[3, p. 28] with a - -1 and b = 1.) Using Euler's identity (Andrews [4, p. 19])
once again, we find that the numerator and denominator on the left side above are,
respectively, (-0; q2)x and (-a2; q2)x. The desired result now follows.
If Q{q) denotes the left side of B1.1), then, by using Entries 22(i), (ii) and
25(vii) of Chapter 16 (Part III [3, pp. 36, 40]), we can easily show that
Q\q) = *V> = J_
?>4(a) 16a
in the notation of Entries 5 and 6 of Chapter 17 (Part HI [3, pp. 100-102]). Thus,
modular equations for Q(q) can be trivially derived from any of Ramanujan's
modular equations.

50 Ramanujan's Notebooks. Part V
Entry 22 (Formula F), p. 290). For\q\ < 1,
(<?;<?8)oc(<?7;я%)оо _ l я+я2 ^
to3;?8),*^5;*8),*, l + l + i +
l
+ l +
The first published proof known to us is by Selberg [ 1, eq. E3)]. Other proofs
have been given by Andrews [5] and Ramanathan [4]. Entry 22 also appears in
Ramanujan's "lost notebook" [11]. Another continued fraction for the left side of
Entry 22 has been found by Andrews [1] and Gordon [1]. Chan and Huang [1]
have developed an extensive elegant theory for these continued fractions, including
modular equations and explicit evaluations.
Entry 23 (p. 373). For \q\ < 1,
/(-<?, V) 1 ^
Я^
+ 1 +
+ i +
By the Jacobi triple product identity, we may rewrite Entry 23 in the form
?WH
я + я2
*3+«6
{q^~, q*)oc(.Я^' Я^Уео } + 1 + 1 +
Hence, Entry 23 is equivalent to Entry 22.
+ 1 +
3. Continued Fractions Arising from Products of Gamma
Functions
For the first result, we quote Ramanujan [9, vol. 2, p. 281]. We have
2/3 23 - 2
43 - 4 53 - 5
+ 3x2 +
73-7
+ ~5x~2~ +
B4.1)
x2 + 6 + 3x2 + 6
The symbol ? l/дс denotes ?,к<х * A- However, B4.1) is clearly incorrect with
this interpretation, because the left side is discontinuous for positive, integral x,
while the right side is continuous for such x. Now if x is a nonnegative integer,
then (M. Abramowitz and I. A. Stegun [1, p. 259])
iK* + l) + K = ?l/*. B4.2)
k<x
where у denotes Euler's constant and ^U) = Г СО/ Г(дг). As in Chapter 6 (Part
I [1, p. 138]), we shall then take the left side of B4.2) as our interpretation of
X! 1/jc for all positive numbers дс.
We now reinterpret B4.1).
32. Continued Fractions 51
Entry 24 (p. 281). If Rex > 0.
*(x + I) - Ф & + 1) + - -log 3
2/3 23 - 2 43 - 4 53 - 5 73 - 7
3x2
5jc2
B4.3)
Qk- lK-
Bk+\)x2
Proof. If I, m, or n is an integer or if Re л > 0, then by Entry 35 of Chapter 12
(Part II [2, pp. 156, 157), Jacobsen [4]),
1- P
l + P
limn
x2-l2-m2-n2
oo 4(l2 -
. К
=\ Bj + \){x2 -Е2-т2-п2 + 2J + 2j + 1}'
B4.4)
where
p =
Dividing both sides of B4.4) by m and letting m tend to 0, we find, upon an
application of L'Hospital's rule, that
{{x + I - n + 1)) + * (±[x - I + n + 1})
-* {\[x + ^ +n + 1}) - ф Q[x -l-n + Щ
2tn ~ -4j2(?2 - J2)(n2 -
B45)
2 - e2 - n2
j=\
\){x2 -t2-n2+ 2j2 + 2y + 1}'
We must justify the limiting procedure. Let I, n, and x be fixed with Re x > 0.
Of course, if the continued fraction terminates, no justification is necessary. So
assume that I and n are not integral and that \m\ < 1. Let K(ak(m)/bk(m))
denote the continued fraction in B4.4), but with the first partial numerator divided
by m, and let fk(jn) and /(m) denote the continued fraction's /fcth approximant
and value, respectively. Without loss of generality, we assume that /(m) Ф oo
in a neighborhood of m =0. (Otherwise, consider 1/A + K(ak(m)/bk{m))).)
Furthermore, let gk denote the Jtth approximant of the continued fraction in B4.5).
We want to prove that
lim gk = lim f(m) =: /@).
/fc-юо m-»0
B4.6)
Suppose that the convergence of fk(m) to f(m) is uniform with respect to m
in a neighborhood of m = 0, i.e., for some e > 0 and |w| < e,
B4.7)

52 Ramanujans Notebooks, Part V
where kk tends to 0 as к tends to oo. Also suppose that fk(m) tends to gk as m
tends to 0, uniformly with respect to к in a neighborhood of к = oo, i.e., there
exists an integer ко such that for к > ко,
|*t - fk(m)\ < rim). B4.8)
where r(m) approaches 0 as m tends to 0. Now,
1/@) - gk\ < 1/@) - /(m)l + |/(m) - fk(m)\ + \fk(m) - gk\.
Thus, B4.6) follows provided that B4.7) and B4.8) hold. Indeed, these two state-
statements of uniform convergence follow from the uniform parabola theorem (W. J.
Thron [2]).
Next, the even part (see F4.1) below) of the continued fraction in B4.3) is given
by
CFE(x) : =
6B3 - 2)D3 - 4)
23 - 2 + 6x2 - 6{43 - 4 + 53 - 5 + 6 ¦ 3x2)
62E3 - 5)G3 - 7)
- 6{73 -7 + 83-8 + 6-5л:2}
62(C* - IK - Qk - l))(Cfc + IK - Ck + 1))
- 6{C* + IK - Ck +[) + Ck + 2K - (ЗА: + 2) + 6BJfc + \)x2}
43-4 5 -7E2 — 1)G2 - I)
4/6
- 43-4 + 53-5+ 18x2 - 73 - 7 + 83 - 8 + 3Ox2
9k2(9k2 - l)(9k2 - 4)
- 3BJfc + 1)(9*2 + 9k + 2 + 2л;2}
8/27 4
4(*2 + l)/9 - 3{2 ¦ I2 + 2 • 1 + 4(x2
4-22B2-l)B2-|)
- (IJ)
- Bk + I){2fc2 + 2fc+4(x2-
Hence, by B4.5), with ? = \, n = \, andx replaced by 2x/3,
CFE(x) =\{{f (\(x + 1)) + ф (i(x + 2)) - ф A* + 1
Since (Abramowitz and Stegun [1, p. 259])
1 !_V
,A /t-1+гУ
(\x)}.
B4.9)
32. Continued Fractions 53
where у denotes Euler's constant, we find that
* (i(x + 1)) + * {\(x + 2))-i, {\x + 1) - V [\x)
^л/ 1 1 1 1
f lk + 2 Зк+l Ък++
^V 3k + x-2 Зк+x-l 3k+x 3/t+x
1 1 1__ 3
3fc + x - 2 ~ 3k + x - 1 ~ 3it + x + 3fe + x
1 1 1111
Зк-2
+ x ~ 3k-2J + f^\3k + x 3k-\)
1\
1\
= 3f(x +
r
^x + Л + -
3*-2 3*-l
= Зф(х + 1) - Зф 1-х + 1 ) + - + 3 lim (log /V - log 3/V)
\3 / X Л/-Ю0
- 3^(x + 1) - 3^ Г^х + Л + | - 3 log 3.
Using the foregoing calculation in B4.9), we complete the proof.
Entry 25 (Formula E), p. 292). If Rex > 0, then
1
1
- I + _L |J_ i_ il
~ x 2л2 I 3x + 5x + 7x
BA

54 Ramanujan's Notebooks, Part V
Proof. Replacing x by 2x in Entry 30 of Chapter 12 (Part II [2, p. 149]) we find
that
k=0
- л 4- 2fc + 1 2x+n
+ 2k + 1 J
(и/2) 12A2-п2)/4 22B2 - n2)/4
x + 3x + 5x +
(и/2)
(\-n)F(n,x)'
where
22B2-n2)/4
'v' ' 3* + 5jc +••
After some elementary algebra, we find that
n-2xT(n,x)
2T(n,x)(l-n)
= F(n,x).
Letting n tend to 1, applying L'Hospital's rule, and using the facts T(l,x) =
ЭТ
\/Bx) and — (\,x) ф 0, we find that
an
lim F(n,x) - lim
ЭТ
\-2x — (n,x)
дп
-л)-— (п,х)
on
1-2*^A.*)
an
-l/x
= -x + 2x'
El—L
2kJ Bx+2k + 2J
= -¦* + -
2
Since
2 ' ?f(* + *)r
22B2 - l)/4 32C2 -
'у1'л'~23х+ 5х + lx +•••
we see that the proof is completed after a little algebraic manipulation and an
appeal to uniform convergence as in the proof of Entry 24.
Entry 26 (Formula F), p. 293). If Rex > 0, then
i ^ i _ i _L|I El Si El Si I
where, far к > 1, pk = k2(k + l)/Dfe + 2) andqk = k(k + lJ/D* + 2).
32. Continued Fractions 55
Entry 26 is, in fact, due to T. J. Stieltjes [1], [3, pp. 378-391]. It is also given in
Wall's book [1, p. 37], where 4г2 should be replaced by 4г3.
Entry 27 (p. 325). Let n be a complex number such that Re л > - j. Then
1 / IM 1 (IIJ (IIJ B-3J
3 + 5(n2 +n) + 7
J C • 5J
B - 3J C ¦ 5J C ¦ 5J 1
+ 9(n2 + n) + 11 + 13(я2+и) +•••]¦
B7.1)
Proof. From the corollary to Entry 30 of Chapter 12 of the second notebook (Part
II [2, p. 150]), we find that, for* = 2n + 1 and Rex - ReBn + 1) > 0,
1
= 2I' '- 21 Ъ- 1
I x + 3x + 5x + lx + ¦ ¦ • j
I
I4
lDn2+4n+ i) + з + 5Dи2+4/| + j) + 7 +..
B7.2)
We shall use the Bauer-Muir transformation A3.7) to prove that this continued
fraction converges to the same value as the one presented in this entry.
Choose a>ik = 0 anda>2*+i = -Bk + IJ for each nonnegative integer k. From
A3.6),
X, = 1, X2k = 4*2B/fc - IJ, and Xzjt+i = 16it4,
for each positive integer k. Moreover,
= 4(*
IJ,
and
= D*+1)D«2 + 4л).
Hence, the continued fraction in B7.2) is transformed by A3.7) into the continued
fraction
1 4A IJ 4A П2 4B-3J 4B • 3J
4и2
4A ¦ IJ 4A ¦ IJ 4B ¦
3 + 5Dл2 + 4л) + 1
A-1J B ¦ ЗJ
A ¦ IJ
+ 9Dи2 + 4и) Н
B - ЗJ
3 +5(п2+п)+
+9(и2+п)+ ¦••
Since the even approximants of the two continued fractions coincide and since both
continued fractions converge for Re n > - ^, the proof of Entry 27 is complete.

56 Ramanujans Notebooks, Part V
In the foregoing proof we have seen that the even approximants of the continued
fractions in B7.1) and B7.2) are identical. Thus, an alternative proof can be derived
by showing that the even parts of B7.1) and B7.2) agree. Such a proof would be
shorter and simpler but not as instructive as the constructive approach via the
Bauer-Muir transformation.
If we let n = 1 in Entry 27, we find that
ИЛ
• 32 22 • 32
3 + 10 + 7 + 18 +
where < denotes the Riemann zeta-function.
Letting и = 1 in Entry 28 below, we deduce that
log 2
-i_MI i_i Li Ы. lil \
~ 2D+ 1 + 4 + 1 + ~4~ + • j
Entry 28 (p. 325). Let n be a complex number such that Re и > - j. Then
^ (-l)l+1 / 1Л f 1 11 11 2-3
*=!
n+k
-(-01
2n2 + 2n + 1 + 2и2 + 2n + 1
2-3 3-5 3-5
2n2 + 2и
2n2 + In
Proof. From the corollary to Entry 29 of Chapter 12 of Ramanujan's second
notebook (Part II [2, p. 149]), it follows that
^ л: + 2* - 1 ~ x + x + x + x +'
for Re* > 0. Setting x = 2и + 1, we find, via equivalence transformations, that
(-!)*+' 1 I2 22 32
2n + 1 + 2n + 1 + 2n + 1 + 2л + 1 + • • •
1 1^ 22 32
"" }
2« + IJ + T + Bn + IJ + 1 +
= V + 27J2n2 + 2n+i
1 л2 I i!
2 l 2 ' 3
2 + 1 + 2и2 + In + i + 1 + • ¦
B8.1)
for Re я > -1.
We apply the Bauer-Muir transformation A3.7). Let o>2* = 0 and &>2*+1 =
-Bk + l)/2 for each integer к > 0. Then from A3.6),
A., = 1,
— 1), and \2k+\=2k2,
32. Continued Fractions 57
for each positive integer k. Furthermore,
=kBk- 1),
1),
and
= 2fl
Thus, the Bauer-Muir transformation transforms the continued fraction in B8.1)
into the continued fraction
1 11 11 2-3 2-3
2w2
2n2 + 2w
1 + 2и2 + 2w + • ¦ •'
Since the even approximants of the two continued fractions coincide and since
both continued fractions converge for Re n > -\, the proof is complete.
Alternatively, the even part of the last continued fraction in B8.1) is precisely
the continued fraction in Entry 28. This gives an even shorter proof.
Entry 29 (p. 343). Let x and n be complex numbers such that either Re л: > О or
n = {2k + l)j for some integer k. Then
2 ,
(jc + 3 + 4*J
4
JC +
2x
(x + 1 + 4*J
+
2x
2x
Proof. Replacing n by in in Entry 25 of Chapter 12 (Part II [2, p. 140]), we find
that, under the conditions specified above,
n2+52
B9.1)
~x+ 2x + 2x + 2x +••'
However, from Euler's product formula for the gamma function,
Г (
Г (IU-/
П
Using this formula and an analogous formula in B9.1), we complete the proof.
(A product representation for |Г(х + iy)\2 is given in Gradshteyn and Ryzhik's
tables [1, p. 945, formula 8.326, no. 1].)

58 Ramanujan's Notebooks, Part V
Entry 30 (p. 343). For all complex n,
tanhGrn/4) _ 1 n2 + I2 n2 + 32 n2 + 52
n ~T+ 2 + 2 + 2 +•••'
Proof. Setting л: = 1 in Entry 29, we deduce that
2
4ikTT)
«2+52
1+ 2 + 2 + 2 +¦•
However, by a familiar product representation for tanh г, the left side above equals
Entry 30 may also be found in O. Perron's book [1, p. 36, eq. B3)]. Entry 31
below is also in Perron's text [1, p. 33].
Entry 31 (p. 343). Let x and n be complex numbers such that either Re x > 0 or
n = Bk + \)ifor some integer k. Then
x+2k-\
2 ?<-!)¦
t+i
22 «2
42 n2
x + x + x + x +x+ x +•••¦
Proof. Replacing и by in in Entry 29 of Chapter 12 (Part II [2, pp. 147,148]), we
find that, under the conditions given above,
v^ u. i x +2k - 1
2^(-l)* + l ,x + 2k_ 1J + „2
1 1
*=1
+ ni + 2k - 1 x-ni + 2k-l
t  + 32 ?
д:+ л; +*+ л:
Entry 32 (p. 344). For all complex numbers n,
l 4и2+12 г2
4w2 + 32
Proof. Set д: = 1 and replace n by 2и in Entry 31.
In the continued fraction of Entry 32, Ramanujan inadvertently wrote n2 for
4n2.
32. Continued Fractions 59
Entry 33 (p. 344). Let x and n be complex numbers such that either Re x > 0 or
n = Bk + \)i for some integer k. Then
2y !
?Z(x + 2k+
n2 + 22) 32(n2 + 32)
Proof. In Entry 30 of Chapter 12 (Part II [2, p. 149]) merely replace n by in, and
the desired result immediately follows.
Entry 34 (p. 343). For every complex number n,
— coth (—\ = 1 + — l2(+J) 2V + 22) 32(л2 + 32)
2 V 2 / I + 3 + 5 + 7 +•¦••
Proof. It is clear that the identity holds for и = 0. Thus, assume that n ф 0.
Setting л: = 1 in Entry 33, we find that
= l \2(n2 + l2) 22(я2 4- 22)
1 + 3 + 5 +•••'
Upon multiplying both sides by n2 and rearranging, we complete the proof.
Entry 35 (p. 344). Let x and n be complex numbers such that either Rex > —^
with x $ (-5,0], or n = ki for some integer k. Then
til
I2 „2+22
+jc2+x+
x2+x
Proof. In Entry 31 of Chapter 12 (Part II [2, p. 150]), replace n by 2i n and x by
2x + 1. Then, under the proposed hypotheses we find that
(-U*
22
42
4л2 + 4дт + 1 + 4л2 4- 4лг + 1 + Лх^+Лх +
which is equivalent to the continued fraction displayed in Entry 35.

60 Ramanujan's Notebooks, Part V
Entry 36 (p. 344). Let m, n, and x denote complex numbers such that either
Rex > 0, or m — ij for an integer j and x ф — Bk + 1) for any nonnegative
integer k, or n = ij for an integer j and x =/= -Bk + 1) for any nonnegative
integer k. Furthermore, let
Then
u-v
и + v
ш (m2
x +
2)(n2
3x
I2) (m1
+
5x
m-n
• 22)(и2 + 22)
Proof. Assume that x, m, and n are positive. In Entry 33 of Chapter 12 (Part II
[2, p. 155]), replace m and n by im and in, respectively. Employing a product
formula for |Г(д: + iy)\2 (Gradshteyn and Ryzhik [I, p. 945]), we find that, under
the given hypotheses,
—тп
X
(т2
+
ГA(д
oo f
П ) 1
111-
*=о 1
оо [
П 1
*=0 1
1/и-
+ 12)(я2 + 1
Ъх
+ \) + \(т
+ 1) + \(т
(т+п)'
1 (х + 2к +
(т+п)'
' (х + 2к +
l/v v — и
2)
+ 1
+ /
IJ
IJ
(т
ч,2
Г-
2 + 22)(п2
Ъх
-|г(*е*
+ \T{\(x
00 f
-П i +
+ 22)
-\
(т-
(x + 2Jfc
(т-
(х + 2к
и)
+
ч)
+
>И-П)|)|
2 г1
1J1
2 !"'
IJ)
\/v
v + u
This completes the proof for x > 0, m > 0, and n > 0. Since the continued
fraction converges to a meromorphic function of л for Re x > 0, the entry holds in
this half plane by analytic continuation. Furthermore, since the continued fraction
converges to a meromorphic function of m and of n, it follows that Entry 36 is
true for all complex m and n. That the equality holds if the continued fraction
terminates follows by straightforward computation.
Entry 37 (p. 344). Ifm and n are complex numbers with m фп, then
m tanh ({яп) - n tanh {\лт)
m tanh (\лт) - n tanh (^яи)
_mn (m2 + I2)(n2 + I2)
-~Г+ 3
(m2 + 22)(я2 + 22)
32. Continued Fractions
61
Proof. Putting x — \ \n Entry 36, we see that it only remains to show that
(u - v)/(u + v) reduces to the left side of Entry 37.
Using a familiar product representation for sinh z, we see that, when x = 1,
u-v (m +и)~' sinh (jjr(m + n)) - (m - n)~l sinh {^n(m - n)\
и + v (m +л)~' sinh {jjr(m + n)\ + (m - n)~l sinh (jtt(m - и)}
—2nsinh(:jjrm) cosh(j7rw) + 2m sinh (j7rn) cosh (j7rm)
2m sinh (|лт) cosh ( jJT/j) — 2n sinh ({лп) cosh (jjrrn)
m tanh (jti) — n tanh {\лт)
m tanh (jTrm) - n tanh {\nn)'
and so the proof is complete.
Entry 38 (p. 345). Let m and x be complex numbers such that either Re x > 0,
or m = fe(l + i)/2, or m = k(\ — i)/2,for some integer k. Furthermore, set
« —
и +
V
V
V —
2т2
X
"A
+
(х+Ъ
4т4 +
Зл
яН
I4
hi
4-
»г(*<
4т"+
Ъх
х -
24
-2
+
m л
4т
-I))'
1х
and
Then
н—
Note that if m = <сA ±/ )/2 for some integer к, the continued fraction terminates.
Proof. We apply Entry 33 of Chapter 12 (Part II [2, p. 155]) with m and n replaced
by A + От and A - 0"», respectively. We then find that, form > 0 and x > 0,
or for A ± От = к for some integer k,
2m2 4m4 + I4 4m4+24 4ra" + 3"
x + Ъх + Sx + Ix + •¦¦
/ 7m \2t '
1 + 1
)T
where in the penultimate step, we used the same formula for |Г(х + iy)\2 that we
used in the proofs of Entries 29 and 36. The result is then valid for all complex m
and all complex x with Rex > 0 by analytic continuation.

62 Ramanujan's Notebooks, Part V
Entry 39 (p. 345). For arbitrary complex n,
Sinhfrn) - sinfrn) _ In2 4n4 + I4 4nA
sinh(jrn) + sin(jrn) ~ 14-3
4я4 4-34
Proof. Setting x = I and replacing m by n in Entry 38, we deduce that
2n2 4n4 4- l4 4и4 4- 2*
~T +
п
(jrn) sinh(;r>«) —
ГA4-п)ГA-л)
"' зш(тгл)
(зги) sinh(n'w) + (in)
where we have employed a familiar product representation for sinh z and the
reflection formula for the gamma function. This completes the proof.
Entry 46 (p. 345). Let x and n be complex numbers such that either Re x > — j ;
or pn is an integer, where p is a sixth root of unity, and x is not a negative integer.
Furthermore, let
Then
и — v
-I6
-26
3Bх2 4- 2x + 3)
2x + 7) 4-
The constant term within parentheses in the Jfcth denominator is given by к2—к 4-
1. In the notebooks, Ramanujan mistakenly indicated that this constant is equal to
2k - I. Thus, Ramanujan wrote 5 instead of 7 in the third denominator displayed
above. Ramanujan's error can be traced back to a scribal error in recording Entry
40 of Chapter 12. For a discussion of this error, see Part II [2, pp. 163,164]. Note
that if pn is an integer, where p is a sixth root of unity, the continued fraction
terminates.
Proof. We shall apply Entry 35 of Chapter 12 (Part II [2, pp. 156,157], Jacobsen
[4]) with € = ег'1/ъп, т = е4;"/3и, and x replaced by 2x 4- 1. Observe that
(I2 - k2Km2 - k2)(n2 - k2) = n6 - *6
32. Continued Fractions 63
andthatJr2-€2-m2-fl2+2fc2-2t4-listransformedinto4^24-4x4-2Jl:2-2ik4-2.
Thus, the continued fraction that arises from Entry 35 of Chapter 12 equals
2и3 4(и< - I6) 4(w6 - 26)
+ \x 4- 2 4- 3Djc2 4- 4* + 6) 4- 5D*2 4- 4x + 14) 4-
я
-26
D0.1)
2jc2 4- 2x 4- 1 + 3B;t2 4- 2x 4- 3) 4- 5Bя2 4- 2x 4- 7) 4-
We now examine the gamma functions appearing in Entry 35 with the para-
parametric designations given above. Letting w — e27tifi and using Euler's product
representation of the gamma function, we find that, in the notation of Entry 35 of
Chapter 12,
p _ А ГСс 4-1 4- wJn)
lim Л (* + ' + ft - »)(* + 1 4- к - wn){x 4- 1 + к - a>2n)
~ m^°°*=o (* + !+* + и)(д: 4- 1 4- к 4- om)(x 4- 1 4- к
uu
П
t=0
1 -
14-
" у
:+l+k/
Hence, by Entry 35 of Chapter 12, the continued fraction D0.1) equals
1 - P _ 1 - v/u _ и - v
1 4- P ~~ I 4- v/u u + v'
and the proof is complete.
Entry 41 (p. 347). Suppose that m, n, and x are complex numbers such that
Re x > 0, or assume that n is an integer or that im is an integer. Then
m
= tan
\mn (I2
I x +~
3x
+
+ n + 2k +
B2 4- m2)B2 - n2)
5x
4-
Proof. In our proof below, we will temporarily ignore the fact that tan"' z is
multivalued. At the end of the proof, we shall show that the correct branches have
been chosen.

64 Ramanujan's Notebooks, Part V
Replacing m by im in Entry 33 of Chapter 12 (Part II [2, p. 155]), we find that,
for Rex > 0, or n € Z, or im € Z,
D1.1)
x + 3-х + 5.t 4 ¦ • ¦
Suppose first that x, m, and n are positive, and let
a = Г (i(x 4- n 4- 1) + \im) Г (i(* - n 4- 1) - |»m) =: a + ib,
where с and b are real. Multiplying both sides of D1.1) by -i and then taking the
inverse tangent of each side, we see that
tan
(mn
x
- n2) B2 + m2)B2 - n2)
3x
\a+aj\
_ im(ioga)
D1.2)
= Im (log
In order to calculate 7", we employ Euler's product representation for the gamma
function. Hence,
T= -
4=0
tan

mil
\ , / -то/2
I + tan -j
D1.3)
Thus, formally, the proof has been completed.
To complete the proof, we first apply Stirling's formula to show easily that, for
x >0,
lim r {{(.x+n + 1) 4- {im) Г ({{x - n + 1) - ^tw) _ {
n + 1)- \im)
Thus, for the principal branch of tan"' г,
a+a
32. Continued Fractions
65
On the other hand,
n+2k+l
= limf( = 1 + 0( I )\
'-"to \x-n+2k + 1 x+n + 2k + l \(x - \n\ + 2k + 1)V J
/ 1
2+ \(х-\п\ + 2к + \
lim
2тП
for the limit as x -> oo can be taken under the summation sign, since the series
converges uniformly for \n | <x< oo. Thus, our calculations in D1.2) and D1.3)
demonstrate that Entry 41 is correct for* sufficiently large and positive. However,
since both sides of Entry 41 are meromorphic for Re x > 0, the equality of Entry
41 must then be valid for all x with Re x > 0.
If n or im is an integer, the continued fraction terminates. A straightforward
computation shows that the identity still holds for x > 0. Hence, the proposed
result follows by analytic continuation. Furthermore, since both sides are mero-
meromorphic functions of m and n, the entry holds for all m, n € С by analytic
continuation.
Entry 42 (p. 347). Let т.п., andx denote complex numbers. Suppose that either
Rex > 0, or m = 2ir for some integer r, or n = Bs + \)i for some integer s.
Then
E°° , ,.i I _¦ / m + n \ ,/ m-n \|
(-1)* {tan ' I I + tan ' [ I}
t=o I 4*4-2*4-1/ \x+ 2*4-1/1
!m и24- 1
7 + T~
= tan
12 „2
m24-22
m24-42
+ ¦
Proof. As in the previous proof, we temporarily ignore the fact that tan"' z is
multivalued, and assume that x, m, and n are positive.
In Entry 34 of Chapter 12 (Part II [2, p. 156], Jacobsen [4]) replace I by im and
n by in. Thus, under the given assumptions on x, m, and n,
+ 12
m2 4- 22 n2
+ 32
те'
4-42
4-
4-
1 - P
l + P'
D2.1)
where
_
a + ib
a — ib'

66 Ramanujan's Notebooks, Part V
where a = a + ib is the numerator of P. Multiplying both sides of D2.1) by -i
and then applying the operator tan, we deduce that
tan {-
= tan"'
+ x + x + x + x +
D2.2)
Employing Euler's product formula for the gamma function, we find that
T = Im log Г (±(дг + 1 + i(m + и))) + Im log Г (±(* + 1 + i(m - n)))
+ Imlogr(j(jt + 3-j(m - л))) + Im log Г (\(x + 3 - i(m + n)))
OO
*+l +Цт+п)) + к)
+ Imlog (;(* + 1 + Urn - n)) + k)
+ Imlog(±(x + 3 - i(m - и)) + к) + Im log (\(x + 3 - i(m + n)) + k)}
^M . / m + n \ , / от - n \
+ l + 4*/j
-tan
D2.3)
Using D2.3) in D2.2), we formally complete the proof.
To show that we have, indeed, chosen the correct branches in all our calculations,
we use the same type of argument as in the proof of Entry 41 for the case Re x > 0.
Since the details are very similar, we omit them.
If mi/2 or (m - l)/2 is an integer, the continued fraction terminates. The
proposed result again follows, as in the proof of Entry 41.
Entry 43 (p. 347). Let x and n denote complex numbers. Assume either that
Re л: > Oorn = ji for some integer j. Then
1)* tan '
= tan ' { -
+
+
Proof. Set m = n in Entry 42.
4. Other Continued Fractions
Entry 44 (Formula B), p. 276). Let a and b be complex numbers such that
а Ф 0 and | arg(fe/a2)| < Jr. Let Bn denote the nth Bernoulli number. For each
32. Continued Fractions 67
nonnegative integer n, define
An =
Then, for each positive integer N, as x tends to 0 through values such that
Rt(bx2) > 0,
i » • »
a+a+a+a-\
Our version of Entry 44 is slightly more precise than that of Ramanujan. A
proof of Entry 44 has been given by Watson [3] for the case when a > 0, b > 0,
and л: > 0. The extension to b/a2 e С - (-oo, 0] and Re(bx2) > 0 follows
by analytic continuation, since the continued fraction converges to a holomorphic
function of (a, b) for b/a2 e С — (—oo, 0], and the series on the left side converges
to a holomorphic function of (a, b) for Rt(bx2) > 0. The reader should note that
the notations of Ramanujan, Watson, and the authors for Bernoulli numbers are
different.
The next result was communicated by Ramanujan [10, p. 352] in his second
letter to Hardy and is simply the case a — 1, b = | of Entry 44. We precisely
quote Ramanujan below, but, of course, a more accurate version can be formulated
as above. Ramanujan tacitly assumed that x > 0. However, the result holds for all
л € С with Re(x2) > 0, i.e., for | arg jc | < л/Л.
Corollary (Formula C), p. 276). When x is small,
112 3 4
12
360
5040 60480 1710720
nearly.
Entry 45 (Formula D), p. 276). The formal power series
L(x) :=
*=o
has the corresponding continued fraction
я i по a-i
СF{x) :=—. — . — . ,
x + x + x + ¦¦¦
where Ok > 0 for 1 < к < oo. In particular, a\ = аг = 1, ay — 30. a4 = 150,
and as = 493. As before, Bj, 0 < j < oo, denotes the jth Bernoulli number.

68 Ramanujan's Notebooks, Part V
For the definition of correspondence, we refer to the text of Jones and Thron [ 1,
p. 148]. Continued fractions of the form above are called S-fractions or Stieltjes
fractions. They have the property that their even and odd parts converge to analytic
functions both of which have the asymptotic expansion L{x) (Jones and Thron [ 1,
pp. 136, 342]). Moreover, they converge for all x in the cut plane | arg jc| < n if
and only if they converge for one x in this region. It seems to be very difficult to
determine if CF(x) converges.
Proof. To see that CF(x) is an S-fraction, we observe that L(x) can be written
in the form
*=0
where
= 2B4*+2 -
ck = 2B
4 ,<x> U4k+l /-OO и« + 1^м
- п4к+г Jo Sinh и U ~ Jo sinh(TTM)
D5.1)
where we have used a familiar integral evaluation (E. T. Whittaker and G. N.
Watson [ 1, p. 126]). Since sinh(;r u) > 0 for и > 0, the primary assertion of Entry
45 follows from Stieltjes' theory. (See, for instance, Wall's book [1, p. 363].) v
To calculate the numerators a*, we may use Entry 17 of Chapter 12 in Ra-
manujan's second notebook (Part II [2, pp. 124,125]) Alternatively, Viskovatoff's
algorithm (A. N. Khovanskii [1, pp. 27, 28]) can be employed. The calculations
in the second method are somewhat easier, and, in either case, they are routine.
Hence, we omit them.
It is tempting to conjecture that the numerators of С F (x) are integers. However,
ab = 588456/493 and a7 = 10101660478/4029289.
Lastly, we find two functions that have the asymptotic expansion L(x) as x
tends to oo.
From D5.1), for each positive integer n,
tdt
isinhGr()
" (-1)* Г°° t^dt _ 4(-l)" f»
as x tends to oo. Thus, F, (jc) has the asymptotic expansion L(x).
32. Continued Fractions 69
From the Laurent expansions of cot t and coth / about the origin (Gradshteyn
and Ryzhik [ 1, p. 42]), we easily find that
coth t - cot t =
\t\
k=0
Applying Watson's Lemma (E. T. Copson [1, p. 49]), we deduce that
/•O
I
as x tends to oo with Re x > 0. Replacing x by 7y/ix and by Vix, where the
principal branch of the square root is chosen, we find that, for -Ъп/2 < arg x <
n/2.
F2{x) :=
f°
:=2i I
Jo
- eVlJ")(coth t - cot t) dt
as x tends to oo.
Unfortunately, neither F\ nor F2 has been of any use to us in determining the
convergence or divergence of CF(x).
Entry 46 (Formula F), p. 277). For each complex number x.
4-5
23
6-7 , 4-5 ,
~X ~X
XCoth.t = l + ---Г- 1^Г_ 4^_ ±J_ 6_7_
3 9|5+7 + 9+ll + 13 +
Entry 47 (Formula G), p. 277). For each complex number x and each complex
numbern ^0,-1,-2,-3,...,
XXX X
n + n + \ + n +2 + n +3 +
x x \ x a-ix
k* «2*+] X
n n2 \n + \ +П
where, for к > 1,
+ n + 2k + л + 2* + 1 H
a2k =
and «2*+l =
Hn+k-l) ~~ "T1 (* + l)(n
We first remark that the continued fraction on the left side of Entry 47 is equal
to
for all complex x, where Jv denotes the ordinary Bessel function of order v. See
Wall's book [1, p. 349] or Part II [2, p. 133, Entry 19].
Next, we show that Entry 46 readily follows from Entry 47.

70 Ramanujan's Notebooks, Part V
Proof of Entry 46. Recall that (Wall [ 1, p. 349]), for each complex number x,
X2 X2 X2
xcothx = 1 + "^","Т',^г+ > D6.1)
which is due to J. H. Lambert [1]. This suggests that we let я = \ and replace x
by x2/4 in Entry 47. Accordingly, we find that
*2/4
3/2 +
x2/4
3/2
x2/4
5/2 +
x2/4
9/4
x2/4
5/2 +
кг/4
7/2 +
2-5/2
1-3/2
7/2
xl/4
9/2
X2
4
+
+
1
2-
3/2
5/2
9/2
X2
4
3
2
•7/2.
¦5/2
11/2
X2
4
2
3
\-
•5/2.X2
¦7/2 4
13/2 +•••
which is equivalent to
2 I 3 + 5 + 7 +
2 3 2 9
5 +
45 2 2-3 2
2^3* 4^f!
7 + 9
6-7
11
4-5
13
Multiplying both sides by 2, adding 1 to each side, and employing D6.1), we
complete the proof.
Proof of Entry 47. We shall employ a lemma of Rogers [2, p. 74]. If /i = eoeu
f2 = e\+e2, hh - e2ei, h + f* =
then
e0 e\x e2x f\x f2x fax
T_ —_ — _...=* + — - — - — -...
D7.1)
in the sense that both continued fractions correspond to the same (formal) power
series. This means that if both continued fractions converge in a neighborhood of
x = 0, then they converge to the same value (Jones and Thron [1, p. 181]); that
is, D7.1) expresses an identity between their values. Writing the left side of Entry
47 as the equivalent continued fraction
x x x
n(n + 1)
(n+ !)(/!+ 2)
i
1 - 1
we see that, in the notation D7.1),
x
e°~n' Bk~ (n+k-lKn
1
к > 1.
32. Continued Fractions 71
We now calculate fk,k > 1. Straightforward calculations show that
By induction, we shall show that
(Jfc+lH/i
f2k+\ = —
- l)(n + 2k- 1)(я
*(я H- Jfc - 1)
-, k > 1, D7.2)
-, it>l. D7.3)
and
We assume that D7.2) and D7.3) hold for * = 1,2, ...,m. Simple algebraic
calculations show that
(м + 2)(л+и + 1)
(m + l)(n + ш)(л + 2m + l)(n + 2m + 2)
and
, e2m-t2e2m+l (jn
J2m+3 — -
¦ m ¦
This completes the proof.
For the next result, we again quote Ramanujan.
Entry 48 (Formula A), p. 290).
x x2 x2
2m + 2)(n
3)
2n n - I n + l n -2 n + 2
~ + 1 - ~T~ + 1 - x
— 1 nearly.
Proof. From our remark after Entry 47, it is not difficult to show that
4n + 2 + 4« + 6 + 4n+10H Jn-\/2(ix/2) x
in the sense that the continued fraction converges to the function on the right side
for all (n, x) 6 C2, x Ф 0. It will be convenient to write the right side in terms
of Bessel functions of imaginary argument (Watson [15, p. 77]). Thus, comparing
Entry 48 with D8.1), we must show that
/„_V2(*/2) 4л - 2 ,
2л п - \ n+ \ n-2 n+2
D8.2)
nearly,
x+l-x + l-x +
where /„ denotes the Bessel function of imaginary argument of order v.

72 Ramanujan's Notebooks, Part V
As x tends to oo (Watson [15, p. 203]),
/Ддс/2) ~ -Ц^
Re* > О,
where we have ignored the exponentially decreasing terms in the complete as-
asymptotic expansion. Using this expansion in D8.2), we find that the left side of
D8.2) is asymptotically equal to the quotient
2 *=0
; l/x)
2n2F0(l -л.и
x 2F0(l-n,n;\/x)
k% Ь\ Г(п-к)хк
where 2FQ{a, b\ z) denotes the (divergent) hypergeometric series
v 1 и \
2F0(a, b. z) =
*=o
Jfc!
From the book of Jones and Thron [ 1, p. 212], it follows that
In n-\ л + 1 n-2 n+2
~x + 1 ~ x + 1 - x +¦¦¦
2n/x (l-n)/x (л + 0/дс B-n)/x (n + 2)/x
1-1-1
2«2FOA -И.И+ 1; l/x)
1
1
x 2F0(l -n,n; l/x) '
D8.3)
in the sense of correspondence. From Jacobsen's paper [4, Theorem 2.3(iii)], it
follows that the continued fraction converges for x € С - [0, oo). For positive x,
it is likely to diverge.
We are now able to properly interpret the word "nearly" in Entry 48, or, equiv-
alently, D8.2). Replacing the left side of D8.2) by a quotient of asymptotic series
as x tends to oo, with Re x > 0, we see, from D8.3), that the continued fraction
on the right side of D8.2) equals this quotient of asymptotic series in the sense of
correspondence of C-fractions.
Observe that, if и is an integer, the power series and continued fraction in D8.3)
each terminate. Thus, in such an instance, we have equality in D8.3) in the usual
sense.
Entry 49 (Formula B), p. 292). Let x and n denote complex numbers such
that Rex > 0, or such that Re* = 0 and 0 < I Im x\ < 1. Furthermore, let
у = (A +x2)l/2 - \)/xandm = иA + дг2Г1/2. Then
\-2x2 2•3x2 3¦4x2
n+ 4 + n + 6 + n + 8 + n +••
= у + т
m + 2k
32. Continued Fractions 73
Proof. Let x Ф 0. In Entry 22 of Chapter 12 (Part II [2, p. 136J), Ramanujan
offers a continued fraction for a certain quotient of ordinary hypergeometric series.
Setting о = и - 1,0 = о, and у = и, and replacing x by a/0 with |a/0| < 1, in
that theorem, we readily deduce that
-v,
2(u + v + l)a0
D9.1)
«0 - av + M0 - av + 1@ - a) + м0 - av + 2@ - a) H '
(This last result was also established by Preece [3].) We now set r = A + x2)'/2.
a = (r- I)/*2, 0 = (r + \)/x2, u = {pr + n)/Br), and v = (pr - n)/Br),
where p will be specified shortly. Then |a/0| = \(r — l)/(r + 1)| < 1, since
Re г = Re(l + x2)l/2 > 0, and D9.1) lakes the simplified form
-v,\;l+u;-a/p)
r+\
px
2(p + l)x2
2)x2
D9.2)
p + n + p +n + 2 + p + n + 4 + p + n + 6 H '
By a fundamental result on hypergeometric series (Bailey [ 1, p. 2, eq. B)]),
, u; 1 +u; -y2). D9.3)
-2Fl(\-v,\;]+u;-y2)= -A +y
Thus, D9.2) may be recast in the form
r + l
2F\(u + v, u; 1 + и; —y2)
px2 2(p+l)x2
D9.4)
P + n~/> + /l+Z~t~ р + П + 4 г p + n + O r'*'
We now put p = 2, and so и + v = 2. Since y(r + 1) = x, we find from D9.4)
that
yd+y2)
2F,B, m; 1 +u\-y2) =
2-3x2
. D9.5)
и - ' ' 2 + n + 4 + n + 6 + n +
An elementary calculation shows that
A -ы)A +v2JF|A,m; 1 +«;->>2) = A+/JF,B,m; 1 + м; -у2) - и.
It follows that
y(l + У2)
, u; 1 +u; -y2)
= У + —-— [У + -J y2iFdl, u; 1 + и; -у1)
D9.6)

74 Ramanujan's Notebooks, Part V
Combining D9.5) and D9.6) and then simplifying somewhat, we complete the
proof.
Entry 50 (Formula D), p. 292). Let x, p, and n be complex numbers such that
either Rex > 0, or Re x = 0 and 0 < | Im x \ < 1, or p is a nonpositive integer.
Furthermore, let у = {A +х2Уп - l)/x and let m = n{\ + *2Г1/2. Then
x \px2 2{p + l)x2 3(p + 2)x2
p + n + p + n + 2 + p + n + 4 +
n+6 +
Proof. Let x2 e С - (-oo, 0]. Multiply both sides of D9.4) by y. Note that
y(r + I) - x, и — (m + p)/2, and 1 + y2 = 2ry/x. After some elementary
algebraic simplification, we deduce Entry 50 in this first case. If, in addition, p
is a nonpositive integer, then both the continued fraction and series terminate.
We therefore have an identity between two rational functions of x for Re* > 0.
Hence, the identity holds for all complex x by analytic continuation, when p is a
nonpositive integer.
In his second letter to Hardy, Ramanujan [10, pp. xxix, 353] asserted that
.. a a2 BaJ CaJ
/о ' {A+а2)'/2 + И
which is a particular case of the continued fraction
a pa2 2(p + l)a2
p + n + p + n +2 + p + n+4 +¦¦'
which is a particular case of a corollary to a theorem on transformation of integrals
and continued fractions." In Ramanujan's Collected Papers 110, p. 353], the third
denominator above appears incorrectly as p 4- n + 3.
Now С. Т. Ргеесе [3, p. 99] showed that, for n, p > 0,
pa<-
\)a2
p + n + p + n + 2 +
= 2"a(\ + a2)(p-1)/2
'о (Ш
+a2I/2 -
32. Continued Fractions 75
To see that this result is equivalent to Entry 50, replace a by x and write the right
side above as
{(I + x2)^ + \}p Jo
_ 2»x(]+x2)<
XP
/•' t"-1+mdt
Jo (ThW
(-»kiphy* f\M
Jo
k=0
A!
Entry 51 (Formula A), p. 292). Let x and n be complex numbers such that
Re x Ф 0, or such that Re x — 0 and 0 < | Im x | < 1. Furthermore, let у =
{A + дс2)|/2 - l)/x and let m - n(\ + *2)~l/2, where the principal branch of
A +х2У'2 is chosen. Then
BxJ QxJ
1 +л + 3 + n + 5 + n + 1 + n +
'^m + 2k+\
An elementary calculation shows that |y| = 1 if and only if Rex = 0 and
|Im x\ > 1.The choice of the principal branch of A+jc2)i/2 ensures that |>>| < 1,
and so the series on the right side above converges.
Entry 51 is simply the case p = 1 of Entry 50.
Entry 52 (Formula C), p. 292). Let n and p denote complex numbers such that
either Re n > 0 or p is a nonpositive integer. Then
4(p + 3)
n + n +
+
+
Proof. Replace n by nx in Entry 50; thus, now m = nx{\ +x2)~>/2. We then find
that
x I px2 2(p+l)x2 3(p + 2)x2
p + nx + p + 2 + nx + p + 4 + nx + p + 6 + nx + ¦¦•
1 1 P 2Q> + 1) 3(p + 2)
n + p/x
2I x
4)/x
6)/x +¦¦¦
k\(m
2k)'
E2.1)
Now let x tend to oo. Then у tends to 1 and m approaches n. Thus, we see that
the left and right sides of E2.1) approach, respectively, the left and right sides of

76 Ramanujan 's Notebooks, Part V
Entry 52. To see that equality still holds, we apply the uniform parabola theorem,
just as we did in the proofs of Entries 24,14, and 19.
Entry S3 (p. 342). Let x and у be complex numbers with Re x > 0 and Re у > 0.
Then
x +
(y + IJ + n (y + 3J + n (y + 5J + n
2x
2x
2x
(x+lf+n (л: + ЗJ + п (х + 5J + п
= у+ Ту + 2у + 2у +•••
(у+1J + я (х + IJ + п (у + ЗJ + п (х + ЗJ + п
= х+ х + у+2 + х + у + 4 + дс + у + 6 + х+у + % +
We remark that, by symmetry, each of the continued fractions above is also
equal to
У +
3J+n (y + 3J
+ ) y
x + y + 2 + x+y + 4 + x + y + 6 + x+y + 8 +••
The first equality in Entry 53 is actually the same as Entry 27 of Chapter 12
(Part II [2, p. 146]), for a- > 0 and у > 0. As we remarked there, this elegant
identity is found in Ramanujan's [10, p. xxix] second letter to Hardy. The first
proof in print is by Preece [2], and the result can also be found in Perron's book
[1, p. 37, eq. C1)]. This result has also been proved by Ramanathan [6]. Since
both continued fractions converge locally uniformly for Re x > 0 and Re у > 0,
the identity follows by analytic continuation for Re x > 0 and Re у > 0.
The first continued fraction diverges for Re x = 0, while the second diverges
for Re у = 0. The identity does not hold if Re x < 0 and/or Re у < 0, since the
first continued fraction is an odd function of x but not of y, whereas the second is
an odd function of у but not of x.
The third (and fourth) continued fraction converges to a meromorphic function
of x and y, since it is equivalent to a continued fraction К(с*/1), where
fc2 1
C<~B^ = 4>0'
as* tends to oo.
Proof. As just indicated, it suffices to establish the second equality.
In Gauss's continued fraction, Entry 20 of Chapter 12 (Part II [2, p. 134]), we
set x = 1 and then replace a, fi, and у by (х — n — l)/2, (x + n — l)/2, and
32. Continued Fractions 77
(л + у)/2, respectively. After some simplification, we find that
2fi (g(y + 1 +я), \(x - 1 +n); [(x+y); -l)
X + У * {{( +
У
(i(y + l+/j), |(jr + l +n);i(x+y + 2);-l)
(у+\)г-п2 (x + \J-n2 (y + 3J - n2
дс+у + 2 + дс+у+4 + x+y + 6 E3])
2 n2
(x + 3J - n
+ x + y+8 H '
Second, we use Euler's continued fraction, Entry 22 of Chapter 12 (Part II [2, p.
136]), when x = 1 and a, /3, and у are replaced by-(у + l+n)/2, (*- 1 +n)/2,
and {x + y)/2, respectively. Upon simplification, we deduce that
2F| (|(y + 1 + n), \(x - 1 + я); \{x + y); -l)
+ У)-
2
= 2y
lJ-n2 {x + 3J - n2 (x + 5J - n2
Ь + 2y + 2y +
E3.2)
Comparing E3.1) and E3.2) and replacing n by iy/n, we deduce the second
equality of Entry 53.
It is interesting to note that the third continued fraction in Entry 53 can be
obtained from the second one by repeated applications of the Bauer-Muir trans-
transformation with modifying factors wk = л - у + 2k. Also, the first continued
fraction can be obtained from the second one by repeated use of equally simple
Bauer-Muir transformations.
Entry 54 (p. 342). For all complex numbers x and n,
1 v^ (-*)* 1 x 1 x 2 x 3
2x
3x
n+x — n + x + 1 — n + x + 2 — n+x + 3 ~ ¦¦
Proof. The latter continued fraction in Entry 54 is merely the even part of the
former continued fraction, a fact immediately seen from F4.1).
To establish the first part of Entry 54, replace * by x/f) and set у = п in
Part II [2, p. 134, Entry 21, eq. B1.2)]. Since the continued fraction converges
uniformly with respect to p in a neighborhood of p = oo, we may let /3 tend to
oo to complete the proof. Alternatively, we can replace x by — x in the second
continued fraction of Corollary 1 of Entry 21 of Chapter 12 (Part II [2, p. 136]) to
immediately achieve the desired result, since both continued fractions converge
for all x and n.

78 Ramanujan's Notebooks, Part V
Entry 55 (p. 343). For every complex number x,
x л1_х2хЗл
l-e~*~ 1 + 1 + 1 + 1 + 1 + 1 + 1 +
Proof. Setting n = 1 in Entry 54, we deduce that
1 ~ (_*)* A (-x)k
1 -e~
x
x k=l к.
1 x 1 * 2 x 3
T+T+T+T+T+ i + !+
Taking the reciprocal of both sides, we complete the proof.
Entry 56 (p. 342). For all complex x,
X X X X X
Proof. If we set и = 1 and replace x by 2x in the first continued fraction of
Corollary 1 of Entry 21 of Chapter 12 (Part II B, p. 136]), we find that, for all
complex Jt,
x 2x 2x 4x Ax 6x 6x
T-T + T-T + T-T + T
x x x x x x x
Entry 56 also readily follows from a continued fraction for ez found in Wall's
book [1, p. 348].
Entry 57 (p. 343). For all complex numbers x and n,
n x n x n+1 x и + 2
x-n
Ё
1 + 1 + 1+ 1 +1+
Proof. By a straightforward calculation, it is easily shown that the left side of
Entry 57 is equal to
F{x).- * iFiB;n + 2;Ac)
32. Continued Fractions 79
To show that F(x) has the given continued fraction, we require the continued
fraction
2F,(a,P;y;x)
_ 1
P(y-a)x
- 03 - a + l)x + у + 2
(p + 2)(y -a + 2)x
E7.1)
- (P - a + 2)x + у + 3 - •
due to E. Frank [1] and valid for \x \ < 1. Replacing x by *//?, letting /9 -*¦ cx>,and
using the fact that the resulting continued fraction in E7.1) converges uniformly
with respect to p in a neighborhood of P = oo, we find that, for all x,
(y-a)x (y-a + l)x (y-a + 2)x
y-x + y + \~ x + y + 2 - х+у + Ъ
Putting a = 1 and у = n + 1, we deduce that, for all x,
(n + l)x (n + 2)x
nx
1 1
— F(x) —
x n + 1 - x +n +2 - x +n + 3 - x + n + 4 -
By F4.1), this last continued fraction is the even part of
CF(x)-- П Х
W~ 1 + 1 + 1 +
1 + 1 + 1 + ••
It remains to show that CF(x) converges to F(x)/x.
The odd part of CF(x) is
1 л (п + \)х (n + 2)x (n + 3)x
Т-дс+n+l -x+n+2-x+n+3-x+n + 4 '
which converges for all x and n. Thus, the even and odd parts of CF(x) both
converge to meromorphic functions of x and n. The even part converges to F(x )/x,
and so we want to show that the odd part also converges to F(x)/x. Now CF(x)
and thus the even and odd parts converge to the same values for jt > 0 and n > 0.
Therefore, by analytic continuation, they are equal for all * and n. This completes
the proof.
Frank's continued fraction E7.1) can, in fact, be derived from Euler's continued
fraction, Entry 22 of Chapter 12 (Part II [2, p. 136]).
Entry 58 (p. 343). Let xbea complex number such that Re x2 > -1. Then
4x2 4A +x2)
+ ~T + ' i н—'
sinh ' x
x 2x2 2A + jt2)
I + ~T +
1

80 Ramanujan's Notebooks, Part V
Proof. Using a familiar transformation for 2F, (Bailey [1, p. 2, eq. 2]), we find
that (Gradshteyn and Ryzhik [1, p. 60])
i i; §; -x2) = x(l +^I/22F,A> 1; \; -x2).
sinh x = x 2F,(i, i; §; -x2) = x(l +^)
We now apply Entry 21, eq. B1.2), of Chapter 12 (Part II [2, p. 134]) with 0 = 0,
у = 1/2, and x replaced by x2. Hence, for Re x1 > - 5,
sinh x
(l+x2)'/2
x/2 л
3A
~ 1/2 + 1 + 1/2 + 1 + 1/2 + 1 + 1/2 H '
which is easily seen to be equivalent to the proposed continued fraction.
Entry 59 (p. 343). Let x be any complex number such that Rex2 > -5. Then
, x *2 W+x2) ^1 4(\+x2)
, _ 2
tatl X~~\ + 1
2
~\ + 1 + 1 +1+ 1
Proof. We know that (A. Erdelyi [1, p. 102])
ил~1 x = x
We again apply Entry 21, eq. B1.2), of Chapter 12 (Part II [2, p. 134]) but now
i i replaced by x2. Thus, for Re хг > - \,
with ?=-i,y = i,
' V _
c/2 x2/2 l+x2 Ъх2/2 2A +x2)
X~ 1/2 + 1 + 1/2 + 1 + 1/2 +•¦'
which is equivalent to the proposed continued fraction.
5. General Theorems
Entry 60 (p. 339). For 1 < к < n, assume that ak Ф 0. Then
ak
~ a,,-\ + a,,
F0.1)
In fact, we have stated a finite version of Ramanujan's claim, i.e., Ramanujan's
statement is for "n = 00."
Proof. Entry 60 is easily established by induction on n. In fact, Entry 60 is a
version of an identity
" b\ b2 Ьз bn
l—j l — 1 -+- b2 ~ 1 + b$ ~ '' ¦ ~ 1 + on
F0.2)
32. Continued Fractions 81
due to Euler (Jones and Thron [1, p. 37)), where bk Ф 0, 1 < к < п. То derive
F0.1) from F0.2), set b\ = \/a\ and bk = ak-Jak, 2 < к < п. After a simple
equivalence transformation, we deduce F0.1).
In the following three entries, Ramanujan examines the convergence and diver-
divergence of limit i-periodic continued fractions of the form
a^ 02 03
P + P + P +•'
F1.1)
where Нт„_юо akn+J = a* 6 R, for 1 < j < k. The convergence behavior for the
special periodic case, akn+J = a*, 0 < n < 00, has been known since the 1880s
(O. Stolz Jl], Jones and Thron [1, p. 46]). If we think of the continued fraction
F1.1) as being generated by the linear fractional transformations
p + w
n = 1,2,3,
such that
Sn(w) := S[ оs2 о ¦ ¦ ¦ оsn(w)
А„
Bn
F1.2)
we may deduce the following information:
F1.3)
A) For к = 1, the approximants Sn @) of the continued fraction
a a a
p + p + p + ¦¦¦
are just iterations of the linear fractional transformation s\ (ш) evaluated at w — 0.
Hence, F1.3) converges if si(w) has one attractive fixed point and it has no
repulsive fixed point at 0. The fixed points of s\ (w) are p(,±y/\ +4a/p2 - l)/2.
Hence, F1.3) converges if and only if
4a/p2 €<C- (-oo.-l). F1.4)
B) For fe > 1, we regard the periodic continued fraction as iterations of Sk(w),
given by F1.2), evaluated at the points
0,e, =
F1.5)
Hence, the Л-periodic continued fraction converges if and only if Sk(w) has an
attractive fixed point and it has no repulsive fixed points at any of the points F1.5).
It was first pointed out by T. N. Thiele [1 ] in 1879 that if Sk (w) has a repulsive fixed
point at one of the points F1.5), then the periodic continued fraction diverges. This
phenomenon is therefore called Thiele oscillation (Perron [ 1, p. 87]).
For more details, we refer to Jones and Thron's book [1, p. 47]. The results
quoted above were probably known to Ramanujan who most likely derived them
himself, because they are not found in the books of G. Chrystal [1] or G. S.
Carr [1], the two primary sources of information about continued fractions for

82 Ramanujan's Notebooks, Part V
Ramanujan. He then must have realized that he could generalize these results to
limit it-periodic continued fractions, and Entries 61 and 63 below are the results
of his investigations. His first result is on limit 1-periodic continued fractions.
Entry 61 (p. 339).
1 a] a2 ay a4
T-7-7-7-7—
E intelligible or not according as limn_»ooan < or > 1/4.
F1.6)
Here we have precisely quoted Ramanujan. By "intelligible," Ramanujan evi-
evidently means "convergent." In the periodic case а„ = a, Entry 61 is true, since
the condition F1.4) then reduces to a e С — (i, oo). It is also true that the limit
periodic continued fraction converges if a € С — Ц, oo). This was first proved by
E. B. Van Vleck [1] and is beautifully presented in Perron's book [1, p. 93]. In the
case lim,,-»oo a,, = \, the point wisely omitted by Ramanujan, the continued frac-
fraction may converge or diverge, according to how а„ tends to \. But what happens
if Ишп-юо an = a > д ? It is easy to prove that if an tends to a "fast enough," then
the continued fraction F1.6) diverges (J. Gill [1]). That it may converge otherwise
was also shown by Gill [1], if one allows complex elements а„.
In our Memoir [2] with Andrews, Jacobsen, and Lamphere, we asked if there
exist convergent continued fractions F1.6) with an > 0 and lim,,-*,*, an = a> \.
In a lecture at a conference on continued fractions in Trondheim, Norway, on May
31, 1997, L. J. Lange answered this question. In particular, in 1985, he and N. J.
Kalton [1, Theorem 8.1 ] had proved the following theorem.
Theorem61.1. Ifan is real, М-
ч„ = a > \, and
-an\ < oo,
then F1.6) diverges.
Moreover, if the last hypothesis above is dropped, Kalton and Lange [1, Theo-
Theorems 6.1,6.2] found specific classes of sequences {а„) for which F1.6) converges.
For further results relevant to our question, see Theorems 2.1 and 3.2 of their pa-
paper [1]. Lange also raised a more difficult question. If a is any real number such
that a > 5, does there exist a real sequence {а„} such that an -*¦ a and F1.6)
converges?
D. Masson [1 ] has communicated to us another theorem relevant to our original
question in the Memoir [2]. Using Pincherle's theorem, he has shown that F1.6)
diverges provided that а„ is real and
an = a A + g(n) + o(g(n))),
32. Continued Fractions 83
as n tends to oo, where a > |, and where g(n) is any positive function monoton-
ically decreasing to 0 as n —*¦ oo, for example, g(n) = cn~a, for some constant с
and positive number a.
For the next entry, we again quote Ramanujan.
Entry 62 (p. 340). The continued fraction
1 a\ ai ai
p + p + p + p +•¦•
tends to two limits or one limit according as ? l/y/аЦ is convergent or divergent.
F2.1)
Ramanujan evidently considered an, 1 < n < 00, to be positive and p to be
real. From Stieltjes' classical work [2], [3, pp. 402-566], it follows that F2.1)
converges if and only if
, ai<23 • «2/1-1
Д2Д4 ¦ ¦ ¦
= oo;
F2.2)
otherwise, its even and odd parts converge to two distinct values. This coincides
with the natural interpretation of Entry 62, except for one matter; the condition
F2.2) is not equivalent to Ramanujan's condition
F2.3)
n=l
unless one makes further restrictions. Indeed, F2.3) is a sufficient condition for
the convergence of F2.1) (Perron [ 1, p. 47]), but there exist convergent continued
fractions F2.1) with an > 0, p > 0, and ? l/V^T < 00• For instance, the
continued fraction
1* 1" 34 3" 54 5^
converges since
Ramanujan was not the only person to have made this mistake; for example, see
Khovanskii's book [1, p. 45J.
Entry 63 considers limit it-periodic continued fractions for 1 < к < 5. We
state a rather general version, although Ramanujan probably examined only real
continued fractions.
Entry 63 (p. 340). Consider
at,
CF- 1-1-7-1 -••¦
A) ifCF is limit 1-periodic, then CF converges (/
Ц.оо).
!,, = а е С -

84 Ramanujan's Notebooks, Part V
B) IfCF is limit 2-periodic with limits a and b, then CF converges if
where a + b Ф 1.
C) IfCF is limit 3-periodic with lim,,^^, aj,,+ i = а, Нт„_юо яэл+2 = Ь, and
= c, then CF converges if
abc _ ,
where a +b + с: ф 0, and if\a\ > \c\ whenb = 1, |b| > |a| when с — 1,
a«d |c| > \b\ when a — 1.
D) Suppose that CF is limit 4-periodic with a^+i, а*„+2> fl4*+3. and a^
tending to a, b, c, andd, respectively, as n tends to oo. Then CF converges
if
{1 - (a + b + с + d) + {ac + bd)J € ~l4'OC>'
where a+b+c+d— ac — bd ф 1, and if\ab\ > \cd\ when b + с = 1,
\bc\ > \ad\ whenc + d = \,\cd\ > \ab\whena+d — l,and\ad\ > \bc\
when a + b = 1.
E) Suppose thatasn+\,asn+2, asn+i,ain+4, and а$„ approach a, b,c, d, and
e, respectively, as n tends to oo. Then CF converges if
abcde ,
{1 - (a + b + с¦+ d + e) + a(c + d) + b(d + e) + се}2 в ~[*'°°''
where the denominator above is not equal to 0, and if
\abO -d)\ > \de(l -b)\,when b + c + d-bd=\,
\bc(l - e)\ > |ea(l -c)\,when с + d + e - ce = 1,
\cd(\ -a)\ > \ab(\ -d)\.when d + e + a-da=l,
\de(\ - b)\ > \bc(\ - e)\,when e + a+b-eb- 1,
and
\ea(\ -c)\ > \cd(\ -a)\,when a + b + c-ac=h
The "extra" conditions on the parameters a,b,... are not given by Ramanu-
jan. Thus, for example, in C), Ramanujan says that CF "is intelligible when
A - (a + b + c)J — 4abc is positive." The primary conditions insure that 5*(u»)
has an attractive fixed point and a repulsive fixed point. The "extra" conditions
eliminate the cases where the corresponding jfe-periodic continued fraction di-
diverges by Thiele oscillation.
Such limit ^-periodic continued fractions were first studied by M. von Pidoll
[1] in 1912 and by O. Szasz [1] in 1917. It is interesting to note that Ramanujan
probably made his discoveries in the period 1912-1914.
32. Continued Fractions 85
Note that part A) follows from Entry 61 and the remarks we made following it.
Proof. The corresponding periodic continued fraction is given by
—a b с
"T - I - I '
The corresponding linear fractional transformation
—a b * Ak ¦+¦ Ak-\iv
1 — 1 — • ¦ • — 1 + w Bk + Bk-\W
has an attractive fixed point and a repulsive fixed point if either
(a)
Bk
and
arg 1 +
BkJ
or
(b)
,_,=0 and
(See, for instance, the book by Jones and Thron [ 1, pp. 51-52].) (If Sk is singular,
i.e., one or more of the elements a,b,... are equal to 0, then 5* is a constant
function whose value we regard as the attractive fixed point of 5*. Its "repulsive
fixed point" is then the point w for which S* is not well defined. For more details,
we refer to Jacobsen's paper [3].)
From the work of von Pidoll [1], Szasz [1], and, in more generality, Jacobsen
[3], it further follows that if condition (a) or (b) holds and none of the points F1.5)
is the repulsive fixed point of 5* (Thiele oscillation), then the limit it-periodic
continued fraction converges. The results in Entry 63 arise from the application
of these criteria when к = 1, 2, 3, 4, 5.
As noted earlier, the case к — 1 was examined in Entry 61.
Let к = 2. Then
Si(w) =
For case (a) we require that Bt = 1
4(Л2В,-Я2А|)
-a{\ + w)
0, A| + B2 = 1 - a - b Ф 0, and
arg 1 +
B2J
i.e., ab/{\ - (a + b)J € С - Ц, oo). Note that case (b) is impossible.
If one of the fixed points in F1.5) is the repulsive fixed point of S2(w), then this
fixed point is either 0 or-a. Now Ш[ = OisafixedpointofS2(w)if A2 — -a =0.
Then
„ . , 0
1 -b + w
is singular, and w\ = 0 is the repulsive fixed point if and only if 1 — b = 0. But
this contradicts the requirement 1 - a - b ф 0. Thus, 0 is not a repulsive fixed
point. If w\ = —a is a fixed point of 5з(ш), then b — 0. It is easily seen that the

86 Ramanujan's Notebooks, Part V
other fixed point of S2(w) is w2 = -I.Nowum = -a is the attractive fixed point
of S2(w) (and not the repulsive one) if and only if
(Jones and Thron [1, p. 52]), i.e., if and only if |1 - a\ > 0. This is true since
1 - а Ф 0. Thus, им = -a also is not a repulsive fixed point, and there is no
Thiele oscillation. This completes the proof of B).
Next, let к - 3. Then
a b с А\ + A2w
Si(w) = -- _ - _ ^^ = + B^ ¦
where A3 = -a(l - c), A2 = -a, B3 = 1 - b - c, and B2 = 1 - b, by the
recursion formulas @.5) and @.6), or by direct calculation. For condition (a), we
require B2 = 1 - b Ф 0, i.e., b Ф 1; A2 + B3 = 1 - (a + b + с) Ф 0, i.e.,
+ b + c ф 1; and
a be
6C- ,i
F3.2)
For case (b), we require B2 = 1 - b = 0, i.e., fc = 1, and |A2| ф |B3|> i-e.,
|a| ^ |1 -b- c\ - \c\. Now with b = 1
abc
ас
°
( " (a+cJ 6
if and only if |e| Ф \c\. Hence, S3 has an attractive and a repulsive fixed point (in
the extended sense if S3 is singular, i.e., if abc = 0) if F3.2) holds.
We next need to determine the conditions that yield repulsive fixed points. If
one of the fixed points in F1.5) is the repulsive fixed point of S3(w), then this
fixed point is either 0, -a, or -a/(l - b).
First, w 1 = 0 is a fixed point of S3(u>) if S3@) = A$ = 0, i.e., if-aA -c) = 0.
Case 1. a = 0. Then
0
-b)w
is singular, and им = 0 is "the repulsive fixed point" if and only if 1 - b - с = 0.
But this is impossible by F3.2).
—aw
Case2. c= \,a ^O.Then
and the other fixed point of 53(w) is given by w2 = (b - a)/(l - b). Hence, in
analogy with F3.1), w\ = 0 is the attractive fixed point of S3(w) (and not the
repulsive one) if and only if
|fl3
32. Continued Fractions 87
(Jones and Thron [1, p. 52]), i.e., \b\ > \a\. (The case \Bi + B2vol\ = \ВЪ +B2w2\
is excluded by F3.2).) In conclusion, under the condition F3.2), if с = 1, we
need to require that \b\ > \a\ for the convergence of CF.
Next, w3 = —a is a repulsive fixed point of S3(u>) if and only if w\ — 0 is a
repulsive fixed point of
„(О. -Ьс а
So, by symmetry, we need to require that \c\ > \b\ if a = 1 in order for CF to
converge. Likewise, u>3 = — a/(\ — b) is a repulsive fixed point of 53(ш) if and
only if w\ = 0 is a repulsive fixed point of
_m. . —c a b
which yields the requirement \a\ > \c\ if b = l.This concludes the proof of C).
Cases к = 4 and fc = 5 are proved in exactly the same manner as case к = 3.
However, with increasing k, the details become more laborious. For these reasons,
we shall provide only brief sketches of the proofs when к = 4 and к = 5.
Let к = 4. Then
A4
S4(u0 =
whereA3 = a(c-l), A4 = a(c+d-\), JJ3 = l-b-c,i
For condition (a), we require that b + c ф 1,1 — (a +b + с + d) +ac + bd ф0,
and
abed
(\-(a+b + c+d) + (ac + bd)I G<C ['4'0°)'
In case (b), we need b + с = 1 and |afo| ^ \cd\. Now when b + c = I,
afeed afood
F3.3)
- (a+b
(.ac+bd)}2 = {ab + cd\2
if and only if \ab\ ф \cd\. Hence, 54 has an attractive fixed point and a repulsive
fixed point if F3.3) is valid.
We next need to determine when the points F1.5) are repulsive fixed points.
Now w\ = 0 is a fixed point if A4 = a(c + d - 1) = 0. It is easy to see that the
case a = 0 is impossible. If с + d — 1, then
54(u)
fl4
has the fixed point u>2 = (Л3 - J?4)/B3. Thus, in analogy with F3.1), им is not a
repulsive fixed point if and only if
|B4I > \ВЛ
i.e., if and only if \bc\ > \ad\.

88 Ramanujan's Notebooks, Part V
The remaining three conditions listed in Entry 63 for the case к = 4 arise from
the remaining three possible repulsive fixed points in F1.5) and considerations of
symmetry.
LetJt = 5. Then
where A4 = a(c + d- l),A5 = a(c + d + e-ce-I), B4= 1 -b-c -d + bd,
and Bs = 1 - b - с - d - e + ce + be + bd. For condition (a), we require that
and
abcde
т eC-[i,oo). F3.4)
fl - (a + b + c + d + e) +a(c + d) + b(d + e) + ееJ 4
Forgoing the calculations for condition (b), we conclude that 55 has an attractive
fixed point and a repulsive fixed point if F3.4) holds.
We now examine the five possible repulsive fixed points given by F1.5). Now,
u>i = 0 is a fixed point of Ss(w) if S5@) = A5 = a(c + d + e + ce - 1) = 0.
The case a — 0 is impossible, and so we assume that c + d + e+ce—\ =0. The
remaining fixed point of S5 (w) is (A4 - B5)/ B4. It follows that wx is not a repulsive
fixed point if and only if | B51 > |/44|, i.e., if and only if \bc(\ -e)\ > \ae(\ -c)\.
The remaining four possible repulsive fixed points yield the additional restrictions
listed for the case к = 5 of Entry 63.
It may be remarked that the "extra" conditions in Entry 63 can be eliminated if
we use the notion of general convergence (Jacobsen [2]).
Entry 64 (p. 342). Ifn is even, then
+
+*„
a2
- ап-\Ь„
Ьп-\Ь„)
This is just the finite form of the even part of an infinite continued fraction,
namely (Jones and Thron [1, p. 42]), the even part of
a, a2
is
a\b2
a2
+ b2ia4
F4.1)
33
Ramanujan's Theories of Elliptic
Functions to Alternative Bases
1. Introduction
In his famous paper [3], [10, pp. 23-39], Ramanujan offers several beautiful series
representations for 1/л\ Не first states three formulas, one of which is
where (a)o = 1 and, for each positive integer n,
(a)n =a(a
2)-(a+n- 1).
He then remarks that "There are corresponding theories in which q is replaced by
one or other of the functions
<?, = exp (-nsf2K\/Kx) , q2 = exp (-2лК'г
where
(i, |;
Here K'j = Kj(k'), where 1 < j < 3, k' = УГП^, and 0 < it < 1; it is
called the modulus. In the classical theory, the hypergeometric functions above
are replaced by 2^1E, |; 1; k2). Ramanujan then offers 16 further formulas for
\/jt that arise from these alternative theories, but he provides no details for his
proofs. In an appendix at the end of Ramanujan's Collected Papers [10, p. 336],
the editors, quoting L. J. Mordell, lament "It is unfortunate that Ramanujan has
not developed in detail the corresponding theories referred to in 'ЦН."
Ramanujan's formulas for 1 /я were not established until 1987, when they were
first proved by J. M. and P. B. Borwein [1, pp. 177-188], [2], [3]. To prove these
formulas, they needed to develop only a very small portion of the "corresponding
theories" to which Ramanujan alluded. In particular, the main ingredients in their

90 Ramanujan's Notebooks, Part V
work are Clausen's formula and identities relating 2^1 E, 5; I; x), to each of
the functions 2F, (j, 3; 1;дс), 2F, (±, §; 1; *), and г/7, (±, §; 1; дс). The Bor-
Borweins [4], F] further developed their ideas by deriving several additional for-
formulas for 1 /тт. Ramanujan's ideas were also greatly extended by D. V. and G.
V. Chudnovsky [1], [2] who showed that other transcendental constants could
be represented by similar series and that an infinite class of such formulas ex-
existed.
Ramanujan's "corresponding theories" have not been heretofore developed. Ini-
Initial steps were taken by K. Venkatachaliengar [ 1, pp. 89-95] who examined some
of the entries in Ramanujan's notebooks [9] devoted to his alternative theories.
The greatest advances toward establishing Ramanujan's theories have been
made by J. M. and P. B. Borwein [5]. In searching for analogues of the classi-
classical arithmetic-geometric mean of Gauss, they discovered an elegant cubic ana-
analogue. Playing a central role in their work is a cubic transformation formula for
2^1 E, |; 1; x), which, in fact, is found on page 258 of Ramanujan's second note-
notebook [9], and which was rediscovered by the Borweins. A third major discovery
by the Borweins is a beautiful and surprising cubic analogue of a famous theta-
function identity of Jacobi for fourth powers. We shall describe these findings in
more detail in the sequel.
As alluded in the foregoing paragraphs, Ramanujan had recorded some results
in his three alternative theories in his second notebook [9]. In fact, six pages, pp.
257-262, are devoted to these theories. These are the first six pages in the 100
unorganized pages of material that immediately follow the 21 organized chapters in
the second notebook. Our objective in this chapter is to establish all of these claims.
In proving these results, it is very clear to us that Ramanujan had established further
results that he unfortunately did not record either in his notebooks, unpublished
papers, or published papers. Moreover, Ramanujan's work points the way to many
additional theorems in these theories, and we hope that others will continue to
develop Ramanujan's beautiful ideas.
The most important of the three alternative theories is the one arising from the
hypergeometric function 2 F, E, |; 1; дс). The theories in the remaining two cases
are more easily extracted from the classical theory and so are of less interest.
We first review the classical terminology and theory, which can be found in Part
III [3]. In particular, see Chapter 16, pages 34-37, Chapter 17, pages 101-102,
and Chapter 18, pages 213-214.
The complete elliptic integral of the first kind К = К (к) associated with the
modulus jfc, 0 < it < 1, is defined by
лп/2
JO
d<p
A, I; 1; Jfe2),
A.1)
/o y/l — k2 sin2 <p
where the latter representation is achieved by expanding the integrand in a binomial
series and integrating termwise. For brevity, Ramanujan sets
г :=^К=
A.2)
33. Elliptic Functions to Alternative Bases 91
The base (or nome) q is defined by
q :=
where К' = К (к'). Ramanujan sets x (or a) = k2.
Let n denote a fixed positive integer, and suppose that
A.3)
A.4)
whereO < k,? < I. Then a modular equation of degree n is a relation between the
moduli к and I which is implied by A.4). Following Ramanujan, we put a = k2
and & = I2. We often say that 0 has degree n, or degree n over a. The multiplier
m is defined by
(I, I;!;
A5)
We employ analogous notation for the three alternative systems. The classical
terminology described above is represented by the case r = 2 below. For r =
2, 3, 4,6 and 0 < x < 1, set
z(r):=z(r;x):=
and
qr -=qr(x)
In particular,
r r )
:= exp [ -я cscGr/rJ ' r' / ' '
\ 2М±.?=1;1;*)
A.6)
A.7)
A.8)
and
= exp j -2л -
A.9)
(We consider the notation A.7H1.9) to be more natural than that of Ramanujan
quoted at the beginning of this chapter.)
Let n denote a fixed natural number, and assume that
^; !;!-«) 2F, A, c=i. 1; , _
A10)

92 Ramanujan's Notebooks, Part V
where r = 2,3, 4, or 6. Then a modular equation of degree n is a relation between
a and fi induced by A.10). The multiplier m(r) is denned by
(lid
forr = 2, 3,4, or 6. When the context is clear, we omit the argument r in qr,z(r),
and от (r).
In the sequel, we say that these theories are of signature 2, 3, 4, and 6, respec-
respectively.
Theta-functions are at the focal point in Ramanujan's theories. His general
theta-function f(a, b) is defined by
f{a,b) :=
If we set a = <7<?2lz, fr = qe~2iz, and tj = e"'r, where г is an arbitrary complex
number and Im(r) > 0, then f(a,b) = i?3(z, r), in the classical notation of
Whittaker and Watson [1, p. 464]. In particular, we utilize three special cases of
f(a, b), namely.
A.12)
A.13)
n=0
and
f(-q) := f(-q, -q2) = f] (-1)VC"
+1)/2 =
(U4)
where \q\ < 1. The last equality above is Euler's pentagonal number theorem,
which is most easily derived from Jacobi's triple product identity (Part III [3, p.
35, Entry 19]).
One of the fundamental results in the theory of elliptic functions is the inversion
formula (Whittaker and Watson [1, p. 500]; Part III [3, p. 101, eq. F.4)])
г= 2F,(I.±;1;*)=V(*). A-15)
We set
for each positive integer n, so that z\ = z- Thus, by A.5), A.15), and A.16),
<P2(q)
Z\
m = — =
<p4qn)
A.17)
In the sequel, unattended page numbers, particularly after the statements of
theorems, refer to the pagination of the Tata Institute's publication of Ramanujan's
33. Elliptic Functions to Alternative Bases 93
second notebook [9]. We employ many results from Ramanujan's second notebook
in our proofs, in particular, from Chapters 17, 19,20, and 21.
2. Ramanujan's Cubic Transformation, the Borweins' Cubic
Theta-Function Identity, and the Inversion Formula
In classical notation, the identity
is Jacobi's famous identity for fourth powers of theta-functions. In Ramanujan's
notation A.12) and A.13), this identity has the form (Part HI [3, p. 40, Entry
<p\q) = <p\~q) + \6q*\q2). B.1)
The Borweins [5] discovered an elegant cubic analogue which we now relate. For
со = expB7ri/3), let
a(q) :=
b(q):=
m,n=—oo
m.n=~ oo
B.2)
B.3)
and
m,n=-oo
Then the Borweins [5] proved that
a\q) = b\q) + c\q).
They also established the alternative representations
and
B.4)
B.5)
<z6)
. B.7)
Formula B.6) can also be found in one of Ramanujan's letters to Hardy, written
from the nursing home, Fitzroy House [ 11, p. 93], and is proved by us in [9]. The
identity B.7) is found on page 328 in the unorganized portions of Ramanujan's
second notebook and was proved in Part III [3, p. 462, eq. C.6)] in the course of
proving some related identities in Section 3 of Chapter 21 in Ramanujan's second
notebook. Furthermore, the Borweins [5] proved that
b(q)= ± {3a(q3) - a(q)} B.8)

94 Ramanujan's Notebooks, Part V
and
B.9)
The Borweins' proof of B.5) employs the theory of modular forms on the group
generated by the transformations / -> 1// and t -> / + jV3. Shortly thereafter,
they and F. G. Garvan [1] gave a simpler, more elementary proof that does not
depend upon the theory of modular forms. Although Ramanujan does not state
B.5) in his notebooks, we shall show that B.5) may be simply derived from results
given by him in his notebooks. Our proof also does not utilize the theory of modular
forms.
We first establish parametric representations for a(q), b(q), and c(q).
Lemma 2.1. Letm — zi/гз, as in A,17). Then
_m2 + 6m - 3
a(q) =
b(q) =
and
C-
3(m
4m
m)(9 - m
4m2»
+ \)(m2 -
2I/3
¦ II'3
4m
B.10)
B.11)
B.12)
Proof. From Entry 1 l(iii) of Chapter 17 in Ramanujan's second notebook (Part
HI [3, p. 123]),
f{q2) = iV^(«/<?)'/4 and \K<?')=;sVi7W<73I/4, B-13)
where fi has degree 3 over a. In proving Ramanujan's modular equations of degree
3 in Section 5 of Chapter 19 of Ramanujan's second notebook, we [3, p. 233, eq.
E.2)] derived the parametric representations
(m-l)C+mK
a =
16m3
and
_ (m- lKC+m)
16m
Thus, by A.16), B.7), B.13), B.14), and B.15),
B.14)
B.15)
Am
.m2 + 6m - 3
4m" '
and so B.10) is established. (In fact, B.10) is proved in Part III [3, p. 462, eq.
C.5)].)
33. Elliptic Functions to Alternative Bases 95
Next, from B.7) and B.8)
2b(q) =
and, from B.7) and B.9),
By Entry l(iii) of Chapter 20 (Part III [3, p. 345J), A.16), and A.17),
and
By Entry l(ii) of Chapter 20 (Part III [3, p. 345]) and B.13)-B.15),
(m+3I/3
and
1 -
xjrHq2)
B.16)
B.17)
B18)
B.20)
Using B.13) and B.18)-B.21) and then B.14) and B.15) in B.16) and B.17),
we deduce that, respectively,
2b(q) =
(m + 3J/1
= VZ1Z3
(9-m2)'/3 (m-
тг'ъ 4w~
C-m)(9-m2)'/3
2m2/3
(m + ЗJ/3

96 Ramanujan's Notebooks, Part V
and
2c(q) =
= VZ1Z3-
- II'3 +
3(m2 - l)l/3(m
2m
Hence, B.11) and B.12) have been established.
Theorem 2.2. 77ie cubic thela-function identity B.5) holds.
Proof. From B.11) and B.12),
*з(q) + сз( } = <?l?^! (wC _ m)J(9 _ W2} + 27(m
64
= ?<„,+вИ| 3) «(,).
64m3
by B.10). This completes the proof.
Our next task is to state a generalization of Ramanujan's beautiful cubic trans-
transformation for2^1E. |; u*).
Theorem 2.3. For \x\ sufficiently small,
= A+2jc)v 2F,
1 3c + 5
-; ——
;*3J .
B.22)
Proof. Using MAPLE, we have shown that both sides of B.22) are solutions of
the differential equation
_ A + 2x)[Djt3 - l)Cc + 2x + 1) + 18«]y' - 6cCc + 1)A - xfy = 0.
This equation has a regular singular point at x = 0, and the roots of the associated
indicial equation are 0 and Cc - 1 )/2. Thus, in general, to verify that B.22) holds,
we must show that the values at x = 0 of the functions and their first derivatives
on each side are equal. These values are easily seen to be equal, and so the proof
is complete.
33. Elliptic Functions to Alternative Bases 97
Corollary 2.4 (p. 258). For \x\ sufficiently small,
Proof. Set с = 3 in Theorem 2.3.
The Borweins [5] deduced Corollary 2.4 in connection with their cubic analogue
of the arithmetic-geometric mean. Neither their proof nor our proof is completely
satisfactory, because they depend upon prior knowledge of the identity and differ-
differential equations. Recently, H. H. Chan [4] has given a considerably more natural
proof that depends upon rederiving some of the results in Sections 4-6 of this
chapter without appealing to the theorems here in Section 2. The Eisenstein series
M(q) and N(q), defined at the beginning of Section 4, play key roles. Chan [4]
has also shown that Ramanujan's cubic transformation can be derived from two
cubic transformations due to E. Goursat [1 ].
Our next goal is to prove a cubic analogue of A.15). We accomplish this through
a series of lemmas.
Lemma 2,5. Ifn = 3m, where m is a positive integer, then
2.M b\q)\ a(q)
Proof. Replacing x by A - x)/(l + 2x) in B.23), we find that
Setting x = b(q)/a(q) and employing B.8) and B.9), we deduce that
a(q)-b(q)
a(q) + 2b(q)
ir\
(\ 2
\3' 3' '
= ^2F,(-,-;l;
a(q) ,
,3*3* '
by Theorem 2.2. Iterating this identity m times, we complete the proof of B.24).
The next result is the Borweins' [5] form of the cubic inversion formula.
Lemma 2.6. We have
B.26)

98 Ramanujan's Notebooks, Part V
Proof. Letting m tend to oo in B.24), noting that, by B.2) (or B.6)) and B.3) (or
B.8)), respectively,
lim a(q") = 1 = Hm b(q"),
л->оо n-*oo
and invoking Theorem 2.2, we deduce B.26) at once.
Lemma 2.7. Ifn - 3'", where m is a positive integer, then
l\=«3L2FAX-2-lb^-\. B-27)
')) na(q")
ГЗ' 'a4q),
Proof. By Theorem 2.2, B.25) with * - c(q)fa(q), B.8), and B.9),
3 3 a
3
3a(q)
Л 2 / a(9) -
V3' 3* 'Wq) +
Replacing q by qy in B.28), and then iterating the resulting equality a total of m
times, we deduce B.27) to complete the proof.
Lemma 2.8. Let qi be defined by A.7), and put F(x) = 93. Let n = 3m, where
m is a positive integer. Then
B.29)
a4q))~ \a4q»)
and
\aHq)J \a\q»]
Proof. Dividing B.24) by B.27), we deduce that
B.30)
2 ...b\q)\
. 1 l| -> , v I
Л 2 b\qn)
2 [\3'У >a4q»)
B.31)
Multiplying both sides of B.31) by -2n/y/3 and then taking the exponential of
each side, we obtain B.29).
Multiply both sides of B.31) by ~Jb/Bnn), take the reciprocal of each side,
Theorem 22 and then take the exponential of each side. We then arrive at
Multiply both side o () y /
use Theorem 2.2, and then take the exponential of each si
B.30).
33. Elliptic Functions to Alternative Bases 99
We now establish another fundamental inversion formula.
Lemma 2.9. Let F be defined as in Lemma 2.8. Then
Proof. Letting n tend to oo in B.30) and employing Example 2 in Section 27 of
Chapter 11 in Ramanujan's second notebook (Part II [2, p. 81]), we find that
V + 9aH^
l/n
where in the penultimate line we used B.6) and B.9).
Theorem 2.10 (p. 258). Let F be defined as in Lemma 2.8. Then
z := 2F\ (|, |; 1; jc) = a(F(x)) = a(q3). B.32)
Proof. Let и = u(x) - b\F(x))/a3(F(x)). Then by Lemma 2.6 and Theorem
2.2,
a(F(x))= 2F1 (A, |; 1; 1 -и).
On the other hand, by Lemma 2.9,
F(l - u) = F{x),
or
B.33)
B.34)
By the monotonicity of 2F\ {\, |; 1; дс) on @,1), it follows that, forO < x < 1,
2ММ;1;1-и)= 2fiGJ;i;*). B.35)
(The argument is given in more complete detail in Part III [3, p. 101] with
2Fi E, |; \;x) replaced by 2FX [\, \; 1; x).) In conclusion, B.32) now follows
from B.33) and B.35).
Theorem 2.10 is an analogue of the classical theorem A.15). Our proof followed
along lines similar to those in Ramanujan's development of the classical theory,
which is presented in Part III [3, Chap. 17, pp. 98-102].

100 Ramanujan's Notebooks. Part V
Corollary 2.11 (p. 258). If г is defined by B.32) and <j3 is defined by A.7), then
z2 =
B.36)
where /з denotes the principal character modulo 3.
Proof. In Part III [3, p. 460, Entry 3(i)], it is shown that
B.37)
Since here q is arbitrary, we may replace q by <?3 >n B.37). Thus, B.36) can be
deduced from Theorem 2.10 and B.37).
We conclude this section by offering three additional formulas for г.
Theorem 2.12 (p. 257). Let q=q^andz = zC). Then
Proof. By Theorem 2.10 and B.6),
OO / Jn+1
3n+2
n=0m=l
n=0
oo
,2m '
and the proof is complete.
In the middle of page 258, Ramanujan offers two representations for г, but one
of them involves an unidentified parameter p. Ifq is replaced by -q below, then
the parameter p becomes identical to the parameter p in Lemma 5.5 below, as can
be seen from E.11).
33. Elliptic Functions to Alternative Bases 101
Theorem 2.13 (p. 258). Let z and q be as given above. Put p = (m- l)/2, where
m is the multiplier of degree 3 in the classical sense. Then
<p(q)
B.38)
Proof. Our proofs will be effected in the classical base q. We first assume that
the second equality holds and then solve it for p. Let fi have degree 3. By B.13),
-3
B.39)
by B.14) and B.15). Solving B.39) for p, we easily find that p = (m - l)/2, as
claimed.
Second, we prove the first equality in B.38). By the same reasoning as used in
B.39).
\n
.'/2
У V 2wi / mj
by B.10). Appealing to Theorem 2.10, we complete the proof.
3. The Principles of Triplication and Trimidiation
In Sections 3 and 4, for brevity, we set q = qi, and z = zC; x) (unless otherwise
stated).
In the classical theory of elliptic functions, the processes of duplication and
dimidiation, which rest upon Landen's transformation, are very useful in obtaining
formulas from previously derived formulas when q is replaced by q2 or y/q,
respectively. These procedures are described in detail in Part III [3, Chap. 17,
Section 13], where many applications are given. We now show that Ramanujan's
cubic transformation, Corollary 2.4, can be employed to devise the new processes
of triplication and trimidiation.
Theorem 3.1. Let x, q$ - q = q(x), and гC; х) = z be as given in A.7) and
A.6), respectively. Set x = f3. Suppose that a relation of the form
' <7,г) = 0 C.1)

102 Ramanujan's Notebooks, Part V
holds. Then we have the triplication formula
= 0
and the trimidiation formula
fl| 1-
Proof. Set
Therefore,
By Corollary 2.4,
i —
1 - A - r'3)
= A + 20 2F, (i, f; 1; <3) = (I + 2f)z(r3).
Also, by A.7), C.4), B.25), and B.23),
q :-.
C.2)
C.3)
C.4)
C.5)
C.6)
-« (°-
C.7)
Thus, suppose that C.1) holds. Then by C.5)-C-7), we obtain C.2), but with
t, q, and z replaced by t', q', and г', respectively.
On the other hand, suppose that C.1) holds with t, q, and г replaced by t', q',
and г', respectively. Then by C.4), C.6), and C.7), it follows that C.3) holds.
Corollary 3.2. With q and z as above,
b(q) = {\~
33. Elliptic Functions to Alternative Bases 103
and
c{q) =
Proof. By B.8), Theorem 2.10, and the process of triplication,
*<9) = H3 ¦ 5 (l + 20 " *>'/3}г - г) = A -лI/3г,
while by B.9), Theorem 2.10, and the process of trimidiation,
Theorem 3.3. Recall that f(-q) is defined by A.14). Then for any base q,
C.8)
Proof. All calculations below pertain to the classical base q.
By Entry 12(ii) of Chapter 17 of Ramanujan's second notebook (Part III [3, p.
124]),
<7/24(-<7) = ^j2<*(l-aL. C.9)
It thus suffices to show that the right side of C.8) is equal to the right side of C.9).
To do this, we use the parametrizations for b(q) and c(q) given by B.11) and
B.12), respectively. It will then be necessary to express the functions of m arising
in B.11) and B.12) in terms of a and fi. In addition to B.14) and B.15), we need
the parametrizations
— a =
and
(m+\)\3-m)
16m
C.10)
C.11)
given in E.5) of Chapter 19 of (Part III [3, p. 233]). Direct calculations yield
C.12)
9-m2 = 4m
3 — m = 1m
-0I/8'
A - aK/8
m+ 1 =2-
and
m2 - 1 = 4
C.13)
C.14)
C.15)

104 Ramanujan's Notebooks, Part V
Hence, by B.11), C.12), and C.13),
_ 0I/8
01/24A _ 0I/24
C.16)
By B.12), C.14), and C.15),
Зг,
A -a)'/» «'/24A -
Ciq)~
It now follows easily from C.16) and C.17) that
C.17)
which, by C.9), completes the proof.
Corollary 3.4 (p. 257). Let q = qz> and let z be as in Theorem 2.10. Then
C.18)
Proof. By Theorem 3.3 and Corollary 3.2,
qf\-q) = ^0 ~ ХУ'*z'2-
from which C.18) follows.
Corollary 3.5 (p. 257). With the same notation as in Corollary 3.4,
Proof. Applying to C.18) the process of triplication enunciated in Theorem 3.1,
we deduce that
A+2A-x)'/3)J
33. Elliptic Functions to Alternative Bases 105
as desired.
4. The Eisenstein Series L, M, and /V
We now recall Ramanujan's definitions of L, M, and N, first defined in Chapter
15 of his second notebook (Part II [2, p. 318]) and thoroughly studied by him,
especially in his paper [7], [10, pp. 136-162], where the notations P, Q, and R
are used instead of L, M, and N, respectively. Thus, fot\q\ < 1,
oo 3 n
У]2
oo 3
M(q):= 1+240 У]—
f-Г 1 -
and
JV(,):= 1-
я=1
We first derive an analogue of Entry 9(iv) of Chapter 17 in Ramanujan's second
notebook (Part III [3, p. 120]).
Lemma 4.1. Let q = q^be defined by A.7). and let zbeas in Theorem 2.10. Then
D.1)
L(q) = A - 4x)z2
Proof. By logarithmic differentiation,
dx
^1og(*/2V«)) =^log^fj(l -q"A
n=l
On the other hand, by Corollary 3.4,
±log(qf\-q))=q±
D.3)
Now by Entry 30 of Chapter 11 of Ramanujan's second notebook (Part II [2, p.
87]),
d_
dx

106 Ramanujan's Notebooks, Part V
Thus,
dq
dx x(l-x)z2'
Using D.4) in D.3), we deduce that
D.4)
-*)z-^ + 0 -4дс)г2.
dx
D.5)
Combining D.2) and D.5), we arrive at D.1) to complete the proof.
Theorem 4.2 (p. 257). We have
8дс).
D.6)
Proof. From Ramanujan's paper [7], [10, p. 330] or from Part II [2, p. 330, Entry
13],
Thus, by D.4) and Lemma 4.1,
M(q) = L\q) - 12x(l - x)z2^
dx
= A - 4x) Y + 24x(l - Jt)(l - 4x)z3 — + 144t2(l - xJz
dx
3 —
dx
t2(l - xJz2 I ~
ax
where we have employed the differential equation for z (Bailey [1, p. 1])
Upon simplifying the equality above, we deduce D.6).
Theorem 4.3 (p. 257). We have
= z6d -20x-8x2).
D.8)
Proof. From Ramanujan's paper [7], [10, p. 142] or from Part II [2, p. 330, Entry
131,
dq
33. Elliptic Functions to Alternative Bases
Thus, by D.4), Lemma 4.1, and Theorem 4.2,
107
N(q) = L(q)M(q) - !
"I-
-3jc(
= z6{(\
= Ai-
and so D.8) has been proved
4x)z2 -
:i-*)z
-4x)(l
20л-
ixi
f 1
1
+
8*
;i-x)z
2x(l -
4z3(l 4
8лг> -;
2),
2dM
dx
<
8x)-
t
!4x(l
Theorem 4.4 (p. 257). We have
Proof. Apply the process of triplication to D.6). Thus, by Theorem 3.1,
= \z* A + 2A - *)) A - 2A - xI" + 4A - xJ/i)
= |z4(9 - 8x),
and the proof is complete.
Theorem 4.5 (p. 257). We have
Proof. Applying the process of triplication to D.8), we find that
1'
1 +2A -jt)'/3
1+2A-
= ^ (A + 2A - xI/3N - 20A + 2A - x)'/3K A - A - *)l/3K
-8A-A-х)'/3N)

108 Ramanujan's Notebooks, Part V
= |r {-27 + 540A - x) + 216A - xJ}
,6
= ^G29-972л + 216л2)
We complete this section by offering a remarkable formula for z4 and an identity
involving the sixth powers of the Bonveins' cubic theta-functions b(q) and c(q).
Corollary 4.6. We have
Юг4 = 9M(<?3) + M(q).
D.9)
Proof. Using Theorems 4.2 and 4.4 on the right side of D.9), we easily establish
the desired result.
Corollary 4.7. We have
28 [b6(q) -
N{q).
D.10)
Proof. Using Theorems 4.3 and 4.5 on the right side of D.10), and also employing
Corollary 3.2, we readily deduce D.10).
5. A Hypergeometric Transformation and Associated Transfer
Principle
We shall prove a new transformation formula relating the hypergeometric functions
гB) and гC) and employ it to establish a means for transforming formulas in the
theory of signature 2 to that in signature 3, and conversely. We first need to establish
several formulas relating the functions (p(q), tyiq), and f(-q) with a(q), b(q),
and c(q).
Let, as customary,
(a; q)oo := A - «)d - aq){\ ~ aq2) ¦ ,
\q\
For any integer n, also set
(a; q)n =
(a; qn)oo
From the Jacobi triple product identity (Part III [3, pp. 36, 37]),
<p(-q) = (<?; 4H0(9; <?2)oo,
E.1)
E.2)
and
Lemma 5.1. We have
b(q) =
and
Mq2)
C2(q4)
33. Elliptic Functions to Alternative Bases 109
-iq.q)oo- E.3)
E.4)
-. E.5)
E.6)
E.7)
'ZT,- E-8)
f(-q3)'
c{q) = 2qxli^-
= q
= q2
Proof. First, E.4) and E.5) follow directly from Corollaries 3.2, 3.4, and 3.5.
Next, from E.5) and E.3),
с (q) (q ; q )<x(q ; q )?<>
Using E.1), we readily find that the right side above equals <p(,-q)/<pi(-q3), and
the proof of E.6) is complete.
Again, from E.5) and E.3),
which, by E.2), is seen to equal q2^i(q(>)/^(.q2). Thus, E.7) is proved.
Lastly, E.8) follows from combining E.6) and E.7).
Lemma 5.2. We have
c(q)
Proof. By E.8), we want to prove that

110 Ramanujan's Notebooks, Part V
By Entry 10(ii) of Chapter 17 of the second notebook (Part HI [3, p. 122]),
«)'/4 and <р(-Я*) = VHA - /3I/4, E.9)
where p has degree 3 over a. Thus, by B.13) and E.9), it suffices to prove that
Since m = z i /гз, the last equality is equivalent to the equality,
By B.14), B.15), C.10), and C.11), the last equality can be written entirely in
terms of m, namely,
3-m ^m - 1 2
1 -m-
= 2-
#m(m + l) 2 m+f
This equality is trivially verified, and so the proof is complete.
Lemma 5.3. We have
1 +
Proof. By E.8), the proposed identity is equivalent to the identity
By B.13) and E.9), the previous identity is equivalent to the identity,
By B.14), B.15), C.10), and C.11), the last equality can be expressed completely
in terms of m as
3 + m 2 m + 1
1)
1m(m - 1) "m - I 2
Since this last equality is trivially verified, the proof is complete.
Lemma 5.4. We have
Proof. By Entry 1 l(i) in Chapter 17 (Part III [3, p. 123]),
! and
33. Elliptic Functions to Alternative Bases 111
Thus, by Entry 3(i) of Chapter 21 of Ramanujan's second notebook (Part III [3, p.
460]), E.10), B.14), B.15), C.10), and C.11),
M1
1/2 4
+ 3?
w-i
By E.9) and E.10), the right-hand side above equals
The desired result now follows.
Lemma 5.5. If p := p(q) := -2c(q4)/c(q), then
E.11)
and
p- -
c(<?) qy
c(-q) _
c(q) C2(q)c(q4)'
c\q2)
E.12)
E.13)
l+p + p2 = 3-
c4q)
E.14)
Proof. Equations E.11) and E.12) follow from Lemmas 5.2 and 5.3, respectively.
As observed by the Borweins and F. G. Garvan [ 1 ], it follows from the definition
B.4) of c(q) that
E.15)

112 Ramanujan's Notebooks, Part V
Thus, by E.15), E.5), and E.3),
c(-q) _ {-q\
fa6;
Thus, E.13) has been verified.
Lastly, by Lemma 5.4, E.6), and E.7),
Hence, by E.13) and E.16),
V c(
c
which proves E.14).
Theorem 5.6 (p. 258). //
?3B + p)
a :=
and 0:=
T)p\\ + pJ
4A +p + p2)
then, for0 < p < 1,
(l + p + p2) 2^1 C.5; i;«) = У1+2р 2fi (i, f; l; /3)
Proof. By Lemma 5.5 and E.8),
г4(п4\,п2(-а^ Ф2(й2)
a = -16
c4q) <P2(-q)
E.16)
E.17)
by Jacobi's identity for fourth powers, B.1), with q replaced by q3. Thus (Part III
[3, p. 98, Entry 3]), with q replaced by -<?\
2F,(i,I;l;«)=^2(-93)- E.19)
33. Elliptic Functions to Alternative Bases 113
Also, by Lemma 5.5,
Thus, by Lemma 2.6,
By Lemma 5.5, E.19), and E.6),
C2(q)
= y/\+2pa(q2)
E.20)
by E.20).
Lastly, we show that our proof of E.17) above is valid for 0 < p < 1.
Observe that
da 6p2(l+pJ
dp (\+2pJ
and
dp _ 27p(l -p2)(l+2p)B-
rfp ~ 4(l+p + p2L
Hence, a(p) and 0(p) are monotonically increasing on @,1). Since a@) = 0 =
0@) and <x(l) = 1 = /3A), it follows that Theorem 5.6 is valid forO < p < 1.
We now prove a corresponding theorem with a and fi replaced by 1 — a and
I - p~, respectively.
Corollary 5.7. Let a and 0 be as defined in Theorem 5.6. Then, forO < p < 1,
A + P + P2) iFi (i, i; 1; 1 - of) = У3 + 6р>, (I. \; 1; I - fi) . E.21)
Proof. By E.19) and E.20), with q replaced by -q,
a(q2) 2
2. i. сНяг)
3' lZ&j
E.22)
Thus, by E.21) and E.22), it suffices to prove that
E.23)

114 Ramanujan's Notebooks, Part V
By Lemma 2.9,
E.24)
and by Entry 5 of Chapter 17 in Ramanujan's second notebook (Part III [3, p.
100]),
E.25)
q2 =exp| -7Г
Combining E.24) and E.25), we find that
2. ,.
We now see that E.23) follows from combining E.22) and E.26).
Corollary 5.8. With a and /8 as above, for 0 < p < 1,
§; 1;
E.26)
E.27)
Proof. Divide E.21) by E.17).
The authors' first proof of Corollary 5.7 employed Theorem 5.6 and a lemma
arising from the hypergeometric differential equation satisfied by iF\ (a, b, c; x)
and 2^1@. b; c; 1 - д:). We are grateful to Heng Huat Chan for providing the
proof that is given above. He has also shown that still another proof of Corollary
5.7 can be effected by combining Theorem 5.6 with Entry 6(i) in Chapter 19 of
Ramanujan's second notebook (Part III [3, p. 238J). We leave this proof as an
exercise for readers.
Corollary 5.8 is important, for from E.27) and A.7),
\ =¦ q\W) = <72(«) :=
E.28)
where q = q(a) denotes the classical base. Thus, from Theorem 5.6 and E.28),
we can deduce the following theorem.
Theorem 5.9 (Transfer Principle). Suppose that we have a formula
Я (q\a), zB; <*(p))) = 0 E.29)
in the classical situation. Then
= 0, E.30)
33. Elliptic Functions to Alternative Bases 115
in the theory of signature 3.
If the formula E.29) involves a, in addition to its appearance through q and z,
it may be possible to convert E.29) into a formula E.30) involving only /9,93,
and zC). This good fortune is manifest in the next three instances, as we offer
alternative proofs of Corollary 3.5, Theorem 4.4, and Theorem 4.5.
Second Proof of Corollary 3.5. By elementary calculations,
(l-p)(l+pK
1 -a =
and
l+2p p
E.31)
From Entry 12(iii) of Chapter 17 (Part III [3, p. 124]), Theorem 5.6, E.28), and
E.30),
i-q]) = fi-Я2) =
Д1 {«A - «
im
J \
2K)
1-PJV
by Theorem 5.6 and E.31). This completes our second proof of Corollary 3.5.
Second Proof of Theorem 4.4. By Entry 13(i) of Chapter 17(PartIII [3,p. 126]),
Theorem 5.6, E.28), and E.30),
f) = M(q2) = Z4(l - a + a2)
(l+
2pJ 4 / _ p3B + p) p6B+pJ
+ p2)*Z{ 4 l+2p d+2pJ
z
~z(i)
P2K
= z4C) A-1/8).
Second Proof of Theorem 4.5. By Entry 13(ii) of Chapter 17 (Part III [3, p. 126]),
Theorem 5.6, E.28), and E.30),
ql) = N(qz) = г6A + a)(l - |a)(l - 2a)
z6C)
(l+2 + 'V

116 Ramanujan's Notebooks, Part V
x A + 2p - 2p\2 + p))
= 2A /ff ргN № + P
4(l + pL}
Having thus proved Corollary 3.5, Theorem 4.4, and Theorem 4.5, we may use
the process of trimidiation to reprove Corollary 3.4, Theorem 4.2, and Theorem
4.3.
6. More Higher Order Transformations for Hypergeometric
Series
The first theorem will be used to prove Ramanujan's modular equations of degree
2 in the theory of signature 3.
Theorem 6.1 (p. 258). //
a := a(p):-
pJ
2A
and p := fi{p):=
then, for 0 < p < 1,
F.1)
F.2)
Proof. We first prove that
a(q)-a(q2) =
F.3)
From Entry 3(i), (ii) of Chapter 21 of Ramanujan's second notebook (Part III
[З.р.460]),
F.4)
Thus, by F.4), Theorem 2.13, A.17), B.13), B.14), B.15), E.10), and E.7),
, x .^(q2) 15^3(<?3) <p*(q)
(
4<p(q)
1/4 ,c 3/2 3/2
4z
1/2
3/2
<-3
33. Elliptic Functions to Alternative Bases 117
c{q) '
which completes the proof of F.3).
Second, we prove that
a(q) +
F.5)
By F.4), Theorem 2.13, A.17), B.13), B.14), B.15), C.10), C.11), E.9), and
E.6),
2<p(q) 2<p(<j3)
I* o ,3/2 ,3/2
mJ 3 1
4z
= 3-
1/2 '
which proves F.5).
Now let
F.6)
(Note that p tends to 0 as q tends to 0, by B.2).) Then by F.6), F.3), and F.5),
- 1 1 I ^4- + 2^
=
2A + pK 2 Va(<?2) 7 V«(<?2) ' 7 <>4q)
_ (a(<?) - a(q2)) {a(q) + 2a(q2)J
= 2-
2aHq)
c\q) 1 c\q)
c{q)
F.7)
Also, by F.6), F.3), and F.5),
4c4(g2) c\q)
c(q2) a
F.8)

118 Ramanujan 's Notebooks, Part V
The desired result now follows immediately from Lemma 2.6, F.6), F.7), and
F.8).
We now determine those values of p for which our proof of F.2) above holds.
By F.1),
da_ 3C
dp'
and ^ =
dp
2A + pL dp 4
Thus, a(p) and j8(p) are monotonically increasing on @,1). Since a@) = 0 =
/3@) and a(l) = 1 = /3A), it follows that F.2) holds for 0 < p < 1.
As functions of p, the left and right sides of F.2) are solutions of the differential
equation,
p(l - p)(l + pJB + p)C + p)u" + 2A + p)C - 4p - 6p2 - 4p3 - р4)м'
= 2(\-p)Q + p)u.
Corollary 6.2. Let a and /J be defined by F.1). Then, for 0 < p < 1,
2F, (i, 2; 1; 1 _ a) = 1A + p) 2F, A. |; 1; 1 - fi) .
Proof. By Lemma 2.9 and F.7) and F.8), respectively,
ifi(i,§;l;l-of)
and
q=e\p\-
я2-
F.9)
F.10)
F.11)
The desired result now follows easily from F.2), F.10), and F.11).
Corollary 6.3. Let a(p) and /3(p) be given by F.1). Then, for 0 < p < 1,
22F|E,3;l,l-«) = 2^1C,3.1.1 -P) FJ2)
mC) = l + p, F.13)
where mC) is the multiplier of degree 2 for the theory of signature 3.
Proof. Divide F.9) by F.2). Since F.12) is the defining relation for a modular
equation of degree 2 in the theory of signature 3, F.13) follows from A.10) and
F.2).
The next transformation is useful in establishing Ramanujan's modular equation
of degree 4.
33. Elliptic Functions to Alternative Bases 119
Theorem 6.4 (p. 258). Let
27 p(l + pL
27p4(l + p)
P PV ( рРУ
F.14)
Then, for 0 < p < 1,
B + 2p - p1) 2F, (|, 2; i; „) = 2A +4p + p2) 2F, (I, |; 1; j8). F.15)
Proof. For brevity, set z(x) = 2Fi (^, |; 1; дс). In view of Theorem 6.1, we want
to find л and v so that
УC
27p(l
2A+
27р4A+р)
F.16)
F.17)
F.18)
2{1+*P + Pl). F.19)
2 + 2p - p2
Solving F.18) for x, or judiciously guessing the solution with the help of F.19),
we find that
and
2 + 2p-p2'
Substituting F.20) into F.19) and solving for y, we find that
F.20)
У=-. v F.21)
1 + p + p2
Substituting F.21) into the left side of F.17), we see that, indeed, F.17) holds.
Lastly, it is easily checked that, with x and у as chosen above,
xC -\-X) v \^ 4™ v)
2A+лK ~ 4 '
i.e., the middle equality of F.16) holds. Hence, F.15) is valid, and the interval of
validity, 0 < p < 1, follows by an elementary argument like that in the proof of
Theorem 6.1.
Corollary 6.5. Let a and /8 be defined by F.14). Then, for 0 < p < 1,
2B+2p-p2JF! (|,|; 1; 1 -a) = (l+4p+p2JF, (\, \; 1; 1 ~ p) . F.22)

120 Ramanujan 's Notebooks, Part V
Proof. The proof is analogous to that for Corollary 6.2.
Corollary 6.6. Let a and 0 be defined by F.14). Then, for 0 < p < 1,
F.23)
Proof. Divide F.22) by F.15).
7. Modular Equations in the Theory of Signature 3
We first show that Corollary 6.3 can be utilized to prove five modular equations
of degree 2 offered by Ramanujan.
Theorem 7.1 (p. 259). Iff) has degree 2 in the theory of signature 3, then
(i) (a/})'/3 + l(l-«)(l-/9)}l/3= 1,
'а2\1Р |A-«J|'/3
Kfi) I 1-0 J 
2 _ ^
¦ p m'
-aJ
(iii)
(iv)
and
(v)
Proof. From F.1), by elementary calculations,
»2'
= m2.
Mld
A-p)B+pJ
i
Thus, from F.1) and G.1), respectively,
Hence,
2A + p)
(d -
and
p2+2-p-p2 =
G.1)
2A + p)
33. Elliptic Functions to Alternative Bases 121
Similarly, by F.1), G.1), and F.13),
3 3+p 1-p 2 2
P ) I 1-0 J A+PJ (l+pJ l + p
[1 - Pf ]l/3 _ (Pl\ft = B + p)(l + p) _ p(l + p)
1-« J \a ) 2 2
№
Л1/3
+
1-P
(l+pJ ' (l + pJ
+
= 1 + p = m,
4 4
2 m2'
pJ m
and
B
р)
р)
Thus, the proofs of (i)-(v) have been completed.
Theorem 7.2 (p. 259). Let (} be of degree 4. and let m be the associated multiplier
in the theory of signature 3. Then
Proof. From F.14), we easily find that
(l-pLB + p)(l+2p) (l-p)B + pL(l+2p)
I - a = — . _ _ _1л% and 1 - /3 =
2A +4p + p2
Thus, from F.14) and G.2),
2B + 2p - p2I
p(l+4p
+
B
G.2)
4p+p2)
«/ M-«/ (l+p)B + 2p-p2) A-р)B + 2р-р2)
_ 2(l+4p + p2)(l+2p)
From Theorem 6.4,
Hence, by G.2) and G.4),
m =
2p-p2)(l+p)(l-p)'
2(l+4p + p2)
2 + 2p-p2 '
2p-p2) pB + p)(l+4p + p2J
-p)B + 2p-p2J
G.3)
G.4)
2pB+p)(l
p7)
-p)B + 2p-p2)'
G.5)

122 Ramanujan's Notebooks, Part V
Therefore, combining G.3) and G.5), we deduce that
®"+№T-i
2A + 4p + p2)(l + 2p)
B + 2p - p2)(l
2(l+4p + p2)
- p)
2pB + p)(l +4p + p2)
-p)B + 2p-p2)
= m,
2 + 2p-p2
by G.4), and the proof is complete.
Theorem 7.3 (p. 204, NB1). Let a, p", andy have degrees 1,2, and4, respectively.
Let m\ and тг denote the multipliers associated with the pairs a, /3 and 0, y,
respectively. Then
- fi))U6
OT(
Proof. In F.14), replace p" by y, so that, for 0 < p < 1,
и —
27p(l
and у =
27р4A + p)
2(l+4p+p2K
From the proof of Theorem 6.4, ft has the representations
G.6)
and
where x and у are given by F.20) and F.21), respectively. In either case, a short
calculation shows that
n 21 p\\ + pJ
fi 4A +P + p2)y
Using G.6) and G.7), we find that
2p6 - 3p5 - 6p4 + 14p3 - 6p2 - 3p + 2
G.7)
1 -a =
2(l+4p
1-/9 =
(p-D4Bp2+5p + 2)
2(l+4p + p2K '
4p6 + 12p5 - 3p4 - 26p3 - 3p2 + 12p + 4
4A + p + p2K
(p-lJBp2+5p + 2J
4A
2K
p2)
G.8)
G.9)
33. Elliptic Functions to Alternative Bases 123
and
\-y =
-2p6 - 15ps - 39p4 - 32p3 + 24p2 + 48p + 16
2B + 2p - p2K
2B + 2p - p2)
G.10)
Hence, from G.6)-G.10),
27p2(l
%
ч4A
+ p2)
P?^) ,„
11 - PAP
5p
p2K 2B + 2p - p2K
27p4(l+p) (p-lLBp2 + 5p +
B + 2p - p2K 2A + 4p + p2K
(l+4p + p2)B + 2p-p2)
p + p2)((l + p)B + p) - p(l - p)\
4p + p2)B + 2p - p2)
G.11)
On the other hand, from the proof of Theorem 6.4, and from F.20) and F.21),
m, \+y A + 4p + p2)B + 2p - p2)
m2 1+jc 2A +p + p2J
Combining G.11) and G.12), we complete the proof.
G.12)
Next, we show that Ramanujan's beautiful cubic transformation in Corollary
2.4 yields the defining relation for modular equations of degree 3. We then iterate
the transformation in order to derive Ramanujan's modular equation of degree 9.
Lemma 7.4. //
then
G.13)
G.14)
Furthermore, the multiplier m is equal to 1 + 20l/3.
Proof. In B.23) and B.25), set x = plf\ Dividing B.25) by B.23), we deduce
G.14). The formula m = 1 + 2p"l/3 is an immediate consequence of B.23).

124 Ramanujan's Notebooks, Part V
Theorem 7.5 (p. 259). Ifm is the multiplier for modular equations of degree 9,
then
2/3|/3
1+2A -a)'/3'
where p has degree 9.
Proof. Let a be given by G.13), but with p replaced by
G.15)
, •= i -
Applying B.23) twice, we find that
We want to express the multiplier
G.16)
entirely in terms of a and p. Solving for /3|/3 in G.13) and then replacing p by t,
we find that
2A-a)
Thus,
2A -«*)'/'+ Г
Using this in G.16), we deduce G.15) to complete the proof.
Theorem 7.6 (p. 259). If P has degree 5, then
(a/3)l/3+ {(!-«)(!-,
G.17)
Proof. By Corollary 3.2, we may rewrite G.17) in the form
b(q)b(q5) + c(q)c{qs) + l^/b(q)c(q)b(q5)c(q5) = a(q)a(q ).
From Theorem 2.2 and E.4) and E.5) of Lemma 5.1, we find that
a i /l/3
G.18)
G.19)
33. Elliptic Functions to Alternative Bases 125
(In fact, G.19) is given by Ramanujan in his second notebook (Part III [3, p. 460,
Entry 3(i)J.) Thus, by E.4), E.5), and G.19), G.18) is equivalent to the identity
Cubing both sides of G.20), simplifying, and setting
A = f{-q), B = f(-q\ C = f(-qs), and D = f(-qiS),
we deduce that G.13) is equivalent to the proposed identity
45q2A6B6CbDb + 10^ A8C8B4D4 +90?3/44C4B8D8 + Al0Ci0B2D2
+ Slq4A2C2Bl0DiO = q*At2Dn + Bl2C12. G.21)
Setting
f(-15)
P =
and Q =
and dividing both sides of G.21) by q2(ABCDN, we find that G.21) can be
written in the equivalent form
!
or
(PQY
•№-&'¦
By examining P and Q in a neighborhood of q — 0, so that the proper square
root can be taken on the right side of G.22), we find that G.22) is equivalent to
the identity
Now G.23) is stated by Ramanujan on page 324 of his second notebook and has
been proved in Part IV [4, p. 221, Entry 62]. (See also a paper by the author and
L.-C. Zhang A ].) This therefore completes the proof of G.17).
Theorem 7.7 (p. 259). IfP has degree 7, then

126 Ramanujan's Notebooks, Part V
Proof. Using Corollary 3.2 and recalling that m = zi/zi, we find that G.24) is
equivalent to the equality
b(q) c(q) Hq)c(q) \ b(q)c(q)
Employing E.4) and E.5) in G.25) and then multiplying both sides of the result-
resulting equality by f(.-q)f{-q*)/ (<?/(-<?7)/(-?21)). we find that G.25) may be
written in the equivalent form
?3) qfH-q)f\-qlx) , /2(-<73)/2<-<77)
-7
/(-q)f(-q3)
If4-q)f4-q2x)
- - 3. G.26)
If we set
P =
П-q)
and Q =
we deduce that G.26) is equivalent to the identity
7 P2 Q2
G.27)
However, G.27) can be found on page 323 of Ramanujan's second notebook and
has been proved by the author and Zhang [1, Theorem 4]. See also Part IV [4, p.
236, Entry 68]. This completes the proof of G.24).
Theorem 7.8 (p. 259). Iffi has degree 11, then
+ {A - 1/3
{A
G.28)
Proof. Employing Corollary 3.2, we find that G.28) is equivalent to the identity
c(q)c(qn) + b(q)b(qtl)+6jb(q)b(q")c(q)c(qn)
= a(q)a(qn). G.29)
By E.4), E.5), and G.19), G.29) can be transformed into the equivalent identity
,
33. Elliptic Functions to Alternative Bases 127
f\-qr)f\-q") , f4-q)f\-q")
G.30)
Setting
= f(-q"), and D = /(-?")-
and multiplying both sides of G.30) by A BCD, we find that it suffices to prove
that
9q4B4D* + A4C4 + lSq2A2B2C2D2 + 27333
{ }'/3{С12 + 279"О12}1/3. G.31)
We next cube both sides of G.31), simplify, and divide both sides of the resulting
equality by 27(qABCDN. After considerable algebra, we deduce that
+ D ¦ 35 + 2 ¦ 34 + 8 • 33 + 4 • 32)
6 1 (BC\6
G.32)
Now set
P =
nrl
where »;(г) = ql/24f(-q) denotes the Dedekind eu-function, q = e\pBniz),
and Im(z) > 0. Then G.32) is equivalent to the identity

128 Ramanujan 's Notebooks, Part V
-GN?)'-
G.33)
The beautiful eta-function identity G.33) has the same shape as many eta-
function identities found in Ramanujan's notebooks, but is apparently not in Ra
manujan's work. In contrast to our proofs of most of these identities, we shall
invoke the theory of modular forms to prove G.33). All of the necessary theory is
found in Part IV [4, pp. 237-239].
Let ГA) denote the full modular group, and let М(Г, r, v) denote the space of
modular forms of weight r and multiplier system v on Г, where Г is a subgroup
of finite index in ГA). As usual, let
¦1С 2)
Then from Part IV [4, p. 237, Lemma 68.1], PQ <s М(ГоC3), О, 1), and from
PartIV[4,p.238,Lemma68.2],(P/QN e Л/(ГОC3),О, l).Letord(g; z)denote
the invariant order of a modular form g at z. Let r/s, (r, s) = 1, denote a cusp for
the group Г. Then, for any pair of positive integers m, n,
отдШтпг); -) =
{mn,s)
24mn
G.34)
where n denotes the Dedekind eta-function.
A complete set of cusps for ГОC3) is {0, 5, ^, 00}. Using G.34), we calculate
the orders of PQ and P/Q at each finite cusp. The following table summarizes
these calculations:
Function g
PQ
P/Q
Cuspf
0
1
3
1
11
0
1
3
1
11
ord(s;О
1
33
1
11
ji
5
198
5
66
5
18
Let L and R denote the left and right sides, respectively, of G.33). Using the
valence formula (R. A. Rankin [1, p. 98, Theorem 4.1.4]) and the table above, we
find that
О > ord(L; oo) + ord(L; 0) + ord(L; \) + ord(L; ±)
= ord(L; oo)-A-JL-f= ord(L; oo) - Ц
G.35)
33. Elliptic Functions to Alternative Bases 129
and
0 > ord(K; 00) - ± - ± - f = ord(tf; 00) - ff.
By G.35) and G.36), if we can show that
L - R = O{qz),
G.36)
G.37)
as q tends to 0, then we shall have completed the proof of G.33). Using Mathe-
matica, we find that
L = 9-5+6g-4+27<7-3+92g-2+2799~l+756+1913G+453642 + <:>((?:i) = R
Thus, the proof of G.37), and hence also of G.28), is complete.
At the bottom of page 259 in his second notebook, Ramanujan records three
modular equations of composite degrees. Unfortunately, we have been unable to
prove them by methods familiar to Ramanujan and so have resorted to the theory
of modular forms for our proofs. It would be of considerable interest if more
instructive proofs could be found. Because the proofs are similar, we give the
three together.
Theorem 7.9 (p. 259). If a, /J, у, and S are of degrees 1,2,4, and 8, respectively,
and if nt\ and тг are the multipliers associated with the pairs a, f} and y, 8,
respectively, then
G.38)
Theorem 7.10 (p. 259). If a, fi, y, and & have degrees 1,2,7, and 14 or 1, 4, 5,
and 20, respectively, and ifm \ and тг are as in the previous theorem, then
2((crJ) + {A -
тг
{A -
G.39)
Proofs of Theorems 7.9 and 7.10. Transcribing G.38) and G.39) via Corollary
3.2, we see that it suffices to prove that
a(q)a(q*) - c(q)c(q%) -
a(q)a(qu)+2c(q)c(qu) + 2b(q)b(.q14)
1) + 2c(q2)c(q7) + 2b(q2)b(q7),
and
G.40)
G.41)
a(q)a(q20) + 2c(q)c(q2") + 2b(q)b(q2u)
= a(q4)a(q5) + 2c(q4)c(q5) + 2b(q4)b(q5). G.42)

130 Ramanujan's Notebooks, Part V
Next, employing E.4), E.5), and G.12), we translate G.40)-G.42) into the equiv-
equivalent eta-function identities,
.3 /3(~<?3)/3H?24)
= 9qf(.-q2)f(-q*)f(-g6)f(-qn),
G.43)
18 5
f(-q)f(-q14)
and
7
f(-q)f(-q20) П-я3)П-яео)
I
G.45)
Recall that if ? = ехрBл-»г), where Im(z) > 0, then r/(z) = ql/2<xf(-q) is a
modularform on ГA) of weight i. If I a \ € r(l)andd is odd, the multiplier
system v,, I a ) is given by (Knopp [1, p. 51])
c d
= ± (—
\d\
G.46)
where (jfj)denotes the Legendre symbol,the plus sign is taken ifc > Oord > 0,
and the minus sign is chosen if с < 0 and d < 0. Using G.46), we find that each
of the four expressions in G.43) and each of the six expressions in both G.44)
and G.45) has a multiplier system identically equal to 1, provided, of course, that
33. Elliptic Functions to Alternative Bases 131
с is divisible by 24, 42, and 60, respectively. Hence, both sides of G.43)-G.45)
belong to М(Г0(и), 2,1), where n = 24, 42, and 60, respectively.
If <7oo denotes the number of inequivalent cusps of a fundamental region for
Г0(я), then (B. Schoeneberg [1, p. 102])
d\n
where <p denotes Euler's ^-function and (a, b) denotes the greatest common di-
divisor of a and b. Thus, for n — 24, 42, and 60, there are 8, 8, and 12 cusps,
respectively. Using a procedure found in Schoeneberg's book [1, pp. 86-87], we
findthatfO iilili ool-@ i ! ! I -L JL ool- and lo i i I I i
nnauidl IU, 2, ч, 4, г, r, |2.°°f. \V> 2' 3' 6- 7> 14' jl'00/'3110 IU' 2- 3' 4' 5' 6-
^} M l il
: complete sets of inequivalent cusps for ГоB4),
ГоD2), and Г0F0), respectively. Employing G.34), we calculate the order of each
expression in G.43)-G.45) at each finite cusp. In each instance, we find that each
order is nonnegativc.
Let F24, F»2, and Ffi0 denote the differences of the left and right sides of G.43)-
G.45), respectively. Since the order of F24, F42, and F«j at each point of a funda-
fundamental set is nonnegative, we deduce from the valence formula that
гргм > ord(Fn; 00), n = 24,42,60, G.47)
provided that Fn is not constant, where r is the weight of F,, and
Prow := и[ГA) : Го(и)]. G.48)
Now (Schoeneberg [1, p. 79]),
:Г0(п)] = пП( 1 + - ). G.49)
where the product is over all primes p dividing n. Thus, by G.48) and G.49),
Рг„B4) = 4, РГ(,D2> = 8, and рГо(бО) = 12. Since r = 2 in each case, by G.47),
ord(F24; 00) < 8, ord(F42; 00) < 16, OTd(FM; 00) < 24, G.50)
unless F24, F42, or FW)> respectively, is constant.
Using the pentagonal number theorem, A.14), i.e.,
00
Лг,\ — \ ' / 1\»|„п(Зп-1)/2 i 2,5,7 12 15 , 22 ,
~q) = / (-l)q ' = i — q —q + q' + q — q — q +q + • ¦ ¦ ,
n——oo
and Mathematica, we expanded the left and right sides of G.43)-G.45) about the
cusp oo (q = 0). We found that the left and right sides of G.43)-G.45) are equal
to, respectively,
9q - 9^3 - 18c?5 + O(q9),
90c?15 + O(qn),
G.51)

132 Ramanujan's Notebooks, Part V
and
3 + 18<?3
+ 5V
36qll
+ 36qn
I26q2{
36q12
90?
24
O(q2i).
G.52)
Thus, F24 = O(q9), F42 = 0(?17), and Fm = O(q25), which contradicts G.50)
unless F24, F42, and F^o are each constant. These constants are obviously equal to
0, and hence G.43M7.45) are established. This completes the proofs of Theorems
7.9 and 7.10.
On page 328 in his first notebook, Ramanujan gives another modular equation of
degree 8 in the theory of signature 3. This equation is quite interesting, because it
is the only known modular equation of Weber type (H. Weber [ 1 ]) in the alternative
theories.
Theorem 7.11 (p. 328, NB 1). Let
P := 1-(а0I/3-{A-<*)A-/9)}|/3,
and
Then
Рл - RPEP + 97") - 2Я2 = 0.
G.53)
Proof. As with the proof of Theorem 7.9, we utilize the theory of modular forms.
Transcribing G.53) via Corollary 3.2, we determine that it suffices to prove that
4 - rpEp + 9r) - 2r2 = 0,
G.54)
where p := z\Z«P, ( := ZiZ»7\ and r - z\z\R. Employing E.4), E.5), and
G.19), we find that G.54) is equivalent to an eta-function identity in the spirit of
G.43H7.45). Because the identity contains the same eta-function products and
quotients as G.43), it follows from our previous work that each term has multiplier
system identically equal to 1. Furthermore, it is easy to see that r = 8; recall also
from the proof of Theorem 7.9 that proB4> = 4. Thus, from the valence formula,
in order to prove G.54), it suffices to show that
pA-rpEp + 9t)-2r2 = O(q*) G.55)
as q tends to 0. Indeed, we have used the pentagonal number theorem A.14) in
connection with Mathematica to prove G.55). This completes the proof.
33. Elliptic Functions to Alternative Bases 133
8. The Inversion of an Analogue of K(k) in Signature 3
Theorem 8.1 (p. 257). Let q = q^ be defined by A.7), and let z be defined by
A.6) with r - 3. For 0 < <p < л/2, define 9 = 0(<р) by
9z =
Then, for 0 <9 < я/2,
°5, sinBn0)
[
0
+2cosh(ny))
¦E, 5;5;xsin t)dt.
% sinBn0)?"
?-< n(\ + q"
(8.1)
=: ФF>), (8.2)
where q =: e~y.
Recall from Entry 35(iii) of Chapter 11 (Part II [2, p. 99]) that
2F,(i+n, \-n; \;x2) = A -^2)"l/2cosBnsin-'jt), (8.3)
where n is arbitrary. With л = \ in (8.3), we see that the integral in (8.1) is
an analogue of the incomplete integral of the first kind, which arises from the
case л = 0 in (8.3). Since 2F| (|, |; +; x sin2 /) is a nonnegative, monotonically
increasing function on [0, я/2], there exists a unique inverse function ц> = (p(9).
Thus, (8.2) gives the "Fourier series" of this inverse function and is analogous to
familiar Fourier series for the Jacobian elliptic functions (Whittaker and Watson
[1, pp. 511-512]). The function <p may therefore be considered a cubic analogue
of the Jacobian functions. Theorem 8.1 is also remindful of some new inversion
formulas in the classical setting which are found on pages 283, 285, and 286 in
Ramanujan's second notebook and which have been proved by the author and S.
Bhargava [1]. (See also Part IV [4, Chap. 26].)
When <p=0 = 0, (8.2) is trivial. When <p = я/2,
rip
/
Jo
г/2
Thus, в = я/2, which is implicit in our statement of Theorem 8.1.
We now give an outline of the proof of Theorem 8.1. Returning to (8.3), we
observe that
$(*>:= 2MM;?;*2) = <l-xV/2cos (isin-'x), UK 1, (8.4)
is that unique, real-valued function on (-1, 1) satisfying the properties
S(x) is continuous on (-1,1), (8.5)
5@) = 1, (8.6)

134
and
Ramanujan's Notebooks, Part V
4A - x2)S\x) - 3S(x) -1=0.
(8.7)
Properties (8.5) and (8.6) are obvious, and (8.7) follows from the elementary
identity 4cos3 в = 3 cos в + cosC0). To see that S(x) is unique, set у = S(x) in
(8.7) and solve for x2. Thus,
2=
4-уЭ
у
Since S(x) is real valued, either у < 0 or у > 1. Since S@) = 1 and S is
continuous, we conclude that у > 1. Hence
where
«W :=
Now, g(y) is monotonically increasing on [1, oo). Thus, g '(•*) exists, and if
0 < x < 1, у = g~l(x), while if — 1 < x < 0, we have у = g~l{—*)¦ Thus,
S(x) is uniquely determined.
We fix q, 0 < <? < 1. Set x = c\q)/a2(q), so by B.5), 0 < x < 1. Then,
by Lemma 2.6, Z := 2Fi E. §; 1; x) = a(^). (We use the notation Z instead of
z = zC), because in this section г will denote a complex variable.) With Ф(в)
defined in (8.2), we shall prove that
0 < • < я/2,
(8.8)
and
4x sin"
1 /rf<i>\ 3 (db
By (8.8), we may define 0 := Ф~' . @,л-/2]
Z d®/d<p, we see from (8.9) that
4S3(l - jc sin2 <p) - 3S - 1 = 0.
Hence, (8.7) holds with x2 replaced by x sin2 <p. Now, from (8.2),
[0, tt/2]. Setting 5 :=
Ф'@) = 1 +
by Theorem 2.12. Thus,
S@) = Z/<D'@) = 1.
Hence, by (8.4M8.7), we conclude that
rd&
d<p
2Ft (Ц; {; x sin2 <p), 0<<p<rt/2,
33. Elliptic Functions to Alternative Bases 135
and so
since 0@) = 0. It follows that 0(^) = 9(<p) and that <p = Ф(9). Hence, the
desired result (8.2) holds.
= 2^1E,5; 3; ^ sin2;) rff, 0<<p<n/2,
Jo
Proof of Theorem 8.1. For \q\ < \z\ < l/\q\, define
(8.10)
n=\
We note, by Theorem 2.12, that v(l, q) = a(^), and, by (8.2), that
(8.11)
ao
By expanding 1/A — <73") in a geometric series and inverting the order of sum-
summation, we find that
,3n+2
z<?J
\-zq
in+2
(8.12)
As a function of г, u(z, g) can be analytically continued to С \ {0}, where the
analytic continuation v (г, q) is analytic except for simple poles at z = qm, where m
is an integer such that m#0 (mod 3). Using (8.12), we find, by a straightforward
calculation, that
v(zq\q) = v(z,q). (8.13)
M. Hirschhorn, F. Garvan, and J. M. Borwein A] have studied generalizations
of a(q), b(q), and c(q) in two variables. In particular, they defined
Hz,q):= J^ a)>»-"q»'2+»>»+»2z°
m,n=-oo
and showed that [1, eq. A.22)]
Ь(г. я) = (.; я)^; *V /Г^УЛ
. Я
= nd- <7"K Л
И 1_9з- И
and [l,eq. A.17)]
We next show that v(z, q) can be written in terms of b(z, Я) '¦
(8.14)
(8.15)

136 Raman ujan 's Notebooks, Part V
Lemma 8.2. //
«(<?) :=
О?;
and
then
(8.16)
The case г = 1 of (8.16) follows from the paper by Hirschhom, Garvan, and
Borwein [1, eq. A.29)]. To prove Lemma 8.2, we employ the following lemma
due to A. O. L. Atkin and P. Swinnerton-Dyer [1].
Lemma 8.3. Letq,0<q < I, be fixed. Suppose that f (z) is an analytic function
ofz,exceptforpossiblyafinitenumberofpoles,ineveryregion,Q < z\ < \z\ < Z2-
f{zq) = Azkf(z)
for some integer к (positive, zero, or negative) and some constant A, then either
/(г) has к more poles than zeros in the region \q\ < \z\ < I, or /(г) vanishes
identically.
Proof of Lemma 8.2. Define
F(z) := Ь(г, q)v{i, q) - \a{q)b(-z, q) + \fi{qMz.q). (8.17)
Examining (8.16), we see that our goal is to prove that F(z) = 0. From (8.13) and
(8.15),
F(.zqy) = Z-2q-yF(z).
From our previous identification of the poles of v(z, q) and from the definition
(8.14) of b(z, q), we see that the singularities of F(z) are removable. Thus, by
Lemma 8.3, to show that F(z) = 0, we need only show that F(z) = 0 for
three distinct values of г in the region |<?|3 < |г| < 1. We choose the values
z — —\,w,a>2, wherew = expBjri/3).
For г = -1, by (8.17) and (8.14), we want to prove that
But this has been proved by N. J. Fine [1, p. 84, eq. C2.64)].
For the values z — a>,w2, we need the evaluations
v(a>, q) = v(,co2, q) = b(q).
(8.19)
33. Elliptic Functions to Alternative Bases 137
The first equality follows from the representation of v(z, q) in (8.12). To prove
the second, we first find from (8.10) that
by Theorem 2.12. Since v(l,q) = a(q), by Theorem 2.12, we deduce that
By B.8), we conclude that b(q) = v(<o, q) to complete the proof of (8.19).
Now setting z = to, we see from (8.17) that we are required to prove that
(8.20)
where we employed (8.14), much simplification, and E.1). From Entry l(iii) of
Chapter 20 of Ramanujan's second notebook (Part III [3, p. 345]),
3!) .,_(,*??_ Л"\
-q) \ V\~q) )
Thus, by E.9), A.17), C.10), and C.11),
(
-Я9) _ Л = <p4-q) ( v\-q3) _
-q) J 2v(-q>)\ <p*{-q)
= zf A - a)** (9 1-/8 _
'/2 V2
2г'/2A -
z?/2C-mJ/9
C -
- Hq),
/8 _ \
a /
1/3
1/3
(8.21)
by B.11). (See also the Borweins' book [ 1, p. 143, Theorem 4.11 (b)].) Thus, (8.20)
has been proved. (Note that in (8.21) a and fi are squares of moduli and are not to
be confused with the definitions of a(q) and fi(q) in Lemma 8.2.)
In conclusion, we have shown that F(z) = 0 for z = — l,u>,a>2, and so the
proof of Lemma 8.2 is complete.

138 Ramanujan's Notebooks, Part V
We shall need some further relations among a(q), c(q), ot(q), and fi{q). First,
from (8.21),
Since, from (8.14) and (8.16), with z = 1,
3a2
it follows from B.5) that
27a4
(8.22)
(8.23)
Recall from (8.11) that V@) = db/dB, where Ф@) is defined by (8.2). Our
next task is to derive an infinite product representation for d V/d6. To do this, we
employ Bailey's 6^б summation (G. Gasper and M. Rahman [1, p. 239]).
Lemma 8.4. Let
n[
*>¦ ,bn
Ь„\ q)o
and, for |z| < 1.
-"
a, —q.s/a, b, c, d, e
Г ^-Ja,~q^j
L Ve. ~\/a, qa/b,qa/c, qa/d,qa/e' ' bcde\
- П [aq' aq^bc^ a(il(bd), aq/(be), aq/{cd), aq/{ce), aq/(de),q,q/a"I
L q/b,q/c,q/d,q/e,aq/b,aq/c,aq/d,aq/e,a2q/(bcde) J
(8.24)
Lemma 8.5. With z = еъ\ we have
dV _d2b •у.п%т{2пв)дп
(zV;<?3)oc(z-243;^3),
- 2cosDfl)g
3"
-q")O -
-
- 2cosB0)<?3n-1
(8.25)
33. Elliptic Functions to Alternative Bases 139
Proof. From (8.10),
zdv _ yi n(zn - z-")q"
3dz~ ?* 1+ q" + q2" '
(8.26)
By expanding 1/A - qin) in a geometric series and inverting the order of sum-
summation, we find that
-Зл+2
O-z<73n+2J
Replacing n by -n - 1 in the second and fourth sums and then combining the first
and fourth sums and the second and third sums, we find that
Z dv
= , _
„f^oo <! - г«3"+|JA - z-'?31J
q(z- l/z)(l-g2) Г q\-q\zq,zq,Z-lq,Z-[q ,
Z, 92Д, 92Z, «2Z, q4/z, q4/z, q4Z, q4Z,
by (8.24). With z = em, we thus have shown, by (8.26) and (8.27), that
dV dv SS.
-7T = 2'Z— - -
(827)
rf 1 + q" + q2"
'" +q6")(l -<?")(! -^3"K
which is (8.25).
Next, define
(8.28)

140 Ramanujan's Notebooks, Part V
Thus, (8.9) is equivalent to the identity
*@) = sin2D>@)).
As a first step in proving (8.29), we establish the following lemma.
Lemma 8.6. With Ф defined by (8.28),
(8.29)
(8.30)
Proof, Differentiating (8.28), we find that
^=-^(V
d9 4xZ*
dV
)
(8.31)
Putting (8.31) in (8.30), recalling that V(9) = d<t>/d$, and simplifying, we see
that it suffices to prove that
(8.32)
Employing (8.25) above and using (8.14), we now find that it suffices to prove that
— 1
= b\z, q){v - Z) D*Z3 - (Z -
4l0
2ZJ) ,
(8.33)
where v = v(z, q). By a direct calculation, it can be shown that the left side of
(8.33) satisfies the functional equation
F(zq\q) = z-*q-l2F(z,q). (8.34)
By (8.13) and (8.15), it is obvious that the right side of (8.33) is also a solution
of (8.34). As observed in (8.12), v(z, q) has simple poles at z = qm, where m is
an integer such that m ф 0(mod3). From (8.14), we see that fc(z, q) has simple
zeros at these same points. Thus, the singularities on the right side of (8.33) are
removable. In view of Lemma 8.3, it suffices to show that (8.33) is valid for at
least nine values of z in the region |<?3| < \z\ < 1.
It is clear that (8.33) holds for z = 1, since v(l,q) = a(q) = Z. In fact, this
zero is of order at least 2 on each side, since, by (8.10), 9u(z, q)/dz vanishes at
z = l.
Next, we show that (8.33) holds for z = q. Since b(z, q) has a simple zero at
z = q, and v(z, q) has a simple pole at z = q, we see that we must show that
lim
z, q))* =
V;
(8.35)
33. Elliptic Functions to Alternative Bases 141
But, by (8.12) and (8.14),
which establishes (8.35). A similar argument shows that (8.33) holds for z = q2.
Next, we examine the case z = a>. We shall need to use the equality
a(q) - b(q) = 3c(<?3), (8.36)
which is a consequence of B.8) and B.9). We shall also need the equalities
b(<v, q) = b(q*) = b((o2, q), (8.37)
which are readily verified by means of (8.14).
Recall that x = ci(q)/ai(q) (see the beginning of the paragraph immediately
prior to the proof of Theorem 8.1) and that Z = a(q). Also, by (8.19), v(co,q) =
b(q). Hence, by B.5),
4*Z3 - (Z - v((o, q)) BZ + v((o, q)J = 4c3 - (a - b)Ba + bf
= 4(д3 - b3) - (a - b)Ba + bJ
= 3b2(a - b).
Thus, by (8.36) and (8.37), the proposed equality (8.33) for z = a> reduces to the
equality
-243? V; 49)Uq; <7&,(<73; q')l - -27*VV(<7 V(«).
But this equality is readily verified by using E.4) and E.5).
An almost identical argument shows that (8.33) also holds for z = <o2.
From (8.16), (8.22), (8.23), and considerable algebra, we find that
b\z, q) (v(z, q) - Z) Dc\q) - (Z - v(z, q)) BZ + v(z. q)f)
81a2
= - tttt (<*b(z, q) - fib(-Z, q)) (aH-z. q) ~ PHz, q))
x {b(z, q)b(-z, q) {3a2 - p2) + a0 [b2(z, q) + b2(-z, q)}) .
Hence, both sides of (8.33) are even functions of z. Therefore, we have shown that
(8.33) holds for 12 values of z, namely, ±1 (with multiplicities 2), ±q, ±q2, ±w,
and ±to2. Thus, (8.33) holds for all values of z, and the proof of Lemma 8.6 is
complete.
We are now ready to complete the proof of Theorem 8.1.
Proof of Theorem 8.1 (continued). From (8.25),
dV .
0 < 9 < я/2.
(8.38)

142 Ramanujan's Notebooks, Part V
From (8.18),
It follows from (8.38) and (8.39) that
0 < fi(q) = V(>r/2) < V№ < V@) =
Observe that (8.8) follows from (8.11) and (8.40).
Combining (8.31) and (8.38), we find that
0 < Ф@) < *(тг/2).
We now calculate Ф(*?г). By (8.39) and (8.22),
Z - V(tt/2) = a(q) -
За^_ 3
26 2*
and
Thus, by (8.28), (8.42), and (8.43),
1 /3a2 3/J\ 9a4
3a2
—.
(8.39)
0 < 9 < 7Г/2.
(8.40)
(8.41)
(8.42)
(8.43)
21a4
by (8.23). So, from (8.41),
0<Ф(в)<1, О<0<тг/2. (8.44)
Now, by (8.38) and (8.31), dV/dO > 0, and again by (8.11) and (8.40), d<t>/dO >
0. Also noting (8.44), we conclude from (8.30) that
1 dV d<i>
= — = —> 0 <0 < л/2.
(8.45)
Since V@) = Z, from (8.28), we see that *@) = 0. By definition, Ф@) = 0.
Hence, by (8.45).
2 Л) T^fv^Wl </'
*<*" du
= i f
2 Jo
>\ -.
- arcsin
i.e.,
= sin2
This proves (8.29). Thus, (8.8) and (8.9) have been proved, and this finally com-
completes the proof of Theorem 8.1.
33. Elliptic Functions to Alternative Bases 143
Theorem 8.1 will be used to prove Ramanujan's next result. Theorem 8.7 below.
For each positive integer r, define
с -
"V
Ramanujan evaluated S2r in closed form for r = 1,2, 3, 4. Ramanujan's claimed
value for Sg> namely,
Sg = ^x(l + &x)z9,
is actually incorrect. We shall prove a general formula for S2r from which the
values for S2, S4, Se, and Sg follow. J. M. and P. B. Borwein [5] have evaluated S2
but give no details.
Theorem 8.7 (p. 257). Forr> 1,
(~lY
where the polynomials s2r(V, x, Z) are defined by (8.49)-(8.S2) below. In pa
ular
ular,
rtic-
rticz.
s, -
+ 648jc + 80л2) „9
37
z9.
Proof. Define
Then, by (8.32),
x, Z) := -| DxZ3 + (V - Z)(V + 2ZJ) (V - Z).
By a straightforward calculation,
(8.46)
= - I B*Z3 + (V - Z)(V + 2Z)BV + Z)).
(8.47)

144 Ramanujan 's Notebooks, Part V
From (8.46),
dVd2V dq(V,x,Z)dV
and so, from
Forn > 2
by
and, for л >
Sn(V,X,
(8.47),
, define two
3,
Z) :=/„-,(
d9 d92 dV d9'
d2V
^-„(V.x.Z).
sequences of polynomials sn(V, x,
s2(V,x,Z):=p(V,x,Z),
t2(V,x,Z):=O,
V,x,Z)p(V,x,Z)+-^tn-dV,x
aV
Z) and и
,Z)q(V,.
(8.48)
,(V,x,Z)
(8.49)
(8.50)
x,Z)
(8.51)
and
By using (8.48)-(8.52), we may prove by induction that
d"V dV
— =sn<V,x,Z)+,n{V,x,Z)-,
n>2.
(8.52)
(8.53)
Now, from (8.2), (8.11), and the definition of S2r, we readily see by induction
on r that
d2rV(9)\
d92r
= 6(-l)'2"S2r.
(8.54)
e=o
Since V'@) = 0 and V@) = Z, we conclude from (8.53) and (8.54) that
S2r = ^-^S2r{Z,x,Z).
A program was devised in MAPLE to calculate S2r. This completes the proof.
We also calculated S2r by using Theorem 8.1 in another way. From P. Henrici's
book [1, p. 102],
00 ,
,p = ?] - Res (Ф )""<?",
(8.55)
n=l
where Res (Ф")"" denotes the residue of (Ф~')~Л at <p = 0. By using (8.2), it is
easy to prove that
<p = 9 + 3
r=O
Br + 1)!
с
S2r.
(8.56)
33. Elliptic Functions to Alternative Bases 145
Equating coefficients of 92r+l in (8.55) and (8.56), we find that
3 •
We then used Mathematica to expand BФ ' (<p)) r in powers of <p so that the
desired residues could be calculated.
After our proof of Theorem 8.1 was completed, L.-C. Shen [1] found another
proof based on the classical theory of elliptic functions.
9. The Theory for Signature 4
The theory for signature 4 is simpler than that for signature 3, primarily because
of Theorem 9.3 below.
Theorem 9.1 (p. 260). ForO < x < 1,
Theorem 9.1 is precisely Entry 33(i) of Chapter 11 of Ramanujan's second
notebook (Part II [2, p. 94]).
Theorem 9.2. For 0 < x < 1,
Proof. With x complex and \x \ sufficiently small, by Entry 33(iv) of Chapter 11
(Part II [2, p. 95]),
Replacing л by A -jc)/A +*)in(9.3), we find that, for 11 -x\ sufficiently small,
(9.2) holds. By analytic continuation, (9.2) holds for 0 < x < 1.
Theorem 9.3 (p. 260). Let q =: q(x) denote the classical base, and let q* =:
q4(x) bedefinedby A.8). TkenJorO < x < 1,
(9.4)
Proof. Dividing (9.2) by (9.1), we find that, for 0 < x < 1,

146 Ramanujan's Notebooks. Part V
Replacing x by Jx~ and recalling the definitions of q and qA, we see that (9.4)
follows.
Theorem 9.4 (p. 260). For 0 < x < 1,
Proof. Applying (9.1) and then (9.3) with x replaced by 2x/(\ 4- *), we find that,
for x complex and \x\ sufficiently small,
13
2*
Thus, (9.5) has been established for \x | sufficiently small, and by analytic contin-
continuation, (9.5) is valid for 0 < x < 1.
We now describe a process for deducing formulas in the theory of signature
4 from corresponding formulas in the classical setting. Suppose that we have a
formula
Sl(x,q\z)=0. (9.6)
Let us replace x by 2y/xf(\ + ^/x). Then by Theorem 9.3, q1 is replaced by q4.
i.e., z is replaced by \J\ + ->/xzD). Hence, from (9.6), we deduce the formula,
= 0. (9.7)
In each proof below, we apply (9.7) to results from Chapter 17 of Ramanujan's
second notebook, proofs of which are given in Part III [3].
Theorem 9.5 (p. 260). We have
M{q4) = Z4D)(l + 3x).
Proof. By Entry 130) of Chapter 17 (Part III [3, p. 126]),
= zA{\ -x + x2).
33. Elliptic Functions to Alternative Bases 147
Thus, by (9.7),
= z4D) {A
Theorem 9.6 (p. 260). We have
4*} = z4D)(l + 3x).
Proof. By Entry 13(ii) of Chapter 17 (Part HI [3, p. 126]),
Using (9.7), we deduce that
= z6D)(l +
Theorem 9.7 (p. 260).
= г6D)A - 9x).
= z4D) A - |
Proof. By Entry 13(v) of Chapter 17 (Part HI [3, p. 127]),
Hence, by (9.7),
= \z\4) {4A
Theorem 9.8 (p. 260). We have
- Зле).
l) = г6D) A - |
Proof. By Entry 13(vi) of Chapter 17 (Part III [3, p. 127]),
Thus, from (9.7),
{8A
V*") - д:} = |z6D)(8 - 9x).

148 Ramanujan's Notebooks, Part V
It is interesting that the two previous results have simpler formulations in the
theory of signature 4 than in the classical theory.
Theorem 9.9 (p. 260). We have
Proof. From Entry 12(iii) of Chapter 17 (Part III [3, p. 124]),
Hence, by (9.7),
1/12
Theorem 9.10 (p. 260). We have
Proof. By Entry 12(iv) of Chapter 17 (Part III [3, p. 124]),
Thus, by (9.7),
Recall the definition of L(q) at the beginning of Section 4. The following
intriguing formula does not appear in the second notebook but can be found in the
first notebook [9].
Theorem 9.11 (p. 214, NB 1). We have
1F?({,\;\;x)=2L(q24)-L(q4).
= q4 and z = zD). From D.2) and Theorem 9.9,
Proof. For brevity, set q
d
^Tq
'dq
dx
= <7
dx
(9.8)
33. Elliptic Functions to Alternative Bases 149
By Entry 30 in Chapter 11 of Ramanujan's second notebook (Part II [2, p. 87]),
with q = ??4,
dq q
dx jcA-jc)z2"
Using (9.9) in (9.8), we deduce that
(9.9)
,„_,„>
dx
(9.10)
Repeating the same argument, but with q replaced by q2 and with an application
of Theorem 9.10 instead of Theorem 9.9, we find that
1 /12dz 2 1 \ 2
= -(—— + -- -j I x(l - x)z2
2 \ z dx x 1 - x/
= 6x(\ -*)^ + 5<2-
Our theorem now easily follows from (9.10) and (9.11).
(9.11)
We conclude this section with a new transformation formula for 2 F\ (?, |; 1; x)
found also on page 260. We need to first establish some ancillary lemmas.
Lemma 9.12. We have
and
(9.12)
2a(q2) - a(jg) =
(9.13)
Proof. By Entries 4<iii), (iv) of Chapter 19 of Ramanujan's second notebook (Part
HI [3, pp. 226-227,229]),
and

150 Ramanujan's Notebooks, Pan V
Using B.6) to calculate the Lambert series for \ \a(q) + a(q2)] and comparing
it with the Lambert series above, we deduce (9.12). The proof of (9.13) is similar
and follows by substituting x = q3"^, qin+1 in the elementary identity
2x2
1-х2 1-х
Lemma 9.13. We have
jt6{q)
$4q3) '
and
V
V
V) -
(q2) (9.14)
(9-15)
Proof. For brevity, seta := a(q) and A := a(q2). Then, squaring (9.12), we find
that
From E.7), F.3), and (9.12),
and
36qz
= (a - AY.
(9.16)
(9.17)
(9.18)
Hence, from (9.16H9.18),
Next, from (9.16) and (9.18),
and
From E.6) and F.5),
Thus, from (9.13) and (9.22),
(922)
= \&A - a)\a + 2Л) (9.23)
3
33. Elliptic Functions to Alternative Bases 151
and
p9(-q") <p
(-q") <p4-q) l
<pH~q) Vi-q*) 27
Lastly, we need the elementary identity (Part III [3, p. 40, Entry 25(iv)])
(9.24)
(9.25)
Hence, from B.1), (9.25), (9.23), and (9.20),
[<p\q) + 16,^ V)}2 = {/(<?) - 16<?tfr V)}2
= \BA - aK(a + 2A) + |(a + Af(a - A)
= a4 + 4A4 + 4aA(fl2 - 2A2)
= (a2 - 2Л2J + 4aA(a2 - 2A2) + 4a2A2
= (a2 + 2oA - 2A2J. (9.26)
Equality (9.14) now follows from (9.19) and (9.26), upon taking the square root
of both sides of (9.26) and checking agreement at q = 0 to ensure that the correct
square root is taken.
The proof of (9.15) is similar. Thus, from (9.16И9.18),
2flA - a2). (9.27)
Proceeding as in the proof of (9.14), from B.1), (9.25), (9.24), and (9.21), we
find that
V
VV)
- ±ia + 2AKBA - a) + ±(a - A)\a + A)
V)
= | ((a2 - 2A2J - 4aA(a2 - 2A2) + 4a2A2)
= ±(a2-2A2-2aAJ. (9.28)
Equality (9.15) now follows from (9.27) and (9.28).
Lastly, we need the following lemma connecting 2^1C. \\ 1; jc) with theta-
functions.
Lemma 9.14. We have
,2 Л 3 (К/Я±
1 \4'41 ]\<p4(q)^
). (9.29)

152 Ramanujan's Notebooks, Part V
Proof. Let
x =
Then
2х \ЬдУ(д4)<р1(д1) _
where we have employed (9.25) and the elementary identity
<p2(.q2) + 44^2(<?4) = v2(q)<
(9.30)
(9.31)
which is achieved by adding Entries 25(v), (vi) of Chapter 16 of Ramanujan's
second notebook (Part III [3, p. 40]). Hence, from Theorem 9.1, (9.30), (9.31),
B.1), and E.19), with -?3 replaced by q,
„2Л .н
V2-2- •
<P4(<7)
Replacing ?2 by <?, we deduce (9.29).
Lemma 9.14 was first proved by the Borwein brothers [1, p. 179, Prop. 5.7(a)],
[5, Theorem 2.6(b)].
Theorem 9.15 (p. 260). If
(9.32)
v/27 - ISp - p2 2F, (i, |; 1; a) = 3^3 + 6p - p2 2F, (i, 2; 1; ^). (9.33)
a:=O + 6p-p2J and Р:=(П-\*Р-Р2J'
then,for 0 </><!,
Proof. Let
1
Then, by (9.14) and (9.15), respectively,
(9.34)
(9.35)
33. Elliptic Functions to Alternative Bases 153
and
21-1&P- p2 =
From (9.32) and (9.35),
r.3/2
(9.36)
by (9.25). Similarly, by (9.32) and (9.36),
by (9.37).
Hence, from (9.36), (9.37), Lemma 9.14, (9.38), (9.35), and (9.34),
„3/4
Thus, (9.33) has been proved.
Lastly, it is easily checked that a =: a(p) and p =: /3(/>) are monotonically
increasing functions of pon@,l). Sincea(O) = 0 = /3@) anda(l) = 1 =/5A),
(9.33) is valid for 0 < p < 1.
10. Modular Equations in the Theory of Signature 4
Page 261 in Ramanujan's second notebook is devoted to modular equations in the
theory of signature 4. In each case, our proofs rely on (9.7). Thus, we will employ
modular equations from Chapters 19 and 20 of the second notebook and convert
them via the transformations
1 -
1 -
A0.1)
Theorem 10.1 (p. 261). Ifp has degree 3, then
(арУ11 + {A - <*)(! - p)I'2 + Л{аРA -a)(l -/3)}1/4= 1. A0.2)

154 Ramanujan's Notebooks, Part V
Proof. From Entry 5(ii) of Chapter 19 (Part HI {3, p. 230]),
By A0.1), (Ю.З) is transformed from the classical theory to
A0.3)
= 1
in the theory of signature 4, or
Squaring both sides yields
Squaring each side once again, we find that
1/2 + 2 {A - l/1
Collecting terms, we deduce A0.2).
Theorem 10.2 (p. 261). Iffi has degree 5, then
{A -
A0.4)
Proof. By Entry 13(i) of Chapter 19 of Ramanujan's second notebook (Part III
[3, p. 280]),
l6 1. A0.5)
/ {A - a)(l - fi))U2 + 2 {16a/*(l - ar)(l
Transforming A0.5) by A0.1) and simplifying, we find that, in the theory of
signature 4,
.1/6 2
= |d + >/»
- {A - >/e)(l -
Squaring each side and collecting terms, we deduce that
2(a/?I/2 + 4 |б4^A - а)A - fi)]1/3 + 8(а^)|/3 {64A -
33. Elliptic Functions to Alternative Bases 155
Further simplification easily yields A0.4).
Theorem 103 (p. 261). Iffi has degree 7, then
{a?I'2 + {A -o)(l - 0)}i/2 + 20{afi(\ - o)(l - p)}{"
+ 2у/2Ш\ - a)(l - 0))l/* ((a^I/4 + {A - o)(l - p)]m) = 1.
A0.6)
Proof. From Entry 19(i) of Chapter 19 of Ramanujan's second notebook (Part III
[3, p. 314]),
(o0)l/8 + {(l-«)(l-0)»I/e= 1. A0.7)
Using A0.1) to transform A0.7) into the theory of signature 4, we find that
A6a0)l/l6 = (A + Va)(l + v/^)I/8 - j 1/S
Squaring both sides, we deduce that
1/8 + 2 (A - o)(l - P))m = (A + v^)
Squaring again and simplifying slightly, we find that
2{A - o)(l -
Squaring one more time, we finally deduce that
4(<*0)l/2 + 4{<1 -o)(l - /3)?/2 + 32{«/9A - o)(l -
+ 8 {o^( l/4
+ 16 {A -
= 2 + 2(or0)l/2 + 2 {A - o)(l - 0)I/2 .
Collecting terms and dividing both sides by 2, we complete the proof of A0.6).
Theorem 10.4 (p. 261). Ifp has degree 11, then
l/2 + {A - a)(l - p)}*'2 + 68 {ctf(l - a)(l - 0)}1/4
16 [afiil - o)(l - ^)I/12 ((a^I/3 + (A -
{A - )
A0.8)
Proof. By Entry 7(i) of Chapter 20 (Part HI [3, p. 363]),
(<*j8)l/4 + (d - и)A - )8)}1/4 + 2 (Ifarftl - or)(l -
= 1. (Ю.9)

156 Ramanujan's Notebooks, Part V
Transforming A0.9) into an equality in the theory of signature 4, we find that
+ 2 J6V^3A - <*)A - 0)I/U - (A + V«)
Squaring both sides and simplifying slightly, we deduce that
8(a/8I/l2 {A - a)(\ - p)\i/f>
= [A + VS)A + V^))'7 + [d - v^)(l - Jp)} •
Squaring both sides again, we see that
4(a,8I/2 + 4{A - a)(l - p))]/n + 64(a/?I/6 {A -
+ 64(ctfI/3 (A - «)A - /3)I/6 + 8 («0A - «)A -
+ 32(a/9I/3 {A - a)(l - 0)I/6 + 32(a/?M/12 {A -
2 {A - o)(l - fi)M/n + SHapI* {A -
- a)(l - 0)}l/4 = 2 + 2(<*j8)l/2 + 2
Collecting terms and dividing both sides by 2, we complete the proof.
The last six entries on page 261 in Ramanujan's second notebook give formulas
for multipliers. By (9.1),
>.}=¦;¦)
Thus, to obtain formulas for multipliers in the theory of signature 4, take formulas
from the classical theory, replace m by y/(l + y/a)/(l + ^/Д) »»D), and utilize
the transformations in A0.1).
Theorem 10.5 (p. 261). The multiplier for degree 3 is given by
\-p\ul _ _9_ //?(!-/?)\1/2
\-a) m2D) \a(\-a)J
m W = I -
Proof. From Entry 5(vii) of Chapter 19 (Part III [3, p. 230J),
and
xn
1/2
1/2
l/2
A0.11)
A0.12)
33. Elliptic Functions to Alternative Bases 157
Using A0.1) and A0.10), we convert A0.12) into an equality in the theory of
signature 4, namely,
1/2
/ II — -»/ D II 1 Г */lil 1
+
-J5)J
or, upon rearrangement,
By A0.13), A0.14), and symmetry,
m2D)
l—
*
Multiplying both sides of A0.15) by vW - 0)/(a(\ -a)), we find that
/^(l-^)\'/2 (p\l/2 = (V0(l-V0)
тЦ4)
Comparing A0.14) and A0.16), we arrive at A0.11).
Theorem 10.6 (p. 261). lfmD) is the multiplier of degree 5, then
,'/4 /1_я\'/4 5
„D)
_ ftI
- a)
A0.16)
Proof. By Entry 13(xii) of Chapter 19 (Part III [3, pp. 281-282]),
and
1/4
l-<x\'/4 /a(l-a
1-p) \Pd-p
A0.19)
Transforming A0.18) into the theory of signature 4 via A0.1) and A0.10), we
find, after a slight amount of rearrangement, that
From A0.19) and symmetry,
mD)
. A0.21)

158 Ramanujan 's Notebooks, Part V
Multiplying both sides of A0.21) by ,//9A - 0)/(or(l -a)) and comparing the
result with A0.20), we readily deduce A0.17).
Theorem 10.7 (p. 261). Let mD) denote the multiplier of degree 9. Then
A0.22)
Proof. The proof is almost identical to the two previous proofs. By Entries 3(x),
(xi), respectively, of Chapter 20 (Part III [3, p. 352]),
and
„o.23>
A0.24)
The transforming of A0.23)and A0.24) viaA0.1) andA0.10) yields the equalities
Г-7ТГ /Vgd + VflV" , /l-lV/8 /у/?A-У?)У/8
^ = UoTvs>J + (j^) 4^<r^s)j (la25)
and
/ пг.,х , &ч!/« /1_„\1/» / nZ(\-.fc\\m
, A0.26)
у^)\1/8 /l-a\1/8 /у^A-у^)
respectively. A multiplication of A0.26) by ^7*A — ^9)/<a(l -a)) and a com-
comparison of the resulting equality with A0.25) gives A0.22).
Theorem 10.8 (p. 261). //mD) denotes the multiplier of degree 7, then
1/2
1/6
Proof. By Entry 19(v) of Chapter 19 (Part HI [3, p. 314]),
and
49
A0.27)
A0.28)
(lo-29)
33. Elliptic Functions to Alternative Bases 159
Transforming A0.28) and A0.29) into the theory of signature 4 via A0.1) and
A0.10), we find that, after simplification,
-8
. '/3
>A -or)/
A0.30)
and
49
1/2
_
respectively. Multiplying both sides of A0.31) by
deduce that
1/3
A0.31)
a(l -a)), we
49
1/2
Combining A0.30) and A0.32), we complete the proof of A0.27).
1/3
A0.32)
Theorem 10.9 (p. 261). ///nD) denotes the multiplier of degree 13, then
mD) = -
mD)
1/12 f /дч 1/12
A0.33)
Proof. By Entries 8(iii), (iv) of Chapter 20 of Ramanujan's second notebook (Part
III [3, p. 3761),
and
13 (a
a(\ - a)
(Ш4)
Aаз5)

160 Ramanujan's Notebooks, Part V
Transforming A0.34) and A0.35) into the system of signature 4by means of A0.1)
and A0.10), we find, after some simplification, that
A0.36)
and
13
1/4
(—Г
A0.37)
respectively. Multiplying both sides of A0.37) by ^A - /3)/(a(l -a)) and
combining the resulting equality with A0.36), we finish the proof of A0.33).
Theorem 10.10 (p. 261). lfmD) denotes the multiplier of degree 25, then
A0.38)
Proof. By Entries 15(i), (ii) of Chapter 19 of Ramanujan's second notebook (Part
III [3, p. 291]),
and
5
_
-лу
П040)
Transforming A0.39) and A0.40) by means of A0.1) and A0.10) into equalities
in the theory of signature 4, we find that
Ja(\-ct)
1/12
A0.41)
33. Elliptic Functions to Alternative Bases 161
and
_
1/12
A0.42)
respectively. Multiplying both sides of A0.42) by ^/3A - j8)/(a(l -a)) and
combining the resulting equality with A0.41), we finish the proof of A0.38).
11. The Theory for Signature 6
The most important theorem in this section is Theorem 11.3 below. This result
allows us to employ formulas in the classical theory to prove corresponding theo-
theorems in the theory of signature 6. To prove Theorem 11.3, we need the following
two results.
Theorem 11.1 (p. 262). //
a :=
p)
l+2p
and /5 :=
27p2(l
then, for 0 < p < 1,
(|, |; V.fi) =
A1.1)
A1.2)
Proof. From Erdelyi's treatise [ 1, p. 114, eq. D2)],
2^1 E. i! >; г) = A - г/4)- 2F, (i, i; 1; -27z2(z - 4)-3), A1.3)
for г sufficiently near the origin. By Example (ii) in Section 33 of Chapter 11 (Part
H [2, p. 95]),
for |z| sufficiently small. Thus, combining A1.3) and A1.4), we find that
1.5)

162 Ramanujan's Notebooks, Part V
Next, recall the well-known transformation (Erdelyi [1, p. 111, eq. B)]),
2fitt.i;l;*JF,(u.fr;O) AL6)
for z in some neighborhood of the origin. Examining A1.5) and A1.6) in relation
to A1.2), we see that we want to solve the equation
27z2(l-zJ
Solving this quadratic equation in x and choosing that root which is near the origin
when z is close to 0, we find that
4A-z + z2K
1 / (l + z)B-5z + 2z2)\
~ 2 I 2A - z + Z2K/2 ) '
Thus, by A1.5)-(l 1 -7). we have shown that
/15 1 (. (l+z)B-5z + 2z2)
2F46'6;;2l, 2A -z + z2)
n
A17)
A1.8)
Now set
z =
p{2 + p)
l+2p '
Then elementary calculations give
2I
2 A+p + p2)
1 -z + z = ,, , л_ч.
and
(l+z)B-5z + 2z2) 27p2(l+pJ
'" 2A-г+z2K/2 2(l + p + p2K'
Using these calculations in A1.8), we deduce A1.2).
Lastly, a and /8 are monotonically increasing functions of p on [0,1] with
a@) = 0 = /3@) and <*A) = 1 = 0A)- It follows that A1.2) is valid for
0< p < 1.
Corollary 11.2. Let a and 0 be defined by A1.1). ForO < p < 1,
Proof. From A11),
and
2pJB
4A + p + P2)
.243
A1.10)
33. Elliptic Functions to Alternative Bases 163
(Recall that 1 - /} was previously calculated in E.31).) Setting z =
A - p2)/(l + 2p) in A1.8), we complete the proof.
Theorem 113. Let a and p* be defined by (ll.l).IfO< p < 1, then
-q2(a):=q2, A1.11)
where q denotes the classical base.
Proof. Divide A1.9) by A1.2) to obtain the equality
2fi(i,|;l;l-0)_2F,(M;l;l-«) A112)
2F,(i,|;l;/») jF,(i,i;l;o) '
valid for 0 < p < 1. From A1.12) and A.9), we immediately deduce A1.11).
From Theorem 11.1, we also can deduce that
v/l+2p zF; 0): = -J\+2p zF)
= v/l + P + A»2 zB) =: УГТрТр2 zB; a).
A1.13)
Hence, from A1.1), A1.11), and A1.13), we derive the following principle. Sup-
Suppose that we have an equality
Q (q2, zB; a(p))) = 0
in the classical theory. Then in the theory of signature 6, we may deduce the
corresponding equality
We now give some applications of this principle.
Theorem 11.4 (p. 262). We have
M(q6) = z4F).
Proof. By Entry 13(i) of Chapter 17 of Ramanujan's second notebook (Part III
[3,p. 126]), (П.11), and(ll.l),
-<•№?)'-<•*>¦
by A1.13).

164 Ramanujan's Notebooks, Part V
Theorem И.5 (p. 262). We have
Proof. By Entry 13(ii) of Chapter 17 (Part III [3, p. 126]), A1.11), and A1.1)
N(q6) = N(q2) = Z6d + «)d - !«)(» ~ 2a>
./, , PB+P)\(. РB +
= z
+ 4p + p2)B + 2p - p2)d - 2p - 2p2)
2A + 2рУ
2A
= г
. (i + p + p*K 2A + p + p2K ~ 27p2(l + pJ
(l+2pK
2A + p + P2)
by A1.13) and A1.1).
Theorem 11.6 (p. 262). We have
1/24
Proof. By Entry 12(iii) of Chapter 17 (Part HI [3, p. 124]), A1.11), A1.13). and
A1.1),
3 {ad - «)}1/12
l+2p
2p)B
7p2(l+pJ (ljlpJ(l + 2pJB + pJ\'/24 f J_V
+p+pv—^тттт^ ) V432;
/j3(l-^)V/M
V 432 ) '
by A1.1) and A1.10).
33. Elliptic Functions to Alternative Bases 165
12. An Identity from the First Notebook and Further
Hypergeometric Transformations
On page 96 in his first notebook, Ramanujan claims that
where "ц can be expressed in radicals of л; and A" and where
2F,(A,1 -A; 1; 1 - x)
У =
A2.1)
A2.2)
2F,(A, 1-A;l;x) '
It is unclear what Ramanujan precisely meant by the phrase, "д can be expressed
in radicals of x and A." If Ramanujan meant that ц is contained in some radical
extension of the field Q(x, A), the field of rational functions in x and A, then his
statement is false for general A. We now sketch a proof.
By Corollary (ii) in Section 2 of Chapter 17 in Ramanujan's second notebook
(Part III [3, p. 90]),
exp(-jry/sin(xh)) = x exp(^(A) +
x 1 +BA2-2A+ \)x-
- A) + 1y)
A2.3)
- I(h-h2)+—(h-h2J)x2+ ¦¦
[ 2 4
where ф(х) is the logarithmic derivative of the gamma function and у denotes
Euler's constant. It should be noted that for A = i, |, \, i the quantity exp(\fr(h)+
rj/(l -A)+2y) isa rational number. In fact, from Chapter 8 of Ramanujan's second
notebook (Part I [1, p. 192]),
ф + 2y) = i.
and
«Р(*ф+ *(|)+2H =jk.
However, for A = | the quantity is transcendental, for (Part I [1, p. 192]),
Г-
+ ^(|) + 2y) = 5 I
which is transcendental by the Gelfond-Schneider theorem. If Ramanujan's state-
statement were true for general A, this would imply that, for A = 5, ц could be
expressed in terms of radicals over some algebraic extension field of Q. Now
V2Fi(A, 1 -A; l;x) and, by A2.3), (p(e-nyl^n{nh)) have Taylor expansions about
x — 0. Thus, A2.1) implies that \x has a Taylor expansion about x = 0. But since
ц can be expressed in terms of radicals in x, it follows that the coefficients in the
Taylor expansion of ц. are algebraic over <Q>. The function *JiF\{h, 1 -A; 1; x)

166 Ramanujan's Notebooks, Part V
has the same property. Hence, by equating coefficients on both sides of A2.1), we
deduce that exp(^(i) + ^E) + 2y) is algebraic, which is a contradiction. Hence,
Ramanujan's statement is false for general h.
However, the statement is true if we restrict the values of h to 5, i, i, i. Let
/х = д(х,*).ВуA.15).
We now describe д(д:, 3) in terms of radicals in x. Recall that m = 21/23.
Theorem 12.1. For 0 < x < 1,
m =
A/ - л/(Л/ + 2)(Af -
A2.4)
A2.5)
is fA« root of the cubic equation
2J
jc =
6K
f/м» и greater then 6. Моле explicitly, M may be given as
M = -
x
8i
- 36* + 27 - 8. 7A - x)? + 3 -
A2.6)
7b ma/ci explicit which cube root is taken, we rewrite A2.6) as
M = - BV9 - 8x cos Q tan (8^0 -x)x\ Zx2 - 36л; + 27)) + 3 - 2л:) ,
X A2.7)
where tan (/}, a) is «Ла« angfe 0 smc/i rfcar -я < в < n and arg(a + 1^) = в.
Proof. By Lemma 2.9 and Theorem 2.10,
c\q) \\ =
for 0 < qi < 1. By Lemma 2.1,
27(m
and
X "~ a*(q) (m2+6m-3K
?>(<?) 2m3/4
**M) =
A2.8)
A2.9)
33. Elliptic Functions to Alternative Bases 167
which is A2.4). From C.12), C.15), and the definition of m, we observe that
m = m(q) maps the interval @,1) monotonically onto the interval A, 3), and if
we view A2.8) as defining x in terms of m, x = x(m) maps the interval A, 3)
monotonically onto the interval @,1). We conclude that x = c3iq)/a3(q) maps
the interval @,1) monotonically onto itself. We see that equation A2.8) is solvable
by rewriting it as
c\q) 27(M + 2J
a\q)
+ 6K
=-x(M),
A2.10)
where
m- Г
A2.11)
In terms of ^-series, M is the sum of two theta-products, namely,
M = m +
m- 1
;
by A.17) and B.13H2.15). Since 1 < m < 3, we see that M > 6. We may solve
the quadratic equation A2.11) for m. The solution is given above in A2.5). We have
taken the negative square root since 1 < m < 3. Since A2.10) is a cubic equation in
M, we may solve it in terms of radicals. Note that д:(-3) = jeF) = 1, x(-2) = 0,
and
dx_ 27(M + 2)(M - 6)
dM ~ (M + 6L
Thus, x(M) decreases from 1 to 0 on (-3, -2), increases from 0 to 1 on (-2,6),
and decreases from 1 toOonF, 00). Therefore, if 0 < x < 1, equation A2.10) has
three real roots, and each of the intervals (—3, —2), (—2,6), and F, 00) contains
exactly one root. Since M > 6, we must take M to be the largest of the roots. The
solution is given in terms of radicals in A2.6) above and more explicitly in A2.7).
For h — 5, д has a particularly simple form.
Theorem 12.2. For 0 < x < 1,
Proof. With h = 1/r, we recall that
qr — q,(x) = exp(-7ry/sinGr/))),
where у is defined by A2.2). We shall prove that
A2.12)
A2.13)
A2.14)

168 Ramanujan's Notebooks, Part V
IfO <q < l.then
q - exp(-7r/)
for some unique t > 0. Set
x :=
In view of A2.15), we consider x = x(t) as a function of t. Define
z@ = <p\q)
Then, by Lemma 9.14,
2Fiil\;U*
We will prove that
and
xB/t)= 1 -x(t)
zB/t) = U2z{t).
A2.15)
A2.16)
A2.17)
A2.18)
A2.19)
A2.20)
We first show that A2.19) and A2.20) imply A2.14). To that end, by A2.2),
A2.19), A2.18), and A2.20),
/zB/l) _ J__
z@ л/2'
A2.21)
Hence, by A2.13), A2.21), and A2.15),
<74(*(O) = ехр(-лл/2у) = exp(-7rt) = q,
which is A2.14). It remains to prove A2.19) and A2.20).
We will need the transformation formulas (Part III [3, p. 43, Entry 27(ii), (i)])
e'"') A2.22)
and
A2.23)
Thus, by A2.23), A2.22), B.1), (9.31), and (9.25),
= if2
33. Elliptic Functions to Alternative Bases 169
= if2 (<p\q) + \6qt\q2))
= \t2z(t).
by A2.17). This proves A2.20).
Next, define
u>@ := 64?xlr\q2)<p\q) =
by (9.25). Hence, by A2.16) and A2.17),
MO
x(t) =
zHO'
Now, by A2.24), A2.22), and B.1),
ш<2/0 = \t\\-q)
= Jf4(«2(O-w(O).
by A2.17) and A2.24). Hence, by A2.25), A2.26), and A2.20),
which is A2.19).
Hence, from A2.1), A2.14), and A2.18),
If we let
then
a - a(q) =
x =
4a
and so, by A2.27),
By B.1) and E.1),
<p\q)
= (g;
(
A2.24)
A2.25)
A2.26)
A2.27)
A2.29)
A2.30)

170 Ramanujan's Notebooks, Part V
Thus, a(q) maps the unit interval @,1) monotonically onto itself. If we define x
as a function of a or q via A2.28), then .v (a) and hence x(q) map the unit interval
@, 1) monotonically onto itself. Hence, given 0 < x < 1,
2-х -
a =
since 0 < a < 1. An easy calculation gives
\+a 2
and putting this in A2.29), we deduce A2.12).
We now relate ц(х, ?) in terms of radicals.
Theorem 12.3. //0 < x < 1, then
JjTTlp + n/1
where
P =
A2.31)
and у is the root of the cubic equation
x =
21y2
4A + yK'
which is between 0 and 2. Explicitly, у is given by
-tan-'tev/U - x)x\ 8*2 - 36* + 27) - — |
3 3 /
A2.32)
у = — FV9-8x
4x V
-4A
Proof. Let M(q) and N(q) be the Eisenstein series defined at the beginning of
Section 4. Let
A2.33)
у = У(ч) =
and define
A2.34)
p = p(q) = y/\-y + y'-y-
We note that both у (,) and piq) map the unit interval @ 1)'»"»?**? ontO
itself. Recall from Entries 13(i), (ii) of Chapter 17 (Part III [3, p. 126]) that
M(q2) = <p\q)(l- a +a2)
33. Elliptic Functions to Alternative Bases 171
and
N(q2) = <pn(qK\ + or)(l - io)(l - 2a),
where, by E.18) and E.19),
Replace q by ^/q, so that a is replaced by 1 — y. We then deduce that
and
From A2.34),
so that
B -
y).
У =
1 + p + p2
It is interesting to note that, by B.1),
pB+p)
which gives an identification fora in A1.1). Thus, by A2.34И 12.38),
x : —
4A - у + у2K'2 2
A2.35)
A2.36)
A2.37)
A2.38)
A2.39)
A2.40)
\\ + pJ
27p2(l + p)
4A
which gives an identification for fi in A1.1). Hence, from Theorem 11.1, A2.40),
A2.39), A2.38), A2.35), E.18), and E.19),
= "<«>¦
A2.41)
We now prove that

172 Ramanujan's Notebooks, Part V
where qe(x) is defined by A2.13) or A.9). It is well known that (Rankin [1, p.
198]) M(q) and N(q) (with q = ехрB7Г(Ч)) are modular forms of weights 4 and
6, respectively, and multiplier systems identically equal to 1 on the full modular
group. Hence, for Im(r) > 0,
iz) A2.43)
A2.44)
A2.45)
A2.46)
A2.47)
and
N(e-2ni/T) =
As in the previous proof, we set
q = exp(—2тгО
for t > 0. Then A2.43) and A2.44) become
and
With л: defined by A2.40), we consider x asafunction of t. Setz(O := M (q), so
that, from A2.46),
A2.48)
and, from A2.41),
2F,(i.|
By A2.40), A2.46), and A2.47),
A2.49)
A2.50)
Hence, from A2.2), A2.50), A2.49), and A2.48),
_ 2Fi(?,|;l;l-x(Q)
У" 2F,(i.|;l;x@)
2F,(if|;l;x(l/0)
1/4
A2.51)
Therefore, by A2.13), A2.51), and A2.45),
= ехр(-2тгу) = ехр(-2тгО = q,
<p(q)
which is A2.42).
It follows from A2.42), A2.41), and A2.1) that
11 \
'б
A2.52)
From Entry 25(vi) of Chapter 16 (Part III [3, p. 40])
4>2(q) = \ (q>
33. Elliptic Functions to Alternative Bases 173
and so, by A2.33),
= 50
Hence, from A2.35), we find that
A2.53)
From A2.40), A2.52), A2.53), A2.37), and A2.38), we find that
as claimed.
We now rewrite A2.40) as
x:=x(y):=
27y2
A2.54)
where
У = РО+Р). A2.55)
Since 0 < p < 1, we have 0 < у < 2. Solving A2.55) for p, we deduce A2.3H).
We have taken the positive square root since 0 < p < I. Since A2.54) is a cubic
equation in y, we may solve it in terms of radicals. Note that jc(-i) = *B) =
l,;r@) = 0,and 4
dx
dy
27y(y - 2)
4A + yL '
Thus, x(y) decreases from 1 to 0 on (-{, 0), increases from 0 to 1 on @,7),
and decreases from 1 to 0 on B, oo). Therefore, if 0 < x < 1, equation A2.54)
has three real roots, and each of the intervals (- \, 0), @,2), and B, oo) contains
exactly one root. We take у to be the unique root that satisfies 0 < у < 2. T'he
root у is given explicitly in A2.32) above.
Some consequences of our derivations of the values of д(лг. Л) are some n<ew
hypergeometric transformations. From B.13) and B.14), we see that
a= I6qijr\q2) = (m- 1)(ю+3K
<p\q) ~ 16w3
A2.56)
Hence, from A2.56), A2.30), E.18), E.19), A2.9), Lemma 2.6, and A2.8),
(m - l)(m-
16m3
4m3^2
= 4>4q) =
m2 + 6m~3
a{q)

174 Ramanujan's Notebooks. Part V
F (I I- i.
' \3'3' '
F (
6m-32 ' \3'3
This transformation is remindful of Theorem 5.6 but is a different transformation.
We have found a generalization of A2.57) via MAPLE, namely,
6 6 16m
1)(т + 3)'\
——
16m3 )
1 .5 TJ(m + lLm> - \)
-;rf+-; ^ + ^ _ 3K
A2.58)
We omit the details. We have also found a generalization of Theorem 5.6 via
MAPLE, viz.,
lFi
A2.59)
1 j 1 27p2(l +
We have investigated the connection between A2.57) and Theorem 5.6 and
found that A2.57) follows from Theorem 5.6 in the following way. Replace m by
1 + 2p, so that A2.57) can be rewritten as
2 '
П_ 1
\2'2' '
A 2 27p(l+pL
2 ЧЗ'З' '2(l+4p + p2)
(p pj
A2.60)
Also, replace p by 2A + p + />2)/B + 2p - p2) - 1 in Theorem 6.1 to obtain
the identity
We now indicate the steps that lead from Theorem 5.6 to A2.60). First, apply
Entry 6(i) of Chapter 17 of the second notebook (Part III [3, p. 238]), i.e.,
to the left side of A2.60). Second, apply Theorem 5.6. Third, invoke A2.61). Last,
after employing Theorem 6.4, we deduce A2.60).
33. Elliptic Functions (o Alternative Bases 175
It is interesting to note that A2.61) and Theorem 5.6 give the transformation
We have found a generalization of A2.62) via MAPLE that is different from
A2.58) and A2.59), namely,
3
_ / 4A+2/» У
-U+2P-P2J)
\\+p)
3' 2' 2B
A2.63)
Using Theorems 4.2 and 4.3 in A2.41) and then employing Theorem 2.10, we
find that
+8дс)
\3 3 /
1 5 1
-, -; 1;
6 6 2
, ; 1; ;
6 6 2 2(l+8x)§
On replacing x by (дг2 - 1)/8 we find that
(*-
2*1
(i-h
We have found a generalization via MAPLE, namely,
. 2 5 (х 3
A2.64)
V 3 6 8/ (,265)
Equation A2.64) has an elegant ^-version; if we replace x by m in A2.64) and
use A2.56) and Lemma 5.5, we find that
13. Some Enigmatic Formulas Near the End of the Third
Notebook
Near the bottom of page 392 (or possibly at the top of page 392, since the page
is reproduced upside down in the published notebooks [9]), Ramanujan recorded

176 Ramanujan s Notebooks, Part V
the following claims:
A3.1)
and
1+4 i
1 -Ix
A3.2)
These two claims were very difficult for us to interpret. First, Ramanujan did
not provide enough terms on the left sides to determine a general term in either
case, and it would appear that the series on the left sides do not converge for any
x, except trivially for x = 0,1. Second, although у is not defined, it is reasonable
to guess (from Chapter 17) thai
A3.3)
Third, Ramanujan had never before used the notation (], and so we did not know
if [4/y] = Л/у and [3/y] = 3/y, or if some other meaning should be attached to
the notation [ ]. (Ramanujan never used [ ] to denote the greatest integer function.)
Fourth, byA3.3), the rightsidesofA3.1)andA3.2) have singularities at* = 0,1,
but if the left sides are convergent series, they must be analytic at x =0,1.
Eventually, we determined that the nth terms of the series on the left sides of
A3.1) and A3.2) have (и!K in the denominators, which is not evident from Ra-
Ramanujan 's formulations. The conjectured definition of у in A3.3) is incorrect, and
the expressions [Л/у] and [3/y] indicate that the results belong to Ramanujan's
alternative theories of signatures 4 and 3, respectively. (See A.8) and A.7), re-
respectively, for the definitions of у in these theories.) Moreover, the expressions
[4/y] and [3/y] should be deleted from A3.1) and A3.2), respectively. For some
inexplicable reason, Ramanujan indicated that his identities arose from the two
alternative theories by placing the "symbols" [4/y] and [3/y] in the midst of the
formulas.
We now offer precise renditions of A3.1) and A3.2).
Entry 13.1 (p.392). Let 0 < x
then
\.lfyisgivenbyA.8)andz = 2^1C- 3; 1;*).
я=0
33. Elliptic Functions to Alternative Bases 177
If У is given by A.7) and z — 2^1 C, 5; 1; x), then
n=0
(и!K
«">
Proof. We first prove A3.4). Recall that
where q = e~y. From the definition A3.6), it is easy to see that
From the proof of Theorem 9.11,
L(q) = A - 3x)z2 + 12*0 - x)z4-
dx
and
L(q2) = 1B - 3x)z2 + 6*0 - x)z^.
Thus, by A3.7H13.9), it suffices to prove that
A3.6)
A3.7)
A3.8)
A3.9)
л=0
(и!K
A3.10)
In Clausen's formula, Entry 13 of Chapter 11 (Part II [2, p. 58]), put a =
-g, fi — -|, and у — I to deduce that
2F2(i,f;l;jc)= jf2(i,|,i;l,l;x). A3.11)
In Entry 12 of Chapter 11 (Part II [2, p. 56]), set x = -|, у = -§, z = 1, and
p = 4*A — *). Accordingly,
f; 1; 4*A - x)) =
Thus, from A3.11) and A3.12),
гР}{\, \\\\x)= 3F2(i, |, i; 1, 1; 4x(l - x)).
A3.12)
A3.13)

178 Ramanujan's Notebooks, Part V
Using A3.13), we see that A3.10) is equivalent to the identity
(и!K
Differentiating A3.13), we find that
2A -2x)dx'
_ x)]n-i D _ 8x),
A3.14)
which i. readily seen to be equivalent to A3.14), and so the proof of A3.4) is
'proof of A3.5) is similar. First, from A3.6), we easily find that
¦ -Ш3). A3-15)
я=1
From Lemma 4.1,
dz
where z = iFdk, \\ U *)• We need a *Ш* rePresentatio"
end, from Corollary 3.5 and D.4), we find that
A3.16)
for L(fl3). To that
Hence, from A3.15H13-
A3.17)
_rfz—. A3.18)
Thus
, by A3.5), A3.15), and A3.18), it is sufficient to prove that
?¦
К
о kV
A3.19:
-"-
33. Elliptic Functions to Alternative Bases 179
Jt = -i,y = -\,z = \,and p = 4дгA -дс) in Entry 12 of Chapter 11 (Part II
[2, p. 56]). Combining these two formulas together, we deduce that
Z2=
(±,§,±;1,1;4*A -x)).
Using A3.20) in A3.19), we find that it suffices to prove that
,.,,_i z dz
„=1 v»., 2(\-2x)dx
Differentiating A3.20), we achieve A3.21) to complete the proof.
At the very bottom of page 392, Ramanujan [9] wrote
. 1 -2
A3.20)
A3.21)
22-(GO'/*
' l " 3
J «*'' A3.22)
The notations G, G', g, and g' were not defined by Ramanujan. In view of the
appearance of 2^E, |; 1;/), it would appear that the latter two equalities pertain
to a new (unknown) type of class invariant associated with the theory of signature
3. Chapter 34 is devoted to class invariants, and in the table of class invariants
in his second notebook, Ramanujan uses the notations G := Gn and g := gn.
However, no dependence on n is indicated in A3.22). Also, the appearance of two
invariants in each equality should be reflected in the appearance of two distinct
moduli on the left side. Thus, we are left with the conclusion that Ramanujan is
claiming two ^-series identities, one for the pair G, G', and the other for the pair
g, g', whatever these "invariants" might be.
Now, by Corollary 3.2 and Lemma 5.1,
{f (i _
a(q)a(q) a2(q)
= qW3 C - 42? + 393?2 - 3240c?3 + ••¦),
A3.23)
where we employed Mathematica to obtain the ^-expansion. Ramanujan, in his
second notebook, defined С by
G1/24 =
21/4^
1/24
and later, in his paper [3], he gave the different definition
Using either definition, and any similar representation for G', we do not obtain an
expansion for the middle expression in A3.22) that is close to that in A3.23).

180 Ramanujan's Notebooks, Part V
More generally, suppose that
GG' =2"(l+bqc+ ¦•¦).
Then
22-(GG'I/4 22-2a/4(l + (b/4)q' +
3 (GG'Y^ ~3 2°'4\ + (b/6)qc + ¦ ¦ •) '
Thus, clearly, if A3.22) holds, we must have a = 4. It then further follows that
с = з and /> = -9 • 22/3. We are unable to use the resulting expansion
GG' = 16A -9 22/У/3 +•••)
to identify GG' in any meaningful way.
14. Concluding Remarks
It seems inconceivable that Ramanujan could have developed the theory of signa-
signature 3 without being aware of the cubic theta-function identity B.5), and in Lemma
2.1 and Theorem 2.2 we showed how B.5) follows from results of Ramanujan. H.
H. Chan [ 1 ] has found a much shorter proof of B.5) based upon results found in
Ramanujan's notebooks.
H. F. Farkas and I. Kra [1, p. 124] have discovered two cubic theta-function
identities different from that found by the Borweins. Let со — ехрBл i/6). Then,
in Ramanujan's notation,
/3 {a>q2<\ wq^) = <»f (-«"*. W)
wq2")
and
/3
^) - /3
, &).
H. F. Farkas and Y. Kopeliovich [1] have generalized this to a pth order identity.
Garvan [2] has recently found elementary proofs of the cubic identity and the pth
order identities, and has found more general relations.
Almost all of the results on pages 257-262 in Ramanujan's second notebook
devoted to his alternative theories are found in the first notebook, but they are
scattered. In particular, they can be found on pages96,162,204,210,212,214,216
218,220, 242, 300, 301, 310, and 328 of the first notebook. Moreover, Theoren
9.11 is only found in the first notebook.
In Section 8, we crucially used properties of b(z, q), a two variable analogui
of b(q). The theory of two variable analogues of a(q), b(q), and c(q) has beei
extensively developed by Hirschhorn, Garvan, and J. M. Borwein [1] and by S
Bhargava [1].
Some of Ramanujan's formulas for Eisenstcin series in this chapter were als
established by Venkatachaliengar [1].
In [3], Garvan describes how the computer algebra package MAPLE was use
to understand, prove, and generalize some of the results in this chapter.
33. Elliptic Functions to Alternative Bases 181
Small portions of the material in this chapter have been described in expository
lectures by Berndt [10] and Garvan [1]. рщ

34
Class Invariants and Singular Moduli
1. Introduction
So that we may define Ramanujan's class invariants, set
oo
(а;д)в0 = ПП-в?я). 1*1 < 1.
я=О
and
x(<7) = (-<?;<?2)oo. (l.i)
if
q = exp(-TrVw), A.2)
where и is a positive rational number, the two class invariants Gn and gn are
defined by
Gn:=2-"V'/24X(<7) and ft, :=2-"VI/24X(-«). A-3)
In the notation of Weber [2], Gn =: 2/4f(,/^ri) andgn =: 2-'^,(У^). The
definition of Gn employed by Ramanujan in his paper [3], [10, pp. 23-39] is not
the same as that used by him in his notebooks [9], while his definition of gn in [3]
is that used in his first notebook but not in his second notebook. More precisely, If
we replace G and g in the second notebook by H and A, respectively, the relations
between the definitions are given by
G?=-±- and ,f = -i-.
As usual, in the theory of elliptic functions, let к := k(q), 0 < к < 1, denote
the modulus. The singular modulus kn is defined by kn := k(e~"^"), where и is
a natural number. Following Ramanujan, set a = k2 and а„ = к*.
It is well known that Gn and gn are algebraic; for example, see Cox's book [1,
p. 214, Theorem 10.23; p. 257, Theorem 12.17]. However, much more is known.
Weber [2, p. 540] and, more recently, H. H. Chan and S.-S. Huang [1], using a
result of Deuring [1], have proved the following theorem.

184 Ramanujan's Notebooks, Part V
Theorem 1.1.
(a) Ifn = 1 (mod 4), then Gn and2<xn are units.
(b) Ifn = 3 (mod 8), then 2-lJl2Gn and 22ocn are units.
(c)Ifn = 7 (mod 8), then 2~l/4Gn and2Aan are units,
(d) Ifn s 2 (mod 4), then gn andan are units.
As G. N. Watson [61 remarked, "For reasons which had commended themselves
to Weber and Ramanujan independently, it is customary to determine Gn for odd
values of n, and g,, for even values of л."
At scattered places in his first notebook [9), Ramanujan recorded the values for
107 class invariants, or polynomials satisfied by them. On pages 294-299 in his
second notebook [9], Ramanujan gave a table of values for 77 class invariants,
three of which are not found in the first notebook. Since the second notebook
is an enlarged revision of the first, it is unclear why Ramanujan failed to record
33 class invariants that he offered in the first notebook. By the time Ramanujan
wrote his paper [3], 110, pp. 23-39], he was aware of Weber's work [2], and
so his table of 46 class invariants in [3] does not contain any that are found in
Weber's book [2]. Except for G325 and G.%3, all of the remaining values are
found in Ramanujan's notebooks; twenty-one of these class invariants are found
in his second notebook. At scattered places in the second and third notebooks,
Ramanujan recorded irreducible polynomials satisfied by four further invariants.
In conclusion, to the best of our tallying, Ramanujan calculated a total of 116 class
invariants, or monic, irreducible polynomials satisfied by them.
In two papers [6], [7], Watson proved 24 of Ramanujan's class invariants from
Ramanujan's paper [3]. In the first [6], Watson devised an "empirical process" to
calculate 14 of the 24 invariants, while in the second [7], he employed modular
equations to prove 10 invariants. In another paper [5], Watson established Ra-
Ramanujan's value for G1353, communicated by Ramanujan [10, p. xxix] in his first
letter to Hardy, and also stated in his paper [3]. In the introduction to [6], Watson
remarked, "It is intended to publish the calculations involved in the construction
of the set N + Q (the invariants appearing in both Ramanujan's paper [3] and
the second notebook) as part of the commentary on the notebooks by Dr. В. М.
Wilson and myself." Although Watson and Wilson's efforts to edit Ramanujan's
notebooks have been preserved in the library at Trinity College, Cambridge, Wat-
Watson's calculations of these twenty-one invariants are not found there. If Watson
actually calculated these invariants, it appears that his work has been lost. The
twenty-one values of n are: 65,69,77,81,117,141,145,147,153,205,213,217,
265,289, 301,441,445, 505,553,90, and 198.
Watson wrote four further papers [9], A0], [12], [13] on the calculation of
class invariants. The values of n considered by Watson depend upon the class
numbers for positive definite quadratic forms of discriminant -.1. In the course of
his evaluations, he determined the class invariants for n = 81 [12], 147 [12], and
289 [13]. Thus, after Watson's work, 18 invariants of Ramanujan from his paper
[3] and notebooks [9] remained to be verified.
34. Class Invariants 185
Five of these invariants are established in Section 3. For each of these five
values, 117, 153, 441, 90, and 198, и is a multiple of 9, and proofs are effected
by formulas relating G<>n with Gn and ggn with gn, which we establish by using
one of Ramanujan's modular equations of degree 3. The latter formula is found
on page 318 of Ramanujan's first notebook, but not in his second notebook, while
the former formula is not found in any of the notebooks.
Since modular equations are crucial in our work on class invariants, we now give
a precise definition of a modular equation. Let K, K', L, and Z/ denote complete
elliptic integrals of the first kind associated with the moduli k, k' := Vl — k2, ?,
and С :— л/1 - i2, respectively, where 0 < k, t < 1. Suppose that
Г V
A.4)
for some positive integer n. A relation between к and I induced by A.4) is called
a modular equation of degree n. Following Ramanujan, set
a = k2 and 0 = I1.
We often say that fi has degree и over a.
As usual, in the theory of elliptic functions, set
q :=
Sincex(q) = 21/61аA-а)/<?Г1/24and X(-q) = ^
III [3, p. 124]), it follows from A.1), A.3), and A.5) that
-а„)
Г1/24
and
A.5)
-<*Г2/<7Г1/24(РаП
A.6)
This formula for Gn will be used in certain modular equations.
In Sections 4-7, we establish the remaining 13 values, each for G,,, n — 65,69,
77, 141, 145, 205, 213, 217, 265, 301, 445, 505, and 553, claimed by Ramanu-
Ramanujan. Quite remarkably, the class number for each of the 13 imaginary quadratic
fields Q(v/—n) equals 8. Moreover, there are precisely two classes per genus in
each case. Our first proofs, given in Sections 4 and 5, employ Kronecker's limit
formula, which is used to find representations for certain products of Dedekind
eta-functions in terms of fundamental units; see Theorems 4.1,4.2, and 4.7. Each
of the 13 values of и is a product of a small prime C,5, or 7) and a larger prime.
Thus, our proofs also crucially employ certain modular equations of Ramanujan
of degrees 3, 5, and 7. It is highly unlikely that Ramanujan was familiar with
Kronecker's limit formula and the arithmetic of quadratic fields, and so our proofs
certainly are not those found by Ramanujan. However, Ramanujan obviously dis-
discerned some unique arithmetical properties in these instances, and it would be
fascinating to discover Ramanujan's approach. The methods in Sections 4 and 5
have been further extended by L.-C. Zhang [2], [3] who has rigorously estab-
established all the invariants in Watson's paper [6] that were calculated there with his
"empirical process."

186 Ramanujan's Notebooks, Part V
Ramanujan used modular equations to calculate only a couple of simple in-
invariants in [3]. This fact and the sentence, "The values of Gn and gin are got
from the same modular equation." [3J, [10, p. 251 are the only clues to his
methods that Ramanujan provided for us. It would seem that if Ramanujan had
employed another type of reasoning, he would have dropped some hint about
it. As mentioned earlier, Watson G) used modular equations to establish some
of Ramanujan's invariants. However, for his calculations of Gn, it is impor-
important that я be a square or a simple multiple of a square. We have been able to
prove six of the remaining thirteen values for Gn, namely, for и = 65, 69, 77,
141, 145, and 213, by using modular equations. As will be seen in our proofs
in Section 6, we need some new ideas to effect proofs of these six invariants
via modular equations. To prove the remaining seven invariants by employing
modular equations, we would need modular equations of degrees 31, 41, 43,
53, 79, 89, and 101. Apparently, only for degree 31 did Ramanujan derive a
modular equation, for he recorded no modular equations for the other six de-
degrees in his notebooks. Thus, Ramanujan's methods appear to be even more elu-
elusive.
Watson [6, p. 82) opined that "I believe that fourteen were obtained by Ra-
Ramanujan by means of the empirical process which I described in the discussion
of G1353" We are not so confident that Ramanujan used this empirical process,
for which Watson offered little explanation. In fact, Watson's "empirical process"
is not rigorous. However, in Section 7 we shall use class field theory to make
Watson's procedure rigorous for a large class of invariants including those 13 in-
invariants mentioned above, and we use the process to calculate two new invariants
as well. Chan [3] has further extended the methods of Section 7 and calculated 27
new class invariants.
Section 8 is devoted to some miscellaneous results on class invariants, including
two entries to which we have not been able to attach any meaning. Here we also
establish Ramanujan's more detailed assertions about the irreducible polynomials
satisfied by G29 and G79.
In Section 9, we turn to Ramanujan's singular moduli. Once Gn or g,, is known,
then it is easy to calculate an from A.6) by simply solving a quadratic equation.
However, this expression for an that one trivially obtains from the quadratic for-
formula is usually not very interesting or attractive. Thus, it is desirable to develop
other algorithms for the calculation of an that will reflect the unit structure of an
described in Theorem 1.1. For the calculation of an, when и is even, Ramanujan
devised a very clever algorithm given in Theorem 9.1. For odd n, we do not have
such an inclusive algorithm, and so we had to develop some lemmas to facilitate
calculations.
In Section 10, a simple function of singular moduli is studied.
In the last two pages of his notebooks, Ramanujan studied the ^'-invariant.
He seems to have quoted some results from the literature. However, Ramanujan
made some remarkable discoveries, including very simple polynomials satisfied
by certain algebraic functions of the /-invariant. Ramanujan's work on the j-
invariant is the topic of Section 11.
34. Class Invariants 187
2. Table of Class Invariants
Both prior to his table of class invariants in his second notebook and at the begin-
beginning of his paper C, eqs. E), G)], A0, p. 23], Ramanujan recorded two simple
formulas relating these invariants, and so we first state and prove these here. See
also Exercise 5c on page 73 of the Borweins' treatise [1].
Entry 2.1 (p. 294, NB 2). For n > 0,
g4n=2l/48nGn.
Proof. This identity is an immediate consequence of the definitions of Gn and gn
in A.1)—A.3) and the elementary identity
(<72; <74)oo = iq: <?2)co(-<?; ?V. B.1)
Entry 2.2 (p. 294, NB2). For n > 0,
Proof. Jacobi's identity for fourth powers of theta-functions (Part III [3, p. 40,
Entry 25(vii)J) can be written in the form (Whittaker and Watson [1, p. 470])
(-*;rt-<»;?2>« =
where we used Euler's identity
(-<?;<?)oc =
By B.1), we can write B.2) in the form
B.2)
1
(q ¦ q2)oo
2,
x B-Vl/3(-«?; q2)l - 2-V1/3(«;
that is, by A.1)-A.3), when q = ехр(-к*/п),
(8nGn)\G*n - gBn) = I.
Recall from Part III [3, pp. 91, 102] the definition
0<x < 1, B.3)
where 2^1E. 5; 1; x) denotes the ordinary hypergeometric function. Recall also
from Part HI [3, p. 36] or Chapter 33 the theta-function
B.4)

188 Ramanujan's Notebooks, Part V
and the fundamental formula (Part III 13, p. 102])
34. Class Invariants
189
B'5)
if , » given by A.5). (The evaluations of X<«> and x (-<?> -«» * 0*>
uponB.5).)If*=«»inB-3)'then
Now from A.6),
and so, since ctn <
*l->
B.7)
B.8)
In the first notebook, Ramanujan frequently records Gn, or equivalently, a,,, in
the form
)) B.9)
C (» ~
( ( ))
by B.6) and B.8). We have indicated such a representation in the tables by placing
"F' after the page number in the first notebook where the corresponding invariant
(or, equivalently, singular modulus) is located.
We next give a table of all the values of Gn and gn found by Ramanujan. We
emphasize that formulas for certain values of Gn and gn may be different in both
the first and second notebooks and Ramanujan's paper [3]. In most cases, it is
not difficult to verify that Ramanujan's formulations are equivalent. In particular,
when Ramanujan employs B.9), a modest amount of calculation is needed. For
example, such calculations are necessary on pages 284, 287, 292, 293, 296, and
311. In all these instances, the calculations are routine, and there is no need to
give them here. In other instances, for example, on page 314 in the first notebook,
Ramanujan records the value for G~24 in the notation KA - p)" that is, he is
using A.6) with а„ replaced by f>.
When the invariant is a root of a cubic polynomial, Ramanujan, as well as
Weber [2], normally, but not always, only gives the polynomial. There are some
instances when Ramanujan calculated the appropriate root but Weber did not. In
particular, we establish Ramanujan's values for G75 and G175 in Section 8. How-
However, Ramanujan's polynomials are those satisfied by 1/BI/4G,,), instead of Gn-
Thus, on page 345 in his first notebook, Ramanujan recorded the irreducible poly-
polynomials satisfied by 1/B1/4С„), for n = 23,31,11.19,27, 43, and 67. These
polynomials are repeated on page 351 together with monic irreducible polynomi-
polynomials satisfied by l/Bl/*G3) and by (erroneously) 1/B1/4G7). In the tables in his
second notebook, Ramanujan explicitly states G,, for и = 3,7, and 27.
Lastly, we remark that the tables of Weber {21 contain some errors. Correction:
have been made by J. Brillhart and P. Morton [1].
J
For convenience, we use the following abbreviations in citing sources for the
listed invariants:
Ramanujan's first notebook: N1,
Ramanujan's second notebook: N2,
Ramanujan's third notebook: N3,
Ramanujan's paper [3]: RP,
Watson's paper [5]: WXIV,
Watson's paper [6]: WI,
Watson's paper [7]: WII,
Watson's paper [9]: W3,
Watson's paper [10]: W4,
Watson's paper [12]: W5,
Watson's paper [13]: W6,
Weber's treatise [2]: We,
Brillhart and Morton's corrections [1]: BM.
Table of Gn
n = 1
2I/12
Refs.: N2,RP,W5,We
Refs.: NlB82F),N2,W3,We
(-
1/4
2l/4
Refs.: N1B87F), N2,RP,We
Refs.: NlB87F),N2,W3,W4,We
1/3
«=11
Refs.: NlB84F,287F),N2,RP,We
2~l'4x, where хг - 2x2 + 2x - 2 = 0
Refs.: NlB95F,345,351)N2,W3,We

190 Ramanujan's Notebooks, Part V
Table of Gn (Continued)
n = 13
1/4
n= 15
n= 17
n= 19
и = 21
л =23
и = 25
и = 27
и = 29
Refs.: NlB92F),N2,RP,We
1/3
Refs.: NlB89F),N2,W4,We
л/17-З
8
Refs.: NlB96F),N2,RP,We
2/4л where хъ - 2x - 2 = 0
Refs.: Nl{295F,345,351),N2,W3.We
'/6
Refs.: NlB93F),N2,We
2|/4дс, where jc3 - л - 1 = 0
Refs.:NlB95F,345,351)>N2,W3)W4,We
Refs.: NlB87F),N2,RP,We
l
,-1/3
Refs.: NlC05F,345,351)>N2,W3,We
= x, where x6 - 9x5 + 5x4 + 2x3 - 5x2 - 9x - 1 = 0
Refs.: N2B63),We
34. Class Invariants 191
Table of С (Continued)
2'/4jc, where x3-x2- 1 =0
Refs.: N1 B96F.345.351 ),N2,W3,W4,We
я = 33
у7
V2 j V V2
и = 37
и = 39
F + V37)
1/4
Refs.:NlC11F),N2,We
Refs.:NlC05F),N2,RP,We
= 41
=x, where
/ IV 5 + V4T/ 1\ 7
чЯз-з\
8 j
Refs.: NlC05F),N2,W4,We
V4T
=0
Refs.: N3C82),WE,BM
n = 45
2"l/4x where x3 - 2x2 - 2 = 0
Refs.: NlC13F,345,351),N2,W3,We
1/3
Refs.: N2,We
n = 47
2'/4jr, where л5 = A + jt)A +
Refs.: NlB34),N2B63),We
Refs.: NlB93F)N2,RP,We

192 Ramanujan 's Notebooks, Part V
Table of Gn (Continued)
и = 55
8 ' V 8
Refs.: NlC15),N2,W4,We
л = 63
2I/4
я =65
л = 67
« = 69
я = 73
и = 75
1/4
Refs.:NlC15),N2,We
Refs.: NlC05F),N2,W4,We
1/2
Refs.:NlC15),N2,RP
where л3 - 2x2 - 2x - 2 = 0
Refs.: NlC45,351),N2,W3,We
3V3
+
1/2
4 V 4
Refs.: N1C14F,315),N2,RP
V73 /1+V73
Refs.: NlC13F),N2,RP,We
3.25/.2
. 51/6 _ Vs -
Refs.: NlC11),We
n = 77
n = 79
я = 81
n = 85
я =93
n = 97
« = 105
n = 117
34. Class Invariants 193
Table of Gn {Continued)
1/2
4 V 4
Refs.: N1C15),N2,RP
where ts - tA +11 - 2/2 + 3/ - 1 = 0
Refs.: N2B63,300)
Refs.: N2.W5.RP
Refs.:NlC15),N2,RP,We
V2 )
13 + V97
1/4
8
5 + V97
8
Refs.: N1C15), N2,We
Refs.: NlC05F),N2,RP,We
1/12
Refs.:NlC17),N2,We
Refs.: N1C15),N2,RP

194 Ramanujan's Notebooks, Part V
Table of Gn (Continued)
и = 121
1
1/3
13
+ 3V3 + 4)'/3 + (ЗуГГ\ - 3V3 + 4)'/3J + 2J
Refs.: N1B94F,317)>RP,WII)W5
n = 133
1/4
я = 141
V47)
Refs.: NlC15),N2,We
1/2
Refs.: NlC20),N2,RP
n = 145
(V5 + 2)
1/2
Refs.: N1C15),N2,RP
я= 147
Refs.: N2,RP,W5
n = 153
E + V17 /л/П-З
37 + 9VT7
J33 + 9VV7\
V 4 )
1/3
Refs.: N1C15),N2,RP
n= 163
n= 165
(¦¦
2- 1/4x where x3 - 6x2 + 4л - 2 = 0
Refs.: N2C00).We
(,/5 + 2)
Refs.:NlC17).N2,We
34. Class Invariants 195
Table of Gn (.Continued)
= 169
n= 175
и = 177
n =205
n =213
/5лЛ+^\
V 2 /
« =217
/9+4V7 , /11+4V7
Refs.: N1B94F,317),RP,WII,W5
3.2'/4
/l+V5\ /3s/5 + V4l\
\ 2 A 2 /
1/2
" = 225
Refs.: NlC16),We
Refs.: NlC15),N2,We
/V4T-l\
V ^ J
V4T ( /У4Т-
8 ¦ V 8
Refs.:NlC14),N2,RP
f21 + 12V3 /l9+12N/3
1/2
Refs.: N1C15),N2,RP
1/2
Refs.: N1C14),N2,RP
Refs.: N1B93F),RP,WII

196 Ramanujan's Notebooks, Part V
Table of Gn (Continued)
«=253
и = 265
и = 273
я = 289
и = 301
л = 325
n = 333
16
1/4
Refs.: NlC15),N2,We
1/2
Refs.:NlC14),N2,RP
±f(^)">
1/4
Refs.:NlC17),N2,We
+
)
Refs.: N1C17),N2,RP,W6
+
42 + 7V43
1/2
4 V 4
Refs.: N1C25),N2,RP
+ 1 =
Refs.. RP.WII
F + л/зту/VV3 + 2л/37)'/6
/^н-гУз + УзГ^Х
Refs.: N 1C14,315),RP,WI
34. Class Invariants 197
Table of Gn (Continued)
n = 345
+ s/5/l + V3\'
2 U/
n = 357
(8
Refs.:NlC17),N2,We
1/6
Refs.: N1C17),N2,We,BM
n = 363
25/12f, where It* - t2 {D + л/33) + \/l 1+ 2Узз
- f {1 + х/ТГ+гТзз} -1=0
n = 385
n = 441
Refs.: RP.WII
Refs.:NlC17),N2,RP,We
V3 + /7
B +
Refs.: N1D6),N2,RP
n = 445
1/4
Refs.: NlC20),N2,RP
2W3T /б + л/ЗМ //11+2V3T /13 + 2V3T
1/2
Refs.: N1C19),RP,WI

198 Ramanujan 's Notebooks, Part V
Table of Gn (Continued)
n = 505
л = 553
и = 765
и = 777
= 897
i/*
/96+11779
16V79
+
/2 + -
Refs.: N1C44),N2,RP
1/2
1/2
Refs.: N1C2O),N2,RP
1/4
1/2
Refs.: N1C43),RP,WI
1/4
B4677 + 107737 j
1/12
Refs.: N1C19),RP,WI
1/4
1/2
Refs.: N1C2O),W1
34. Class Invariants 199
n = 1225
Table of Gn (Continued)
i±^F+7i5)'/^zi±-
/43 + 15>/7 + (8
35 + 15>/7 + (8
n = 1353
1/4
+ Зл/3\ '/4 /l 1 + y^23\'/4 /
)
Refs.:NlD6),RP,WII
1/12
6817 + 321
17 + 3V33
1/2
/569 +99733 \
V 8 )
561+99V33 /569 + 99733
^ +
1/2
Refs.: N1C19),RP,WXIV
я = 1645
/2 + .
/7 + 747\ '/4 /7375 +97329\'
[-ЗГ) { 2 )
/119 + 77329
v г
/743 + 417329
+
+
1/2
Refs.:NlC20),RP,WI
n - 1677
(. \ 3/4 / . .—\ 1/*
7T|+3\ /743 + 739\ х
355 + 54743 + /351+54743
+
1/2
Refs.: NlC20),WI

200 Ramanujan's Notebooks, Part V
Table of gn
1
Refs.:NlC16),N2,We
n= 10
n= 14
= 18
n = 22
и = 26
Rcfs.:NlC16),N2,We
Refs.: NlC16),N2,RP,We
1/3
Refs.: NlC16),N2,We
Refs.:NlC16),N2,RP,We
Refs.: NlC16),N2,RP,We,BM
3^3C
= 30
= 34
B + >/ТЗ)У2 + л/И - ъ/ю^
Refs.: NlC18,344),W5,We
Refs.:NlC16).N2,RP.We
Refs.:NlC16),N2,RP,We
л = 42
« = 46
n = 50
3
и = 58
и = 62
Table of gn (Continued)
where g3
34. Class Invariants 201
+
Refs.: NlC44),We
^ + V7\'
2 /
Refs.: NlC16),N2,We,BM
Refs.:NlC16),N2,We
Refs.: NlC18,344),W5,We
Refs.: NlC16),N2,RP,We
Refs.: N1C19),RP,WII,W6
и = 70
7+V33 . /V33- 1
~8
l/2
Refs.:NlC19),RP,WI
C + л/5)A + л/2)
Refs.: NlC16),N2,RP,We

202 Ramanujan's Notebooks, Part V
Table of gn (Continued)
n = 78
и = 82
= 94
n=98
n = 102
n= 114
Refs.: NlCl6),N2,We
13 + V41 /5 + 741
8 +V 8
l/4
n= 126
Rcfs.: NlC16),N2,We,BM
Refs.:NlC18),N2,RP
8 )
Refs.:NlC19),RP,WII,W6
+
Refs.: N1C18),RP,WII,W6
Refs.: NlC16),N2,We
/23 + 3V57 /15 + 3V57
1/2
У57\
8 V 8
Refs.: N1C19),RP.WI
1/6
Refs.: N1C18),RP,W1
n = 130
n = 138
n = 142
n = 154
n = 158
я= 190
n = 198
n = 210
34. Class Invariants 203
Table of gn (Continued)
3'2
1W5V /3WHV
2 M 2 J
Refs.: NlC16),N2,RP,We
1/2
Refs.:NlC19),RP,WI
9 +
1/4
Refs.: NlC16),N2,We
1/2
'13+2V22 /9 + 2V22
Refs.:NlC19),RP,WI
13V2
Refs.: N1C19),RP,WH,W6
3/2
Refs.: NlC16),N2,RP,We
8 V 8 J
Refs.:NlC18),N2,RP
Refs.:NlC20),We,BM

204 Ramanujan's Notebooks, Part V
Table of gn {Continued)
и = 238
'l+2V2
i v / I ГГ I
Refs.:NlC19),RP,Wl
= 33O
л = 522
Refs.:NlC19),RP,WI
Refs.: NlC20),We
Refs.: NlC18),RP,WI
я =630
Refs.:NlC18),RP,WI
3. Computation of Gn and gn when 9|n
In this section, we establish Ramanujan's class invariants forn = 117 153,441,
90 and 198. Note that for each such n, 9\n. Our starting point is a relation con-
connecting gn and **,. found on page 318 of Ramanujan's first notebook, but not in
his second notebook. K. G. Ramanathan [4] noticed this relation in the first note-
notebook, but apparently he never gave a proof. Also unaware of its appearance in the
34. Class Invariants 205
first notebook, J. M. and P. B. Borwein [1, pp. 145,149], although not stating the
results explicitly, derived a formula connecting gn and g9n, as well as a formula
relating Gn and G9n. We use one of Ramanujan's modular equations of degree
3 to establish the aforementioned formulas connecting Gn and G9n, and gn and
g9n. The former formula is not found in the notebooks, but it can be proved along
the same lines as the latter. The Borweins [1, pp. 145, 146] also derived formulas
connecting G8i,, with Gn and G9n, and g8|n with gn and g9n.
Of course, the theorems described above can be utilized to establish other class
invariants found by Ramanujan when 9\n, in particular, for n = 27, 45, 63, 81,
225, 333, 765, 18, 126, 522, and 630. Undoubtedly, some of these proofs would
be simpler than previous proofs, for example, for 81,225, 333,765,126,522, and
630. In particular, Watson's proofs for n = 333,765,522, and 630 [6] were based
on his "empirical method." In fact, the Borweins [1, pp. 147,149, 150] employed
the aforementioned formulas to calculate the invariants G21, G%\, G225, and g522-
Moreover, previously undetermined class invariants, for example, for n = 171,
189, and 279 can be calculated. However, we shall confine ourselves here to the
cases, n = 117, 153,441,90, and 98.
Theorem 3.1. Let
C.1)
Then
= G,,
Proof. For brevity, we set G = Gn in most of the proof.
We shall employ Entry 15(xii) of Chapter 19 of Ramanujan's second notebook
(Part III [3, p. 231]). Let
and
-a)
where p has degree 3. Then
C.3)
C.4)
Recall from A.6) that, when q = ехр(-я V").
Gn = Da,,(l -а„)Г1/24 and G9n = {4/3„О - &)Г1/24.
Hence, by C.3), P = (G,,G9n) and Q = (Gn/G9,,N. Thus, from C.4),
(G,,/G9nN + (Gn/G9n)-6 + 2V2 ((GnC9n)  - (GnG9,,K) = 0. C.5)

206 Ramanujan's Notebooks, Part V
Set x = (G«n/ GnK. Then C.5) can be rewritten in the form
x4 - 2x/2GV + 2V2G-6x + 1=0. C.6)
Rearranging and using the notation C.1), we find that we can recast C.6) in the
form
Since л > 1,
(x2 - V2G«x + pf = 2(p2 - 1) (g2* - -J=
2 - s/2G6x +p =
- -J= J ,
or
x2 -
(g4 + vV- i)* + p + vV
Remembering that x > 1 when solving for л, we find that
- 1 +p
-l)-2p- 2v/V-l.
C.7)
Now,
and
Thus, squaring and expanding, we find that
G4 (G4 +
+
~ 4) i (p2 -
2 +
- 2 -I- Py/p2 - 4) V^7! + 5 (/» + УР2 - 4) (p2 - 1)
= (p + У/»2 - l) (p2 ~ 2 + vV ~ D(P2 - 4)) C.8)
and
- 2 + pvV - 4) Ур2-1
p + у/Рг - 4) (p2 - 1) - 2p - 2v/pT^rT
= (p + Vp2 - l) (p2 - 4 + У(Р2 - l)(p2 - 4)) .
C.9)
34. Class Invariants 207
Using C.8) and C.9) in C.7), we deduce that
л =
= ~^(p + vV - l) (p2 - 2 + V^P1"- D(P2 " 4))
~ 4 + V(P2 - D(P2 - 4)).
/ C10)
Recalling that x = (G9n/GK, we see that C.10) is equivalent to C.2), and so the
proof is complete.
We next prove the aforementioned result found on page 318 in Ramanujan's
first notebook.
Theorem 3.2 (p. 318). Let
Then
», = ft, (
on 8n •
C.11)
p2 + 2
C.12)
Proof. Set g = g;, throughout the proof.
Using A.3) and A.1), we rewrite C.5) in the form
= 0.
Multiplying both sides by ^fq and then replacing q by -q, we find that
Using A.3) and A.1) again, we find that
-(g/89,,f + (gvn/gf
Setting x — {gsn/g)*. we deduce that
(?. .2,3 ( 3. 6)Э =
= 0.
1=0.

208 Ramanujan's Notebooks, Part V
Recalling the notation C.11), we see that the last equation can be rewritten in the
form
(x1
+ 1)
/ 1 \2
I g2x + -j= \ .
It should now be clear that the remainder of the proof is completely analogous to
that for Theorem 3.1, and so we omit the rest of the proof.
As a bonus, the formulas connecting GSn with Cn and g9n with gn led to
closed form evaluations of Ramanujan's cubic continued fraction at the arguments
± exp(-7rV") by the author, H. H. Chan, and L.-C. Zhang [1].
The cube roots in C.2) and C.12) are not very attractive, and usually in ap-
applications Ramanujan found more appealing expressions for these cube roots.
Ramanujan had an amazing ability for denesting and simplifying radicals, and we
do not have the insights into radicals that Ramanujan had. However, it seems quite
likely that in several instances Ramanujan used the following elementary result
from Carr's book [ 1, p. 52]. Since Carr does not give a proof and since he adds the
extraneous hypothesis that (a2 — b)'/3 be a perfect cube, we provide a proof here.
Lemma 3.3. Suppose that с := (a2 — b)'/J. Then we can write
(a
where
and
= x
Лх2 -Ъсх-а
= х -с.
C.13)
C.14)
C.15)
Proof. From C.13), we easily see that
(a -
x -
x2 — у
Suppose we set с = x2 — y, so that
(a - VbI/3 = x - Vy- C.16)
Cubing both sides of C.13) and C.16) and solving for a, we find that
a = л3 + Ъху. C.17)
But since у = jc2 - c, we deduce C.14) from C.17).
Usually, it is best to solve C.14) by trial or inspection, for if, for example,
Cardan's method is used, the value of л so obtained most frequently is the cube
root that we originally sought to simplify.
34. Class Invariants 209
Since Carr's book [1J was Ramanujan's primary source for learning mathe-
mathematics, it seems likely that Ramanujan employed Lemma 3.3 in simplifications.
However, because most of us do not possess Ramanujan's ability to discern al-
algebraic relationships, we describe another procedure that rests upon elementary
considerations in algebraic number theory and involves less guessing.
We see from Theorems 3.1 and 3.2 that it would be advantageous to find a
number a such that
Then
C.18)
a 3 =
and
Since
set
a3 + а~г = 2y/b+ 1 + cVd.
и = a -\—,
a
C.19)
C.20)
so that, after squaring, C.19) takes the shape
u2(u2 - 3J = 4 (b + 1 + cVrf) . C.21)
Assuming the relevant expressions above are algebraic integers, we find that, upon
taking norms in C.21),
N(u2)N2(u2 - 3) = N D (b + 1 +
Using C.22), we determine u. We then solve C.20) for a.
C.22)
Entry 3.4.
C.23)
Proof. Let n — 13 in Theorem 3.1. From Weber's treatise [2, p. 721J, or from the
tables of the last section,
С,з =
1/4

210 Ramanujan's Notebooks, Part V
By C.i), p = УТЗ. Thus, by a direct application of C.2),
6^1
1/3
+ 6V5 19 + i
-2—+i~^r )
(On page 314 in his first notebook 19], Ramanujan recorded G \ n in the form given
above, which is strong evidence that Ramanujan utilized Theorem 3.1 as we have
done.) It therefore remains to show that
J
+ J4+V3).
C.24)
We apply Lemma 3.3 with
a =
and b =
T
Thus, с — 1. We therefore need to solve
To solve
Thus,
п+бУз
2
this by inspection, it perhaps is best to square both sides and set t = x2.
that t = 1 4- V5/4. Hence, x = \>/4-\- >/3, and, from
It is not difficult to see that / = 1 + Jb/ \
C.15), у = V3/4. Thus, C.24) follows, and the proof is complete.
Alternatively, in the notation C.18) and C.20), by C.21), we want to solve
и2(и2 - 3J = 22 + 12>/3. C
Factoring in Z[>/3] and using C.22), we find that
JV(u2)N2(m2 - 3) = NB2 + 12>/3) = 22 • 13.
Then
±2 = N(u2 - 3) =: N(A + B-fh) = A2 - 3B1
Choose A — 1 = B. We now observe that
C.25)
as required. It is easily checked that, when и
then find that
 + v/4+ч/з) .
= 13,
— V4+ \/3, C.25) holds. We readily
34. Class Invariants 211
Thus, C.24) has been shown once again.
Entry 3.5.
Gm=
8
л/П-З
8
/37
33 + 9VT7
C.26)
Proof. Set« = 17 in Theorem 3.1. From Weber's treatise [2, p. 721 ], or from the
tables in Section 2,
5 + УГ7 , jVvf-3
1
C.27)
Since
4 5 + ^^7 A +
On = : 1
from C.1) we find that p = E + VT7)/2. Also,
5 + VT7
- 1 = z +
5VT7
Thus, from Theorem 3.1,
VT7 /УГ7-3
17 + 5VT7
13 + 5VT7
5VT7)
However, note that
A9 + 5VT7)A3 + 5VT7) = 672 + 160У17 = B0 + 4>A7J.
Hence,
I J5 + V17
GI53 = I l/ ^ 1-
+
/19 + 5Уп\
V 2 J
1/6

212 Ramanujan's Notebooks, Part V
ing the equality above with C.26), by C.27), we see that it remains to
5 + л/П
Comparing
show that
which is rather curious indeed. Note that
6
= G17,
C.28)
= о,,.
Thus, by C.28), it suffices to show that
In the notation C.18) and C.20), we solve
«42(m2 - 3J = 7 +
We now factor in the principal ideal domain Z | A + VT7)/21. Thus,
N(u2)N2(u2 - 3) = NO + л/17) = 25.
C.29)
C.30)
We attempt to solve
±4 = Niu1 -3) =: N (a +
AB-4B2.
Take A = -1 and В = 1. Then u2 = E + Vl7)/2 and Л/(и2) = 2. A simpl
calculation shows that C.30) indeed holds. Lastly, we find that
= j5 + VVJ + /^-^
8 Y 8
By C.27), we conclude that C.29) holds to complete the proof.
Entry 3.6.
G441 =
C.3
34. Class Invariants 213
Proof. We apply Theorem 3.1 with n - 49. From Weber's treatise [2, p. 723], or
from the tables of Section 2,
G49 =
It is easily checked that
>/4+л/7-
2 V 2
After a somewhat lengthy calculation, we find that
(9
±4
Thus, p = 9 + 4-Jl. After a mild calculation,
p + vV-1 = 9 + 4л/7 + 2л/зу 16 + 6л/7
= B + л/ЗМЗл/З + 2V7)
Using C.32)-C.34) in Theorem 3.1, we deduce that
с„,.
+72V7+A
where
C.32)
C.33)
C.34)
1/3
C.35)
A : = 7A92+ 72n/7)A89 + 72>/7)
= 6^2016 + 762n/7 = 6 ¦ 71/4У|C + л/7J.
Using this calculation in C.35), we see from C.35) and C.31) that it remains to
prove that
N
191
•J
+ 72У7 + 6-7'/4У|C-
2
189 + 72^7+6-71/4^
2
Ьл/7J
^|C + VlJ

214 Ramanujan's Notebooks, Part V
C
We apply Lemma 3.3 with
C
C.36)
a =
191 + 72V7 + 6 •
and
189
Again, с = 1. Setting x2 = t in C.14), we see that we must solve
191
tDt - 3J =
_ A6 + 3y/7)C + y/7J + бУб ¦ 7C + V7J
4
We now verify that
Thus,
3 + V7
16
t =
^7
by C.15). Thus, C.36) follows, and the proof is complete.
Entry 3.7.
#90 =
3 + V5
<3'37)
34. Class Invariants
Proof. Set и = 10 in Theorem 3.2. Now from Weber's book [2, p. 721 ], or fr<"n
the tables of the preceding section,
?10 =
An easy calculation shows that p = -s/5. Note also that
I , ,\ 1/6
/1+V5 //l+v
Thus, from C.12) and C.38),
1/3
Comparing C.39) and C.37), we see that we must prove that
1/3
/n/6-1
C.3(8)
C.4Ю)
In the notation C.18) and C.20), we solve
и2(и2-3J= 18+6V6. C.41)
Factoring in the unique factorization domain Z[V6], we find that
N(u2)N2(u2 - 3) = ЛЩ8 + 6\/6) = 3 ¦ 62.
We thus want to solve
±6 = Л/(м2 - 3) =: N(A + B\/6) = A2 - SB2.
Choose A = 0 and В = 1, so that u2 = 3 + л/6 and N (и2) = 3. It is trivial to s»ee
that C.41) is satisfied. Then
a =
h
and the verification of C.40) is complete.
Entry 3.8.
«198 = yl + V2
I9 + V33
8
C.4»)

216 Ramanujan's Notebooks, Part V
Proof. Set« = 22 in Theorem 3.2. From Weber's work [2, p. 722], or from our
tables in the foregoing section,
g22 =
An easy calculation yields p = 4л/2. Thus, from Theorem 3.2,
gm = /l+ V2 ^4x/2 + V33y/6 I /l8 +3V33 + Уп +
By C.42) and C.43), we must show that
In the notation of C.18) and C.20), we seek to solve
C.43)
C.44)
C.45)
Thus,
N(u2)N2(u2 - 3) = NDA8 + 3^33)) = 24 • 33.
Factoring in Z[(l 4- л/33)/2], we attempt to find a solution of
±6 =
- 3) =: N (a + Bl-
AB -
Let A = 1 = fl, so that u2 = (9 + V33)/2 and N(u2) = 12. It is easily checked
that C.45) holds. Hence,
9 + V33 /l+л/ЗЗ
a = i'-T-+V-8—
Thus, we can deduce C.44), and the proof is complete.
4. Kronecker's Limit Formula and General Formulas for
Class Invariants
Let Q(u, v) := y~l(u + vz)(u + vz), where z = x + iy with у > 0. Epstein
zeta-function %q(s) is defined for a — Re s > 1 by
where the sum is over all pairs of integers (к, v) except @,0). It is well known
that $q(s) can be analytically continued to the entire complex s-plane, where
34. Class Invariants 217
?q(s) is analytic except for a simple pole at s = 1. The Kronecker limit formula
provides the constant term in the Laurent expansion about s = 1. More precisely,
^ ) + O(s - l), D.2)
S — 1
where у denotes Euler's constant, and ^(z) 's the Dedekind eta-function defined
by
r)(z):=ql/2\q;q)oo-ql/Uf(-<l), Я = в2"". У > 0, D.3)
where f(-q) is defined in A.14) of Chapter 33.
Next, let К be an algebraic number field over the rational numbers. Let N(Ql)
denote the norm of an ideal 91. Then the Dedekind zeta-function for К is defined
by
where the sum is over all nonzero integral ideals 21 of K. Let С к denote the ideal
class group of К. Then the Dedekind zeta-function for an ideal class A of CK is
denned by
J], a>\.
If x denotes an ideal class character, then the L -series for К is given, for a > 1,
by
LK(s,x) ¦- X>B0(NBl))-s = ?X(ARE, A), D.4)
я л
where the former sum is over all nonzero integral ideals Я of AT, and the latter
sum is over all ideal classes A of С к ¦
In the sequel we assume that AT is a quadratic field. It is well known that (C. L.
Siegel[l,p.58])
= hK, D.5)
where h is the class number of K, i.e., h = \CK\, and where
if К is imaginary,
if К is real.
к :=
w-J—d'
21oge
D.6)
Here w is the number of roots of unity in K, d is the discriminant of K, and e is
the fundamental unit in K.
Let
LAs)
-my-
a > I,

218 Ramanujan's Notebooks, Part V
where (?) is the Kronecker symbol. Then (Siegel [1, p. 58])
D.7)
where f(s) denotes the Riemann zeta-function.
Now let d = d\d2, where d\ > 1 and, for i = 1,2, d, = 1 (mod 4) or di = 0
(mod 4). Let ф denote a prime ideal in AT. Then a Gauss genus character x is
defined bv
( )
defined by
Х(Ф) =
where (д^) a8a'n denotes the Kronecker symbol. Note that Ы(Щ \ d2 if
N(<P)|di. This definition can be extended to all ideals of К by multiplicativ-
ity. It is well known that the genus characters form an abelian group, denoted by
G(K), of order 2*, where к is the number of distinct prime divisors of d.
Next define
Go := (A € CK ¦ X(A) = 1, x € G{K)},
which is named the principal genus. Clearly, Go is a subgroup of CK, and CK /Go
is called the genus group. Furthermore, C*/Go = G(K). Obviously, A\ and A2
are in the same genus if and only if x(A0 = х(А-г) for each x € G(K).
Kronecker (Siegel [1, p. 62, Theorem 4]) proved that, for a genus character x
of К corresponding to the decomposition d — d\d2,
D.8)
Thus, by D.4) and D.8),
For a fixed nonzero integral ideal (Be A
'
{N{k)r
a > 1,
D.9)
where U is the group of units in K. Now assume that К — Q(V-wi) is an
imaginary quadratic field, and so m is a squarefree positive integer. Recalling that
w is the number of roots of unity in AT, we see that, from D.9),
f (s. A) -
w
Let
„-{
a > 1.
if -m = 2,3 (mod 4),
if - m = 1 (mod 4).
D.10)
34. Class Invariants 219
Then
d =
-4m, if - w = 2, 3 (mod 4),
—m, if — m = 1 (mod 4).
It is known (R. Mollin and L.-C. Zhang [ 1 ]) that each ideal class contains primitive
ideals which are Z-modules of the form 55 = [a, b+Q], where a and b are rational
integers, a > 0, a\N(b + Q), \b\ < a/2, a is the smallest positive integer in ЯЗ,
and W(<8) = a.
Let г = {b + Q.)/a. Then, for к = ua + v(b + П),
N(k) = (ua + v(b + п)Киа + v(b + U))
— a2(u + vz)(u + vz)
D.n)
Thus, for г = (b + Sl)/a and у - Im г = vM/Ba),
B(«, w) = (^
And, from D.1), D.10), and D.11),
f E, A) = 1
w
Thus, from D.2),
vz){u + vz).
w
(^=) {-{ logBe) + log Mz)\2) + O(s - 1).
w \vl"l/ D.12)
Since, for any nonprincipal genus character x,
it follows from D.4) and D.12) that
LK(s
D.13)
Recall that in the decomposition d = d{d2 we assume that d\ > 1 and ^2 < 0.
Let K, = Q(Vdi), 1 = 1,2. By D.7),
, X) = -— (-L-) Y\ X(A) (-i loga + log |„(г)|2) + 0{s - 1).
Then, by D.5) and D.6),
logc.
D.14)

220 Ramanujan's Notebooks, Part V
and
2h2n
D.15)
where A/ is the class number of Kt¦, i = 1,2, ei is the fundamental unit of K\,
and wi is the number of roots of unity in K2- Thus, setting s = 1 in D.13) and
using D.8), we deduce that
^ Y, X(A){-{\oga+\og\n(z)\2). D.16)
Thus, setting
F(A) =
D.17)
where г = (b + п)/а, with [a, b + Щ 6 A, we conclude from D.14)-D.17)
that, for x nonprincipal (Siegel 11, p. 72]),
X(A) log F(A),
or
D.18)
We remark that D.18) was utilized by K. G. Ramanathan [1], [31, [4], [5], [7]
to calculate class invariants, values of the Rogers-Ramanujan continued fraction,
and certain other invariants of Ramanujan.
We next prove the three primary theorems that we need to calculate Ramanujan's
class invariants Gm when К = Q(V~ has class number 8. Let r = J—m.
Then, by A.1), A.3), and D.3), it is easily seen that
Equalities D.18) and D.19) are the key ingredients for deriving formulas that will
enable us to calculate Gm. We consider three different genus structures, and the
first two theorems that we prove can be utilized to determine Gm for m = 65,
69, 77,141,145,205,213,265, 301, 445, and 505. For m = 217, 553, the genus
structure is of a third type. In each case, К = QW-m) has class number 8,
and the number of genera equals 4. Thus, each genus contains exactly two ideal
classes. Also note that A and A"' are clearly in the same genus.
Throughout the next two sections, for simplicity, we use the notation for a
primitive ideal to denote the ideal class containing it; this abuse of notation should
not cause difficulty.
Theorem 4.1. Letm = 1 (mod 4), where mis a positive squarefree integer with
prime divisor p. Let К = Qi-J—m) be an imaginary quadratic field such that
each genus contains exactly two ideal classes and such that the principal genus
34. Class Invariants 221
Co contains the classes [ 1, Q] and [2p, p 4- П]. Let G, be a nonprincipal genus
containing the two classes [2, 1 + ?2] and [p, ft]. Then
h/1
-л
*№,)=-1
whereh,h\, and h2 are the class numbers of K,Q(-Jd^), and Q(*/di), respec-
respectively, w and W2 are the numbers of roots of unity in К andtyi^/dl), respectively,
(i is the fundamental unit in Q(y/a\), and the product is over all characters x
(with x(Gi) = -I), associated with the decomposition d = d\d2, and theref
d\,d2,h\,h2, vJ, and(\ are dependent on X-
ore
Proof. Eachoftheidealsfl, J2], [2p, р+п),[2,1+J2], and [p, Щ is ambiguous.
If 21 e A is any one of these ideals, then 21 ~2l~1, A = A~l, and 21 6 A.
For any ideal class В ? Go U G\, it is not difficult to see that (Ramanathan [5,
p. 77])
which implies that
П
where F(B) is defined by D.17). Therefore, by D.18),
П
X(G,)=-1
"= П П
= П
AeCoUG,
since the number of genus characters equals A/2, and so the number of genus
characters with x(Gi) = -1 is A/4.
Let Ao = [1, Q), A'o = [2p, p + J2], A, = [2, 1+ Q], and A', = [p, п]. Then,
by D.20),
П
x(C,)=-i
By D.17) and D.19),
F(A0)
г
Let fi' = Q/p = y/-m/p2. Again, by D.17) and D.19),
F(A'O) п
= oi.
D.21)
D.22)
D.23)
The theorem now follows from D.21)-D.23).

222 Ramanujan's Notebooks, Part V
Theorem4.2. Letm = \ (mod 4), where mis a positive sqmrefree integerwith
prime divisor p. Let К = Q(v^) be an imaginary quadratic field such that
eachgenuscontainsexacdytwoidealclassesandsUchthattheprmapalgenusGo
contains the classes [1. SI] and [p, Cl].LetGx be afionprincipal genus containing
the two classes [2,1 + Si] and [2p, p + Sl). Then
{GmGm/pi) = \ [ C|
Х<С,)=-1
where A. A,, and h2 are the class numbers ofK, <i(V30. and <&/%). respec-
tively w and w2 are the numbers of roots of unity i* К and Q(V?. . respectively,
(, is the fundamental unit in Q(v^), W the product is overall characters X
(with x(Gi) = -W. associated with the decomposition d = d,d2, and therefore
dud2,huh2,w2,and€] are dependent on x¦
The proof of Theorem 4.2 is analogous to that for Theorem 4.1, and so we omit
U' We say that m is of the first kind or second кЫ according as it satisfies the
conditions of Theorem 4.1 or Theorem 4.2, respectively^
It is not difficult to show «hat Ц. Q], [2,1 + SI], \P, Ql, and [2p, p + Q] are
representatives of different ideal classes (Mollin and Zhang [ 1 ]).
Theorems 4 1 and 4.2 need to be combined with three modular equations of
Ramanujan (Part III [3, pp. 231, 282, 315]) in order to calculate R—jan s
class invariants. We have already employed Lemma 4.3 in our proof of Theorem
3.1.
Lemma 4.3 (Modular Equation of Degree 3). i*t
and Q
Then
Lemma 4.4 (Modular Equation of Degree 5). Let
-B))]/n and Q =
Then
a(l
-«)/
Lemma 4.5 (Modular Equation of Degree 7). Let
and
34. Class Invariants 223
Then
Let q = ехр(-лУУп). Since Gn = Gl/n (Ramanujan [3], [10, p. 23]), by
A.6), G,, = Da(l - a)}/24. If fi has degree p over a, then Gn/pi = Gp4n =
{40A - 0))~if7A. In summary, we can express the equalities of Lemmas 4.3-4.5
in terms of Gn and Gn/p2, p = 3,5,7, respectively, by employing the formulas
and
= {40A - 0)Г1/24. D.24)
The genus structures for Q(V-217) and Q(V-553) are different from those
of the eleven imaginary quadratic fields to which either Theorem 4.1 or Theorem
4.2 applies, and so G2\ ? and G553 must be calculated by another means.
Lemma 4.6. Let m denote a positive integer with 7|m. Let z — y/—m/l and
Q = (Gm/Gm/49L. Then
19
Proof. With q = exp(-n</m/7), it follows from D.3) that
D.25)
D.26)
Next, by an entry from Ramanujan's second notebook (Part IV D, p. 209, Entry
55]),
f2(-q)f2(-q2)
+ ¦
fH-q)f4-q2)
-8
qi/2fH-q)fH-qu)
) qV2f\-q)f4-qXA)
f4-q2)/H-q7) f6(-q2)f4-q7)
Multiplying both sides by <?1/2 and then replacing q by -q, we find that
fHq)f4-q2) Anq"f2(q7)f4-qU) /6(-<?2)/6(«7)
/4q)fH-q2) fHq)f4-q14)
, oqf2(-q2)f4q7) „<
Я(9)Я(-914)
fH-q2)f4q7) '
D.27)

224 Ramanujan's Notebooks, Part V
Recall that? = exp(-jrvW7) and recall that Gm/49 is then given by A.3). Thus,
Gm = 2-|/4<Г7/24(-«7; -?'4)oo- Hence,
2 (g7;gu)l
Dividing D.27) by qi/2 and substituting D.26) and D.28) into the resulting equal-
equality, we deduce D.25) to complete the proof.
Theorem 4.7. Let m be a squarefree positive integer with l\m and m = 1 (mod
4). Let К = (QKv/11^) be an imaginary quadratic field such that each genus con-
contains exactly two classes and such that the principal genus Go comprises [ 1, ?2] and
[2, 1 + Q], while [7, Q] and [14, 7 + U]form a nonprincipal genus G\. Then
A/2
= п
И/Л1А2/И/2
D.29)
wAm? h,h\,andhi are the class numbers of K, <Q>(>M). <"»<* Q(>/<h), respec-
respectively, w and w2 are the numbers of roots of unity in К andQ(*/3~2), respectively,
?1 is the fundamental unit in Q(</h\), and the product is over all characters x
(with x(Gi) = ~U> associated with the decomposition d = d\d%, and therefore
d\,d2,h\,h2,u>2, and C\ are dependent on x ¦
Proof. Let M = U, П], a; = [2,1 + Q], A, = [7, J2], and A', = [14,7 + П].
Then by the same reasoning that we used in the proof of Theorem 4.1,
*- П «г
оу/ x(Ci)=-<
D.30)
By D.17),
F(A0) = »?2(S2) = »2Gr),
and
= гJ
Substituting these values into D.30) and recalling that the number of genus char-
characters x with x(Gi) = -1 is equal to A/4, we deduce D.29) to complete the
proof.
The class numbers cited below for \d\ < 500 can be found in tables in the
texts by Z. I. Borevich and I. R. Shafarevich [1, pp. 422-426], H. Cohen [1, pp.
34. Class Invariants
225
503-509J, and for 0 < d < 10,000 in the book by D. A. Buell [1, pp. 224-234].
Lists of fundamental units can be found in the book by Borevich and Shafarevich
(lj(forrf < 101),thebookbyM.PohstandH.Zassenhaus[l,pp.432-435J(upto
d < 299), and the tables of R. Kortum and G. McNiel [1] (up to d = 10,000). In
Cohen's book [1, pp. 262-274], there is a table providing the ideal class structure
for Щ-J^d), d < 97 and for ®(Vd), d < 97.
5. Class Invariants Via Kronecker's Limit Formula
Theorem 5.1.
Proof. The following table summarizes the needed information about ideal classes
and their characters.
d\
1
5
13
65
d2
-260
-52
-20
-4
X
Xo
Xi
X2
Хз
С
Go
G,
G2
G3
С
[1,0]
[10, 5 +Я)
[2,1 + Щ
[5,Q]
[3, 1 + П]
[3,-1+ J2]
[6,1 + Q]
[6,-1+Q]
X(Go) x(Gi)
X(G2) X(G3)
1
1
-1
1
i -1
-1
1 1
_j
A,
1
1
h2
2
2
w2
2
2
л/5+l
2
vf3 + 3
2
Note that 65 is of the first kind. Applying Theorem 4.1 with h = 8 and w — 2,
we find that
Let Q = (G65/G13/5K and P = (G65G,3/5r2. Then, by E.1),
Q =
2>"WI3+ 18)'/2. E.2)
By Lemma 4.4,
p-l=
E.3)

226 Ramanujan's Notebooks, Part V
Now, by E.2),
= (n/5 + 2)EУТЗ + 18) + (V5 - 2)E7l3 - 18) + 18
= E + V65J, E
and, by E.4),
q + 0-1 = У74+10л/б5.
Thus, by E.3H55),
/>-' = 1^74 + Юл/65 + iE + ^/S).
Thus, by E.2) and E.6),
G65 =
1E
Thus, it remains to show that
which is easily shown by a routine calculation.
Theorem 5.2.
Proof. We summarize the needed information in the following table.
E.5)
E.6)
d\
1
92
69
12
d2
-Tib
-3
-4
-23
X
Xo
Xi
X2
Хз
G
Go
G,
G2
G3
С
П.П]
[6, 3 + Щ
[2,1 + U)
[3,0]
[5,1 + fi]
[5, -1+П]
[7, 1 + П]
[7,-1 + П]
X(Go) X(G.)
X(G2) x(G3)
1 1
1 1
1 -1
1 -1
1 -1
-1 1
1 1
-1 -1
hy
1
1
h2
1
1
w2
6
4
24 + 5V23
25 + 3V69
2
34. Class Invariants 227
We apply Theorem 4.1 with Л = 8 and w — 2, as 69 is of the first kind. Thus,
y.
J
E.7)
Let Q = (С69/О2з/зN and P = (G69G23/3)-3. By E.7),
^25 + 3V69x3/4
By Lemma 4.3,
From E.8),
Q
and, from E.10),
(G + Q~'J + 32
Putting these calculations in E.9), we find that
= 7l6A87+108V3),
6л/3J.
-1:(
By E.8),
\ 1/12 / V3W23\
and thus, by E.11), it remains to show that
This can be achieved by a straightforward computation.
Theorem 5.3.
;E8)
E.9)
E.10)
E.11)

228 Ramanujan's Notebooks, Part V
Proof. We compose the following table giving needed information about ideal
classes and characters
We see from the table that 77 ,s of the first kind. Thus, by Theorem 4.1, since
h — 8 and w — 2,
UP = (G77G11/7)-3, then, from Lemma 4.5,
E.12)
E.13)
Now
Q + Q-x +7 = 8Vn + 28 =
Using this in E.13), we find that
/>-' = i BV22 + 7V2 + У182 + 56n/TTj .
By E.12).
E.14)
л/л
1 /8
p-./6)
and thus by E.14) it remains to show that
which is readily shown by a straightforward calculation.
34. Class Invariants 229
Theorem 5.4.
Gl4, =
V47\
1/12
l4+9>/3
1/2
Proof. We record the necessary information in the following table:
d\
1
188
141
12
d2
-564
-3
-4
-47
X
Xo
Xi
X2
Хз
G
Go
G,
G2
G3
С
[1.П]
[6, 3 + fi]
[2,1 + П]
[З.П]
[5, 2 + fi]
[5, -2 + ?2]
[10,3 + fi]
[10, -3 + fi]
X(Go) X(G,)
X(G2) X(G3)
-
1
1
1 -
[
-
-
1
-1
hi
1
1
h2
1
1
w2
6
4
«1
48 + 7V47
95 + 8л/Т4Т
We see that 141 is again of the first kind. Applying Theorem 4.1, we find that,
since h = 8 and w = 2,
E.15)
Let Q = (Ci4i/G47/3N- Then, by E.15),
Q = D8 + 7V47)'/2(95 + SvW" = (^p1) (W3 + V47K'2
= D8 + 7</47I/2 G56^3 + 191 V47)l/2. E.16)
LetP = (Gi4iG47/3)-Then, by Lemma 4.3,
„-1 (е + е'') + У«2 + е-'J + з2
From the last representation of Q in E.16),
Q2 + б + 34 = C6G + 4л/3)J ,

230 Ramanujan's Notebooks, Part V
and so
Q + QTX =
+ Q-1 + 2 = 4д/7855 + 4536>/3.
Using these calculations in E.17), we deduce that
/>-' = —JlS55 + 4536V3 + — G + 4л/3).
Hence, by E.16) and E.18),
1/8
x , -1/7855 + 4536V3 + -^G ¦
It thus remains to show that
which is a straightforward, albeit laborious, task.
Theorem 5.5.
Proof. We compose the following table:
E.18)
1/2
34. Class Invariants 231
Thus, 145 is of the second kind. Thus, by Theorem 4.2, since h — 8 and w = 2,
Hence,
P'1 := (C.45G29/5J = ( 292+
By Lemma 4.4, with ?) = (G145/G29/SK,
Q = P~[
By E.19), we readily find that
and so, by E.20),
E.19)
E.20)
- P = 2л/29 + 5\/5,
C = 2V29 + 5V5 + /24O
Thus, by E.19) and E.21),
1/4
I .GU -1_ S 1
GL5 =
E.21)
/ / \ '/6
|2^^9 + 5л/5 + у240 + 20У145] .
Hence, it remains to show that
207145=
which is readily shown.
Theorem 5.6.
C20S =
л/4Т-
Proof. We record the following table:

232
dx
1
5
205
41
Ramanujan's Notebooks, Part V
dt
-820
-164
-4
-20
X
Xo
Xi
X2
Хз
G
Go
G,
G2
Gy
С
П.П]
[5,0]
[2,1 + 0]
[10,5 + 0]
[11,2 + 0]
[11,-2 + 0]
[13,4 + Q]
[13,-4 + 0]
X(Go)
X(G2)
_
-
X(G.)
X(G3)
1
1
-1
-1
1 -1
I 1
1 1
1 -1
h\
1
2
hi
8
1
tU2
2
4
V5 + 1
2
43 + Зл/205
2
Note that 205 is of the second kind. Applying Theorem 4.2 with h = 8 and
w = 2, we deduce that
P~2 := (G2o5G4i/5L =
Letting Q - (G2o5/G41/5K. we deduce from Lemma 4.4 that
<2 = (P-' -P) +
From E.22),
P -P =
Thus, from E.23),
E.22)
E.23)
G = Ц45 + 7V4T + У4030 +
If follows from E.22) and E.24) that
E.24)
Щ45 + 7 V4T + v/4030 + 630V41 \\ .
It thus remains to show that
i (
45
34. Class Invariants 233
This is more readily accomplished if we first note that
Зл/41
Theorem 5.7.
G213 =
+ У7ТУ/У59 + 7У7Т
2 ) \ ^ )
/21
+
I \ 1/2
/l9+12v^\
V 2—J •
Proof. We have the following table:
di
1
284
213
12
di
-852
-3
-4
-71
X
Xo
Xi
X2
X3
G
Go
G,
G2
G3
С
[1.П]
[6,3,0]
[2,1 + fi]
[3,O]
[7, 2 + 0]
[7, -2 + Q]
[14.5 + О]
[14,-5 + П]
X(GO) x(Gi)
X(G2) X(G3)
— ]
_
-
-
1
1
1
1
[
I
*i
1
1
hi
1
1
U>2
6
4
3480 + 413^
73+5л/213
2
Observe that 213 is of the first kind. Applying Theorem 4.1 with /1 = 8 and
w = 2, we find that
V
VG71/3
1/2
2/3
so that
\3/2
E.25)

234 Ramanujan's Notebooks, Part V
Let P = (G213G71/3). Then, by Lemma 4.3
5.26)
By E.25) and moderate calculations,
(Q + <2~'J + 32 = (l2(87 + 50V3)J
and
~K=(Q + <Г) ~Wq + ^~2 + 2 = 71A35619 + 78300V3).
Thus, by E.26),
P-1 = /±035619+78300л/3) + 4=(87 + 50л/3). E.27)
v2
Thus, by E.25) and E.27),
1/12
783OOV5)
50>/3)
Hence, it remains to show that
лАA35619 + 78300V3)
V 2
which is accomplished by a direct calculation.
Theorem 5.8.
1/2
)
Proof. The following table is easily verified:
J
34. Class Invariants 235
d\
1
5
53
265
d2
-1060
-212
-20
-4
X
Xo
Xi
X2
X3
G
Go
Gi
G2
G3
С
П.О]
[10, 5, П]
[2,1 + П]
[5.Q1
[7, 1 + ?2]
[7, -1+fi]
[14.1+О]
[14,-1 +П]
X(G«) x(G,)
X(G2) X(G3)
-
1
1
-
1 -
-
-
I
1
1
1
Л.
1
1
hi
6
2
w2
2
2
V5+1
2
V53 + 7
2
Note that 265 is of the first kind. Applying Theorem 4.1 with h = 8 and w = 2,
we find that
so that
2)
9/2
\ 2
Let P = (G265G53/5). Then, by Lemma 4.4,
p-y = (Q + и'1) + y/(Q + Q-1J + 16
By using E.28) and the identity Q + Q~l =
that
E.28)
E.29)
in E.29), we find
P~l = -^=^6917 + 425^261 +^(85 + 5^265). E.30)
By E.28) and E.30),
1/4
,2v/2
Hence, it remains to show that
)'"¦
5^265)
/89_+_5_s/265 /8Ц-5л/265

236 Ramanujan's Notebooks, Part V
which is easy to establish.
Theorem 5.9.
G301 = (8
/23У43+57у/7\
I 2 /
1/8
46 + 7V43
Proof. We compose the following table:
l/2
Thus, 301
we find that
is of the first kind. Applying Theorem 4.1 with h = 8 and w - 2,
E.31)
Let P = (Сзо,С43/7)-3. Then, by Lemma 4.5 and E.31),
= 4VT"
= 4= C01
Therefore, by E.31) and E.32),
Cj01 =
E.32)
1/8
34. Class Invariants 237
1/6
A 1 I \ '
-= C01 + 46\/43) + — v/7B5941 + 3956>/43)) .
•Jl V2 /
It remains to show that
4=CO1 + 46V43)
3956^43)
42 + 7V43
which is a routine task.
Theorem 5.10.
f13 + V89
8
У89
Proof. We form the following table:
di
1
5
445
89
d2
-1780
-356
4
-20
X
Xo
Xi
X2
Хз
G
Go
G,
G2
G3
С
[1.П]
[5,П]
[2, 1 + п]
[10, 5 + fi]
[13,6 + П]
[13, -6 + fi]
[19,7 +Я]
[19, -7 + п]
x(Go) x(Gi)
X(fh) X(G3)
1 1
1 1
-1
1
i -1
i -1
1
I -1
hi
1
4
hi
12
1
w2
2
4
n/5 + 1
2
л/445 + 21
2
Thus, 445 is of the second kind. Applying Theorem 4.2 with /1—8 and w — 2,
we deduce that
so that
P~x =
E.33)
Let Q = (G44s/G89/5K. Then, by Lemma 4.4 and E.33),
Q = (/>-' _P) + y(p-i -pJ- l = 189 + 20V89 + /71320 + 7560л/^.
E.34)

238 Ramanujan's Notebooks, Part V
Therefore, by E.33) and E.34),
/л/445+ 21
1/4
(l89 + 20n/89 + yjl 1320 + 7560 V89J
1/6
It thus remains to show that
189 +
20V89 + ^71320 + 7560л/89 = U—^ +у
/5 + V89
By first squaring the binomial on the right side and then cubing the resulting
expression, we can easily verify the desired equality.
Theorem 5.11.
1/4
(¦v/7o!+ioI/4
5V505
+
1/2
Proof. We compose the following table:
1
5
101
505
d2
-2020
-404
-20
-4
X
Xo
Xi
X2
Хз
G
Go
G,
G2
G3
С
11,П]
15, П]
[2,1 + 0]
[10,5 + й]
[11,1+Q]
ui.-i + m
[22, 1 + П]
[22.-1 +?2]
X(GO) x(G.)
X(G2) X(G,)
1 1
1 1
1 -1
1 -1
1 -1
-1 1
1 1
-1 -1
Ai
1
1
h2
14
2
w2
2
2
V5 + 1
2
yioT+io
Hence, 505 is of the second kind. Applying Theorem 4.2 with h = 8 and w - 2,
we find that
P 2 :=
i| (n/ToT+10J,
34. Class Invariants 239
so that
Р1 =
10) = (V5 + 2)
2)
^±1V
+ 10)
E.35)
б = (G5o5/doi/5K- Then, by Lemma 4.4 and E.35),
Q = (/>-' - P) + У(Р-' -PJ- 1
= A3OV5 + 29VI0T) + /169440 + 7540л/505. E.36)
Therefore, by E.35) and E.36),
Г
т30л/5 + 29-s/IoT) + V 169440 + 7540л/505 J
1/6
Thus, it remains to show that
A3OV5 + 29VT0T)
113
) •
which is straightforward.
Theorem 5.12.
'11+4л/7 /9 + 4V7
1/2
5л/7
~l
5V7
1/2
Proof. We set up a table to summarize some information that we need.

240 Ramanujan's Notebooks, Part V
1
124
217
28
d2
-868
—7
-4
-31
X
Xo
Xi
X2
Хз
G
Go
G,
G2
G3
С
И.П]
12, 1 + П]
[7, Q]
[14,7 + П1
111,5 + Й]
[11,-5 + Q]
[13,2 + fi]
[13,-2 + ft]
X(G0) x(G.)
X(G2) x(Gj)
1
1
-
1
1
-
1
j _
I
1
1 -1
1 -1
*i
1
1
A2
1
1
2
4
«i
1520 + 273 V3T
3844063 + 260952 V217
It is clear that Q(V~217) satisfies the conditions of Theorem 4.7. Thus, since
h = 8 and ш = 2, we deduce that
/
7
so that
where
t \ _ A520 + 27з7зУ)(з844063 + 260952ч/2Г7I/2
= A520 + 273>/зТ)E24\/7 + 249>/зТ),
€ = A520 + 273^зТ)|/2E24/7 + 249^зТI/2.
It follows from D.25) that
Q3/2 + 8Ql/2 _ gg-l/2 _ Q3/2 = ?(€ _ f-l}
where Q = (G217/G31/7L. By an elementary calculation,
E.37)
E.38)
= 4A053643 + 398240V7) = 4B7 + 10л/7JC67 + 140v/7).
Let л = б1/2 - Q"l/2. Then E.38) can be recast in the form
x3 + 11л = 14B7 + 10л/7)у367 + \40<Л
= C67 + l40-/l)yJ 367 + 14OV^7 + 11^367 + 140V7.
It is now obvious that
x =
140^7.
E.39)
34. Class Invariants 241
Solving E.39) for б, we find that
Q1/2= 1 ^
367
A4 +
5V7))
E.40)
Now let P = (G217G31/7). Using Lemma 4.5 and E.39), we deduce that
„ „ , 1 , ,
2+9) = -LC76+140V7).
Solving for P~x, we find that
94 + 35V7 /17409+ 6580л/7
+
'11+
E.41)
Thus, from E.40) and E.41),
1/2
/ /16+5V7
x \ : +
1/2
which completes the proof.
Theorem 5.13.
'100+114/79 /96+ll,/79
1/2
143+16V79 /I41
+ j
1/2

242 Ramanujan's Notebooks, Pan V
Proof. We set up the following table to summarize the information that we need.
d,
1
28
553
316
di
-2212
-79
-4
-7
X
Xo
Xi
X2
X)
G
Go
G,
G2
G,
С
[1,П]
[2, 1 + Q]
П,п]
[14,7 + П)
[17,5 + П]
[17, -5 + П]
[19,6 +П]
[19,-6 +П]
X(Go) X(G,)
X(G2) x(G,)
1
1
1 -
-1
1 -
1 -
1 1
-1 -1
1
1
5
1
«'2
2
4
8 + Зл/7
624635 837 407
+26562217704У553
It is clear that Q(V-553) satisfies the hypotheses of Theorem 4.7. Thus, since
h = 8 and w — 2,
= (8 + Зч/7MF24, 635, 837, 407 + 26, 562, 217,704л/553)|/2,
so that
where
E.42)
€ = E14,088+ 194. 307V7I/2B11,227^7+ 62, 876л/79I/2. E.43)
Then an elementary calculation gives
(« -«Г1J = e2 + e~2-2
= 4A43,650,096,411 + 16, 161,898,544^79)
= C91 + 44^/79)A9170 + 2156V79J. E.44)
By Lemma 4.6 and E.42)-E.44), with Q = (G553/G79/7L,
= 7^391+44л/79( 19170 + 2156n/79).
If x = Ql/2 - Q~x/2, then the foregoing equality may be written in the form
x3 + 1 \x = 7V 391 + 44л/79 G2C91 + 44V79) +ll),
34. Class Invariants 243
from which it is obvious that
x = Q - Q~]':
Solving for Ql/2, we readily find that
= 7^391 + 44^79.
E.45)
(98
'100+11n/79 /96+11V79
E.46)
Now let P = (С55зС79/7)~3. Then, by Lemma 4.5 and E.45),
2-/2(P + />"') = Q + 6"' + 7 = x2 + 9 = 19168 + 2156V79.
Solving for P ~', we find that
/»-' = -^= D792 + 539V79 + 745,914,421 + 5, 165, 776^79 j
I /l43+16V79 /l41+16V79\
= h/+ /I (
Thus, by E.46) and E.47),
96+11V79
1/2
143+16V79 /141 + 16V79
1/2
and the proof is complete.
6. Class Invariants Via Modular Equations
In this section we establish six of Ramanujan's class invariants by using tools well
known to Ramanujan, in particular, modular equations.
Second Proof of Theorem 5.1. From A.1) and D.3) it is easily seen that
f(-q2)
F.1)

244 Ramanujan's Notebooks, Part V
Using this equality, we rewrite two of Ramanujan's eta-function identities in terms
of x • Thus (Part IV [4, pp. 206, 211 ])
+ 13
х(-я)
and
f(-q)f(-q2)
,'/6
X(-<7)V
Replace q by -q in F.2) and then set<? = ехр(-я^5/13). If
A '.= € ' —
and
В := e(nl1
then F.2) can be recast in the form
A - 13A = В3 + ЛВ - 4S - B"\
Next, replace q by -q in F.3) and then set q = ехр(-л-^13/5). If
and
B' := e(*
then F.3) takes the shape
A' - 5A'~l = В'г - В'~г.
= В' and
We shall prove that
Now, Gn = 2-1'4еп^'24Х(е-п^) and Gi/n =
A.3). Since Gn = G\/n, we find that
F.3)
F.4)
F.5)
F.6)
F.7)
F.8)
F.9)
F.10)
-by
F.11)
34. Class Invariants 245
(This could also be proved by using F.1) along with the transformation formula
for /.) In particular, if n - 5/13, F.11) yields the equality
/с-л-
F.12)
The aforementioned transformation formula for f{—q) is given by (Part III [3, p.
43, Entry 27(iii)])
where a, b > 0 with ab = л2. If a = тгл/5/13, so that b = ж^/П/5, then we
deduce from F.13) that
r,_е
F.14)
First, from F.5) and F.12),
by F.8). Thus, the first equality of F.10) has been demonstrated. Second, by
F.4), F.1) with q = -ехр(-л-^57ТЗ), F.12), F.14), and lastly F.1) with
q = exp(-jr,/T375),
A =
= ГГгл
5
5 '
by F.7). Thus, the second equality of F.10) has been established.
Employing F.10) in F.9), we find that
= Я3 - B = (B - flK + 3(B - В).
Dividing both sides by и := В - B~] (^ 0), we find that
Solving for и2, we find that u1 = (</65 - l)/2. Thus, since, clearly, В > 1,
B_B-i=

246 Ramanujan's Notebooks, Part V
Now solving for B, we find that
V65-
V65
F.15)
where in solving the quadratic equation we took the plus sign since В > 0.
If q = cxp(-7rV13/5), then q5 — ехр(-7Г\/б5). Hence, from A.3) and
F.5), we readily see that В = Gtb/Gn/ъ- Furthermore, from A.6), G13/5 =
D«A - a)}-|/24. Hence, if 0 has degree 5 over a, then G65 = {40A - /в)Г1/24.
We now employ Lemma 4.4, where it is to be noted that P = (C65C13/5)
and Q = B = (G65/Gi3/5). We already know Q from F.15). To determine
P from Lemma 4.4, we first calculate
Q + Q~[ = В3 + В = (B + B~l){(.B + ВJ - 3)
Thus, using F.16) in Lemma 4.4 and solving for P \ we find that
+10^65") ,
F.16)
F.17)
since P > 0.
Hence, by F.15) and F.17),
x ( I (/74 + 10v/65 + У 90 + ))
V V > > F.18)
We must show that F.18) can be transformed into the form of Theorem 5.1. First,
l + >/65
1 F.19)
Second,
(л/65 + 7
8
= I C + л/65 + Зч/5
34. Class Invariants 247
"V 2
Putting F.19) and F.20) in F.18), we complete the proof.
F.20)
Before commencing our second proof of Theorem 5.2, we establish a general
principle. Let p and r denote coprime, positive integers. Set q — ехр(-Ял/р/г)
andq' = exp(-jr>/pF), and let/5 have degree r over a. Then, by A.6),
Gp/r = {4a(l - а)Г'/24 and Gpr = D)8A - /3)}"l/24.
Furthermore, from A.2) and A.5),
and from the definition A.4) of a modular equation.
F.21)
F.22)
F.23)
If we solve F.22) for r and substitute this in F.23), we find that
From the last equality we conclude:
Iff) has degree r over a, then ft has degree p over 1 — a.
Furthermore, from B.5) and F.22),
K(J) IT
F.24)
F.25)
Second Proof of Theorem 5.2. We need two of Ramanujan's modular equations
of degree 23 (Part III [3, p. 411, Entry 15(i), (ii)]). If/3 has degree 23 over a, then
iap)"* + {A - «)A - P)I'* + 22/3M(l - «)A - P)I'2* = 1 F.26)
and
{A -
г + (A-«)A-/})}1/2)}1/2. F.27)
We also need two of Ramanujan's modular equations of degree 3. The first is given
by Lemma 4.3, while the second is given by (Part III [3, p. 231, Entry 5(ix)])
{a(l-/3)}l/2 + ((l -a)/3}'/2=2{a/3(l-a)(l - p))"\ F.28)
We shall apply F.24) with r = 3 and p = 23. Thus, p has degree 23 over
A - a). Thus, replacing a by A - a), from F.26) and F.27), we find that,

22/3{a/5(l -
248 Ramanujan's Notebooks, Part V
respectively,
{(l-a)p)l/H-\
and
! + {(!-«
= {2A + [A-«)/>}1/2 + {«A-/»)}1/г)}1/2.
F.29)
F.30)
For brevity, in the remainder of the proof, set G = G^andC = С2з/з-ВуF.21),
we can rewrite F.29) in the form
{A - o)/3|1/8 + {«A - /3)}'/8 = 1
Setting и = {GG')~l and squaring both sides, we deduce that
{A - aH)l/4 + {a(l - p)}l/4 = 1 + 2u2 - 2s/2u - Viu3. F.31)
Substituting F.31) into F.30), we find that
2 + 4u2-2V2u-V2u2 = л/2A +{A - a)/9I/2 + {a(l -0)}1/2)'/2. F.32)
Then, using F.28) in F.32), we deduce that
2 + 4u2 - 2л/2и - л/2«3 = \/2(l + л/2и3I/2.
Squaring both sides and simplifying, we arrive at
2 - Sy/iu + 24m2 - 22v/2«3 + 24«4 - 8л/2«5 + 2«6 = 0,
which, with x = и + 1/м, is equivalent to
22л/2 = 2(и3 + и) - 8V2(«2 + и) + 24(« + м)
= 2(x3 - 3x) - 8л/2(х2 - 2) + 24*.
Simplifying, we find that
x3 - 4Vlr2 + 9x - 3>/2 = 0.
By inspection, we verify that \fl is a root. Now Gn is a monotonically increasing
function of n, and it is not difficult to check numerically that the root that we seek
is greater than s/2. Thus,
x2 - 3n/2x + 3=0,
and so x = C + л/3)/\/2. Since x - и + I/u, we find that
+
F.33)
since м < 1.
34. Class Invariants 249
We now apply Lemma 4.3. Noting that P = м3, we see that we want to calculate
« - и3 = v4«-3+w3J-4
- 4
-4
Thus, by Lemma 4.3,
6 / >-t \ 6
~g) +\g') =
Solving for GIG', we deduce that
G / / I \ '
— = ( V748 + 432v^ + V747 + 432n/3 J
F.34)
Thus, by F.33) and F.34),
1/l2
1/2
To complete the proof, it suffices to show that
which is a straightforward task.
Second Proof of Theorem 5.3. We need two of Ramanujan 's modular equations
of both degrees 7 and 11. If p has degree 7 over a, then (Part III [3, pp. 314,315,
Entry 19(i), (viii)])
F.35)
and
7
m = '
m
F.36)

250 Ramanujan's Notebooks, Part V
where m = <p2(q)/<p2(q1). If fi has degree 11 over a, then (Part III [3, p. 363,
Entry 7(i), (ii)])
-/3)J1/l2 - 1 F.37)
and
11
~m'
x D + (a?I/4 + {A - o)(l - Д)}1/4),
where m' = tp2{q)/tp2{qn).
\iq = ехр(-ял/ТТ77I by F.21),
С,,/7 = DаA-а)Г1/24 and C77 = D)8A - ?)Г
Thus, setting w = (GnG\ Xp)~x, we deduce from F.35) that
((«/})'/» - {A - «)A - 0)}1/8J - ((«0)l/8 + {A - cr)(l -
and
(A -
Thus, from F.36),
m - - = 2 (l - 2^2и3I'2 C - V^«3)
m \ '
where m = ^2(г
Letg = ехр(-я%/7/П), and note that м =
Thus, by F.37),
and
(a/})U4 - (A -«)A -
{A -
= A -2u2J-2u
2J2u*
Hence, from F.38),
m' - — = 2 (A - 2м2J - 2u6) ' E -
m'
where m' =
F.38)
F.39)
F.40)
34. Class Invariants 251
From F.25), we see that
Since
m =J^
ЯГ7
we deduce from F.39) and F.40) that
2 (l - 2V2k3)'/2C - V2V) = yX2((i _ 2«2J - 2«6)'/2E - 2«2).
Squaring both sides and simplifying, we find that
4m10 - 11л/2и9 - 98и8 + 327u6 - 322м4 - 66л/2и3 + 210м2 - 19 = 0.
Isolating the terms involving V2 on one side of the equation, squaring both sides,
simplifying, and factoring, we deduce that
(ms-8«6 + 7«4-8«2 + 1)
x A96m12 - 1418m10 + 6044m8 - 13262k6 + 13073м4 - 5092м2 + 361) = 0.
F.41)
Now x := и2 is an algebraic integer (see Lemma 7.2) and so must be a root of a
monic irreducible polynomial. The latter polynomial in F.41) is irreducible, and
so x must be a root of the former polynomial in F.41). Alternatively, we used
Mathematica to check numerically that x is not a root of the latter polynomial on
the left side of F.41). Thus,
x* - 8л3 + 7л2 - 8jc + 1 = x2 ((x + 1/xJ - 8(л + \/x) + 5) = 0.
Since a: + l/x > 1,
Thus,
1
x + -=4 +
д:
-
Since и < 1, we find that
4 ' V ~A ' F-42>
Lastly, we apply Lemma 4.5. Since P = «-з, we deduce> by F 42) ^
е+О-1=2^(И-» + из)-7 = 2^((и + и-)з_зD, + 11-,))_7
= 2V2C + л/ГТ)Уб + УТТ - 7 = 2C +
= 21+ 8vff.
+ УТТ) - 7

252 Ramanujan's Notebooks, Part V
Hence,
Q 2 2
In conclusion, by F.42) and F.43),
G77 = Q~l/Su-1'2
= (8
1/8
2 + VTT
F.43)
1/2
and the proof is complete.
Second Proof of Theorem 5.4. We need two of Ramanujan's modular equations,
one of degree 3 and one of degree 47. If 0 has degree 3 over a (Part III [3, p. 231,
Entry 5(ix)]),
{«A - p)\]/1 + [fid - «)}1/2 = 2{a0(l - <*)(! - /J)}'/8. F.44)
If 0 is of degree 47 over a (Part III [3, p. 444, Entry 23(i)]),
4|/3{а/5A - «)A - /5)}
}1/24
(a/3)l/8 + (A - a)(l -
)
F.45)
Let q - exp(-^V4773). Then, by F.21),
G' := C47/3 = {4«A - a)}/24 and G := G141 - {40A - ?)Г1/24.
Applying F.24) with r = 3 and p = 47, we find that fi has degree 47 over A - «)
when p has degree 3 over a. Thus, by F.45),
Ifu:=(GGT\byF.44),
{«A - 0
{/3A -
)
F.46)
F.47)
34. Class Invariants 253
Hence,
-a)}J
= {«A -
F.48)
and
= {«A -
{/8A - a)}'/4 + 2(a(l - a)fid ~
F.49)
Substituting F.47H6.49) into F.46), we find that
2 AA + л/2«3))'/2 = 1 + (л/2«3 + и6I'2
F.50)
Using Grobner bases, A. Strzebonski denested F.50) and obtained a polynomial
of degree 48 for u. The value of и that we seek is a root of the factor и8 — 32м6 +
15м4 - 32и2 + 1 of this 48th degree polynomial. If x = u2, then
x4 - 32x3 + 15д:2 - 32л + 1 = x2 ((x + \/xJ - 32(д: + l/x) + 13) = 0.
Since x + l/x > 1, we find that
x + - =
д:
Hence,
so that
u+- = у 18 + 9-ч/3,
9л/з /14 + 9V3
F.51)
Lastly, we apply Lemma 4.3 with P = u3 and Q — (G'/GN to deduce, from
F.51), that
Q + 1 = 2V5(«-3 - и3) = 2^2((и - иK + 3(и-' - и))
Solving for 1/Q, we find that
F 52)

254 Ramanujan's Notebooks, Part V
Thus, by F.51) and F.52),
0,4, = C
= C-1/12«-1/2
9n/3)'/2A7 + 9л/3) + /з1419 +
1/12
It remains to show that
18144л/з
,DV3 + V47K/2/lt^V
>/5 /
which is easily accomplished via Mathematica.
Second Proof of Theorem 5.5. We need two modular equations, one of degree
5 and the other of degree 29. The first is found in Ramanujan's second notebook.
If 0 has degree 5 over a, then (Part III [3, p. 281, Entry 13(x)])
1/4 + {/9A - a)I/4 = 22/3{«/Hl - 1/24
{«A -
F.53)
The second is found in Ramanujan's first notebook on page 304, but curiously not
in his second. R. Russell [2] established this modular equation in 1890, but his for-
formulation is imprecise; in particular, it has a sign ambiguity. We give Ramanujan's
formulation as stated in Entry 65 of Chapter 36. Let
P = 1 -
Q = 64
and
Л = 32s/afi(\ -a)(l-0).
Then, if p has degree 29 over a,
>/P(/>2 + 17РЛ|/3 -9Л2/3) = RU6(9P2 + Q- 13PR1P + 15Л2/3). F.54)
Let q = ехр(-д V2975), so that we may apply F.21) and F.24) with r =5
and p - 29. If и = (С145С29/5)~1, then, by F.53),
{«A -
= ({«A -
= 2«2 - u6.
F.55)
34. Class Invariants 255
Thus, by F.55), with a replaced by A - a),
P = I -2u2+u6,
Q= 128и2-64и6- I6m12,
and
- 8„12
Substitute these values into F.54), square both sides, simplify, and factor, with
the help of Mathematica. We then find that
(и2 + l)(tt4 - u2 - 1)(«4 -u2+ ])(«* - 20«6 - 43«4 - 20и2 + 1)
x (и12 - 9м10 + 181u8 - 126u6 - 181m4 - 9и2 - 1) = 0.
In numerically checking the roots of each of these polynomials, we find that
x :— u2 is a root of
дг4 - 2O*3 - 43д:2 - 20* + 1 = x2 ((x + \/xJ - 20(л + l/x) - 45) = 0.
Thus, x + l/x = 10 + л/Т45, and so и - \/u = v/8 + vT45. Hence,
1 _ /8 + vT45 /12+VT45
Н-У~~4 +V 4 '
F.56)
Lastly, we apply Lemma 4.4 with P = u2 and Q - (G29/5/G145K. Then
= 2/12
= 2^241 + 20л/145.
Hence,
- = V 241 + 20^145 + ^240 +
F.57)
From F.56) and F.57),
G.45 = Q'l/6ir^2 = U TAX
^
240 +
^145 /8 + л/145
1/2
To complete the proof, we must show that

256 Ramanujan's Notebooks, Part V
and
+ 207U5 + ^240 + 20л/145.
Both equalities are easily verified.
Second Proof of Theorem 5.7. In addition to the modular equation of degree 3
given by F.44), we need Ramanujan's modular equation of degree 71. If/5 has
degree 71 over a, then (Part III [3, p. 444, Entry 23(ii)J)
i + (a/o1/4 + ш - «hi - |/4 2l/2
= («0)l/8 + {A -
+ 22/3{afi(] - ( )
F.58)
Let r = 3 and p = 71 in equalities F.21) and principle F.24). Thus, fi has
degree 71 over A -a). Replacing a by 1 -a in F.58) and employing F.47)-F.49),
but now with и — (GinGn\i-n)~\ we deduce that
- {A -
^2 F59)
Using resultants, A. Strzebonski and M. Trott denested F.59) and found a poly-
polynomial that factors into several polynomials of degrees 8,12, and 28. Numerically
eliminating all factors except one, we find that и satisfies
m8 - 80m* - 126u4 - 80m2 + 1=0.
Letting u2 =: x, and solving for;c ¦+ \/x, we find that x + \/x = 40 + 24л/3. It
then follows that и + 1/и = \/42 + 2A*fl. Hence,
21 + 12V5 /19 + 12л/3
F.60)
Lastly, apply Lemma 4.3 with P = (C213C71/3) 3 = и3 and Q —
6. So, by Lemma 4.3 and F.60),
Q + - =
- и3) = 4A9 + 12л/3I/2D1 + 24л/3).
34. Class Invariants 257
Solving for l/Q, we find that
F.61)
Hence, by F.60) and F.61),
= BA9 + 12V3)I/2D1 + 24л/3) + ^542,475
1/12
/21 +12л/3 /19+12^3
1\9+пУз\
V 2 )
1/2
It thus remains to show that
2A9 + 12V3)I/2D1 + 24л/3) + ^542, 475 + 313, 2ОО>/3
3/2/59
\
which can be verified via Mathematica.
7. Class Invariants Via Class Field Theory
In [6], Watson employed an "empirical process" to evaluate 14 of Ramanujan's
class invariants. Motivated by Watson's idea, we succeeded in formulating theo-
theorems which give rigorous evaluations of Gpq and Gp/q when p and q are distinct
primes satisfying pq = 1 (mod 4) and h(^/—pq) = 8.
Let К = Q(y/-m) {m squarefree) be an imaginary quadratic field, and let О к
be its ring of integers. By class field theory (J. Janusz [1, p. 228, Theorem 12.1]),
there exists an everywhere unramified extension K(l)\K such that the Galois group
Ga/(ATA) \K)~CK, where CK is the ideal class group of K. The field Km is
known as the Hilbert class field ofK.A Hilbert class field of К is usually defined
as the maximal unramified abelian extension of К.
Let a = [гь т2] be an DK -ideal. Define
= Д[тьт2]) =
, t2])
where
«2([Г,,Т2])=60
(m.n)^(O.O)

258 Ramanujan's Notebooks, Part V
and
*>-MO
(m.n)jMO.O)
It is clear from the definitions of ^([л. *г\) and 8i(\.x\. гг1)
where т = г2/Г|. We also let
Уг(г) - -/Дг)
with the cube root being real-valued when Да) is real.
It is well known that ?<" = К(ДОк)) (D. A. Cox [1, p. 220, Theorem 11.1]).
If DK is the discriminant of К and 3 \ DK, then Km = К(у2(тк)) (Cox [1, p.
249, Theorem 12.2]), where
I</^m, DK s 0 (mod 4),
3 + ^m n , , . ..
-1 , Dk = 1 (mod 4).
2
Lemma 7.1. Let a and b fee two Ок-ideals. Define ао(ДЬ)) fcy
where aa is a principal ideal. Thenaa is a well-defined element of Gal{Kw \ K),
and a »->• ao induces an isomorphism
CK —»¦ Gal{Km | tf).
Proof. See Cox's book [1, p. 240, Corollary 11.37].
Lemma 7.2. L«/AT = Qi^pq), where p and q are two distinct primes satisfying
pq = 1 (mod 4), а/и/ /ef
4, </ 3 f p?,
Y ~
12,
Grr,q is a real unit generating the fie
Proof. From a paper by B. J. Birch [1, p. 290], we find that Gp\ is a real unit of
AT(I). Since (Cox [1, p. 257, Theorem 12.17])
A6G>4-4K
^ • G-2)
we conclude that
G.3)
34. Class Invariants 259
Next, suppose that 3 f pq- Then 3 f ?>*• and y2(jk) generates Km. From the
equality (Cox A, p. 257, Theorem 12.17])
G.4)
and G.3), we find that G\q e Km. Hence, C*v 6 K°\ by G.3).
In [1, p. 290], Birch quoted Deuring's results [1, p. 43] and indicated that Gpq
is a unit when pq = 1 (mod 4). A more elaborate proof of this statement was
given in a paper by Chan and Huang [1, Cor. 5.2] and is contained in Theorem 1.1
in this chapter. In fact, from the treatment given in their paper, one can show that
Gp/4 is also a unit. This fact will be needed in our main theorem.
From class field theory, we know that if Я is a subgroup of С к, then there exists
an abelian and everywhere unramified extension L\K such that
Ga/(tfA)|Z.)~ H.
In particular, when H — C\ .— the subgroup of squares in CK, the corresponding
field M | К is known as the genus field of К. One can show that M is the maximal
unramified extension of К which is abelian over Q (Cox [1, p. 122]).
Theorem 7.3. Let К and у be defined as in Lemma 7.2. If the order of С к is 8,
then
¦ (GpqGp/qy
and
P/4
are algebraic integers which belong to the real quadratic field R, where R e
{Q(Vp). Q(V5). Q(*/pq~)). and where R is afield such that none of the prime
ideals B), (p), or (q) are inert.
Proof. From the hypothesis, we deduce that O| = [1, <S~pq], a2 = [q, *J-pq],
03 = [2, 1 + y/~pq], and a4 = [2q, q + ^/-pq) are O^-ideals lying in distinct
equivalence classes (see Section 4). This implies that С к contains the Klein four-
group generated by the ideal classes [a,] and [Oy] for i > j > I. Using the
isomorphism described in Lemma 7.1, we conclude that Gal(K<l) \ K) contains
a Klein four-group V generated by <та and aa> for 1 > у > 1. To show that ap,q
and f}p,q belong to a field with degree 2 over K, it suffices to show that ста, and aa>
u\ap.q and Pp.,. More precisely, if F := Fix(V) is the field fixed by V, then by
Galois theory (J. Rotmanfl.p. 49, Theorem 63]), \F : K\ = \Gal(K0) | ЛГ) : V|
= 2 (since \CK\ = 8), which implies that F is of degree 2 over K. Since a,,., and
fip,q are real numbers in F, they belong to R := F П R, and R is clearly a real
quadratic field overQ. The fact that they are algebraic integers follows from the
fact that GyP4 and Gyp/q are units (see Lemma 7.2).

260 Ramanujan's Notebooks, Part V
At this stage, we will assume that 3 | pq. From Cox's text [1, p. 257, Theorem
12.17],
У(а2) = Лу/-
Ву Lemma 7.1, we find that
<i*o& -
2i
G.5)
G.6)
14
G.7)
From G.2), G.5), and G.6), we find that
A<<(G%) - 4K _
Simplifying G.7), we deduce that
(a - b)(a + b) {64(a2 + Ь2)а2Ь2 - 48я V + 1} = О,
where a = crO3(C;2) and b = Gj;2/?. But
64(fl2 + Ьг)агЬг - 4&a2b2 +1^0,
for otherwise it would contradict the fact that a and b are algebraic integers. Thus,
we deduce that
aO2(G;2) = ±G\2q. G.8)
Similarly, corresponding to G.8),
oa2(Gl2q) =
or aa
i.e., au2(,Gp2/q) may have the same or opposite sign as aa2(Gp2q). Since cr22 = 1,
the latter is inadmissible. Hence,
aa2(Glp)q) = ±Gi2q. G.9)
From G.8) and G.9), it is now clear that
= <X
p,q
and
Next, from Cox's text [1, p. 263], we find that
and
Да3) = У
У(о4) = j
a.,o,
a..,
34. Class Invariants 261
Now, applying Lemma 7.1 again, we have
By G.10) and G.2), we find that
«
which implies that
Similarly, since 0302 is equivalent to 04,
= ±g;J or
by G.11) and G.5), i.e., (ra, (Gp2q) may have the same or opposite sign as aai (G j,2)
We will show that the latter case is inadmissible. If
aj(G^) = ±G;2 and oai(Gp2) = TG-j2
then
аа7(та>(Ср]) = ±G;}2 and аО)сга2(С) = Т
This is clearly a contradiction since <тп,сга, = оа><та1. Hence,
and
Collecting our results, we see that both aa2 and <гПз fix otPi4 and ^pi<?, and this
implies that ctpq and ^p q are real quadratic algebraic integers.
The proof for the case when 3 f pq is similar. In this case, Gpq generates Kil),
and so <ro, (Gp9) is well defined for 1 > 1. Hence, we may deduce from G.7) that
Simplifying G.12), we have
J - ^4m = 16G", - Ж
G.12)
- b)Da2b + Aab2 + 1) = 0,
where a = oa,(Gpq) and fc = G»,,. But
we
a! W°"ld C°ntradiCt Ae faCt Ла' ° and b m al«ebraic inle8ers- Hence,

262 Ramanujan's Notebooks, Part V
Now, since aai € Gal(K0)\K) and Gpq generates K<» (see Lemma 7.2),
The rest of the arguments are similar to those of the previous case, and we shall
omit them. . _..„_,
We have already seen that ap., and pp.q lie in a real quadratic field R. Our next
task is to give a necessary condition for R. First, we observe that R = F П R,
where F = Ffx(V) is an abelian, everywhere unramified extension of К (see the
paragraph before the statement of Theorem 7.3). Hence, R e [Q(Jp), ®(J<j),
Q(Jpl)) Next wc wil1 show Ла1 none of the prime ideals ^' ^ °r ^ are
inert in R Suppose the contrary holds. Then without loss of generality, we may
assume that (p) is inert in R. This implies that p in К is inert in F, where p\(p)
and the Frobenius automorphism [^ j
pp. 106-107] or Janusz[l, pp. 126-127]).
On the other hand, we know that the Frobenius automorphism <rp =
has order 2 (see the books by Cox [1,
[
II
has order 2 and that
= 1, where
*-"'
Since (Janusz [1, p. 127, Property 2.3])
we find that
])¦
L P JL P J
has order 1. Consequently,
_
This clearly contradicts the last statement of the previous paragraph. Thus, (/>) is
not inert in R.
Our next step is to determine aPA and fip<q using the numerical values of GM
and GPiq. To achieve this, we need the following result.
Theorem 7.4. let R = Q(V»0 be the field which contains ap.q and pp,q. If
2up,q = a\ + агл/т G.13)
and
thenai,a2,buandb2arepositive integers.
34. Class Invariants 263
Proof. Let [о] б А := {[аг], [а3], [а4]}, and let Я be the group generated by
[a]. From the paragraph before the statement of Theorem 7.3, we know that there
exists an abelian and everywhere unramified extension L\K such that
Gal{Km\L)~H.
In fact, from the isomorphism of Lemma 7.1, we find that L = Fix{<ra). Since
Gal(K(l)\K) ~ Z2 0 Z4, the group
Gal(L\K) ~
or Z4
The first case can only happen for exactly one element in Л, and the field L in
this case is the genus field M of K. As for the second case, Gal(L\Q) ~ O8, the
dihedral group of eight elements, since L is generalized dihedral over <Q> (Cox [ 1,
p. 191]). Hence, Gal(L\Q) is nonabelian.
Now, rewrite G.13) as
2(n
П ') =
G.15)
. Therefore, the
where r\ = (GpqGp/q)y. Note that <ra, fixes r\ and oa,(r\) = t]
field L := K(r\) = Fix(< a&2 >) is of degree 4over K.
Suppose L is the genus field of К. Since aai\i = 1, we conclude that the ideal
[аг] lies in an ideal class belonging to the principal genus. Hence, by Theorem
4.2, we may write
П =
П
G.16)
where
Since w = 2 and
2 or 4,
2, 4 or 6,
we conclude that e\ must be of the form e\
rewrite G.16) as
if 3 \pq.
= П
where e\ e N. Hence, we may
G.17)
Now, it is known that a fundamental unit of a real quadratic field takes the form
и + v\/d with u, v > 0 (Borevich and Shafarevich [1, p. 133]). Furthermore, if
¦Ju + vy/d = u'JcT] + v'*/di, then u',v' > 0. Collecting these observations, we
deduce that tj is of the form u{ +u2^/p + Щ^/q + u^^/pq, where и, > О for each
i. Using G.15) and G.17), we conclude that a\ and a2 are positive integers.
Next, suppose L is not the genus field. Then from the beginning of our dis-
discussion, Gal(L\Q) — Og is nonabelian. We claim that there exists an element
a in Gal(L\K) such that a(r/) is complex. Suppose the contrary holds. Then

264 Ramanujans Notebooks, Part V
L П R = Q(r;) would be Galois over Q, and hence Gal(L\<Q(r])) is a normal
subgroup of Gal(L\Q). On the other hand, Gal(Q(r))\Q) ~ Gal(L\K), a normal
subgroup of Gal(L\<Q) (Cox [1, p. 191]). Hence, Gal(L\Q) is isomoфhic to the
direct sum of Ga/(L|Q(r>)) and Gal(Q(r))\Q) and is therefore an abelian group,
and this contradicts our initial assumption.
Next, we will show that a(y/m) = -фп. Suppose that the contrary holds.
Then
and therefore
O{r\) = 1 ОГ T)~
This shows that о (ц) is real, which contradicts our choice of a. Now, applying a
to G.15), we deduce that
From G.15), G.18), and the fact that a(rj) is complex, we find that
(a\ + a2\fmJ > 16
and
(ai — а2л/тJ < 16.
This implies that 4a\a2y/m > 0. Since tj > 0, we deduce that a\ and a2 are
positive. The integrality of Я| and a2 foUowscasily from Theorem 7.3. In a similar
way, we can show that bt and Ьг are positive integers in G.14).
The argument given here for the case when Gal(L |Q) is nonabelian is due to
H. Weber, see Cox's book [1, p. 269].
Let 'УКк be the subset of WH^p), ty(^/q),ty{-J~pq)) satisfying the last state-
ment in Theorem 7.3. Note that, since №к\ is finite and 2apq and 20 pq lie in
a discrete subset of the ring Ъ{фп) for some <Q(*/m) e 94^, we can therefore
determine their exact values, based on the numerical values of Gpq and Gpfq, in
a finite number of steps. This will in turn lead to exact values of Gpq.
Except for ЛГ = Q(V-217) andQ(V-553), in all of our calculations, |9tjf| =
1.
We illustrate our computations with two examples. Before we proceed, we let
2 2
и := GpqGplq
v := Gp
~'J
„/Gp/q, Ui := (и1 + и"'J, and V, •- (Ы + v~'J-
Example 1. Let p = 5 and q = 13. In this case, у = 4. By Theorem 7.3, a5,|3
and 05.13 are real quadratic algebraic integers. Since the primes 2, 5, and 13 are
not inert in Q(\/65), we deduce that they are in Q(\/65).
Now, evaluating и and v using the product representation of Gn (see A.3)), we
find that «5.13 = 81.311288... and fis,n = 57.186772.... We know that these
numbers are of the form a + fcV65, and, by Theorem 7.4, we conclude that
41+5V65 . 33 + 3V65
05. в = r-
and
5.13 =
Therefore,
which implies that
U, =
This further implies that
V65+9
+
V65+1
8
and
and
and
34. Class Invariants 265
37 + 3V65
Yl =
Hence,
and
'n/65 + 9 /V65+1
8 +V 8
Example 2. Let p = 3 and ч = 23. In this case, у = 12. Using the numerical
values of и and i; and Theorems 7.3 and 7.4, we find that
Щ = 281344 + 162432n/3 and V6 = 2992 + 1728л/з.
The first equality implies that
u6 + u~6 = 8D7 + 27>/3).
Since
we conclude that
= 8D7+27V3) =
and
Collecting our results and simplifying, we deduce that
= ( ^7
G69
748
The following table summarizes our further calculations. The values for
and G793 are new.

266 Ramanujan's Notebooks, Part V
и = 77
t/, = 6 + л/ГГ
Gn —
\-
2 + vTT
1/2
1/4
= 141
= 18
V6 = 125680 + 72576^
1/2
1/12
= 145
f/, = 12 + VT45
17
G =
|
1/2
/ /12 + л/М5 /8
х It/—— + -
1/2
л = 205
2025 + 315V4T
/25 + 3V4T
+
1/4
8
+
8 /
ч 1/2

34. Class Invariants 267
n =213
t/, = 42 + 24ч/3
V6 = 2169904+ 1252800л/5
/l9+12v^\
V 2 j
1/2
542476 + 313200V3 + ^542475 + 313200ч/з)
1/12
n = 217
(/, = 22 + 8V7
V, = 16 + 5V7
/22 +
+ 8V7\
—J
1/2
5-4/7
1/2
и = 265
5\/265
G- =
/89 +
/81+5v^65\
V 8^ J
1/2
+
12 +л/265
1/2
и = 301
= 1199+184V43
+
42 + 7V43
1/2
1-
1/4
268 Ramanujan's Notebooks, Part V
n = 445 I U-> = 71325 + 7560^89
I I
( /19163+ 2156ч/79 /19159+ 2156>/79\
\j 4 +V 4 J
/ /769 + 29^f^ /761+29л/697\
~ \i 8^ +V 8 J

34. Class Invariants 269
и = 793
= 704 + 25-ч/793
V, = 452 + 16-/793
I \
25>/793 /700 + 25>/793 \
4 4 4 J
f452+16y/793 /448
- h ,
1/2
8. Miscellaneous Results
In this section we collect together some miscellaneous results of Ramanujan on
class invariants. We have been unable to provide meaningful interpretations for
two of these entries.
Entry 8.1 (p. 311, NB 1). We have
3 ¦ 25'12
(8Л)
3 ¦ 2
G3/25 = -^— ^— . (8.2)
Proof. We apply Lemma 4.4 with q = exp(—n^/n), and so by A.6),
P = Gn-2G2-52n, G = G^/G|5n,
and
^) = 0. (8.3)
Let n = 3. From our table in Section 2, G3 = 21/l2. Thus, by (8.3), with
x = G75.
91/4
^ 2B2 2'^2 = 0. (8.4)
This sextic polynomial has two real roots, and, via @Mathematica, we easily
checked that the expression on the right side of (8.1) satisfies (8.4) and is the
correct real root.

270 Ramanujan's Notebooks, Part V
Alternatively, from Weber's book [2, p. 724], G75 is i root of a certain cubic
polynomial. In fact, (8.4) factors over Q(x/5) into a prodict of two cubic polyno-
polynomials, one of which is Weber's cubic polynomial, and s» we have g.ven another
derivation of Weber's cubic polynomial for G75- 3 _
To prove (8.2), we again use (8.3), but now we set n = 55 • Thus, G25n = Gy =
21/12. Hence, with у = G3/25.
_ll + JiH + 2B-|/6y-2 - 21/6y2) = 0, (8.5)
21/4 уЪ
which is exactly the same as (8.4), but with x replaced byy. We used Mathematica
to verify that the right side of (8.2) is a root of (8.5) and ndeed is the correct one.
Of course, G3/25 is the "other" real root of (8.4) that wementioned above.
Entry 8.2 (p. 316, NB 1). We have
G175 =
1
- 3V2T)
(8.6)
and
G25/7
(8.7)
Proof. We employ (8.3) with n = 7. From our tablesin Section 2, G7 = 21/4.
Thus, with G = G175, we find that
23/4 лз
~^ + ^u + W{/2G-2-?/2G) = 0. (8.8)
G 2J'4
Now, by straightforward algebra, it is easily checked tW (8.8) yields
2G3 - 4 • 21/4G2 + V2G - 3 ¦ 23/4 = ±V5B • 21/4?2 - V2G + 23/*). (8.9)
We shall show that the right side of (8.6) is a solution othe cubic equation in (8.9)
where the plus sign is chosen on the right side. Thus, \ith the plus sign chosen in
(8.9), we find that, after simplification,
2G3 - 21/4D + 2V5)G2 + (л/2 + VlO)G - 23/C + л/5) = О. (8.10)
The form of Ramanujan's formula (8.6) suggests that e set G = 1 /x and solved
for л:. Thus, by (8.10),
23/4C + ч/5)д3 - (л/2 + VTO)jc2 + 2l/4D + ^5)x -2 = 0. (8.11)
34. Class Invariants 271
We solve (8.11) by employing Cardan's method (Hall and Knight [1, p. 480]).
Thus, set
x = у +
3 • 2J/4C + V5)
in (8.11), and, after dividing out the common factor 47 + 21 >/5, we find that
27 • 23/У + 18 ¦ 2l/4V5y + ±(-55 + 23^5) = 0. (8.12)
The solution of the general cubic equation y3 +qy+r = 0 requires the calculation
4/4 + <73/27. With
_ 23л/5 - 55
r ~ 27 ¦ 27'4
we find, after much simplification, that
and
л/Ю
3 '
Thus, from Hall and Knight's text [1, p. 480], the real root of (8.12) that we seek
is
.. I'/3 I ,. — n: /?;. /=¦ .. I'/3
У =
55-23V5
+
55 -
27-211/4 Зл/32"/4 j [ 27-2"/4
= 3ГТП7Т2 (f55 - 23^ + 15V2T -:
Зл/32"/4
J55-23x/5-
Thus,
J8 - 3V5 -
This is easily seen to be equivalent to (8.6).
To prove (8.7), we set n = \ in (8.3). Since Gn = Gi/,,, it follows that
G1/7 -Gn - 2I/4. Thus, with G = G25/7, we deduce (8.8) once again. Hence,
G25/7 is the other real root of (8.8). Therefore, taking the minus sign on the right
side of (8.9), we find that G satisfies the equation
2G3 - 2l/4D - 2V5)G2 + (V2 - VlO)G - 23/4C - у/5) = 0.

272 Ramanujan's Notebooks, Part V
Now repeat the calculations from the proof of (8.6), but with V5 replaced by —V5.
We then deduce (8.7) to complete the proof.
We also calculated G175 from (8.10) by using Cardan's method and found that
3
x |2 +
(8.13)
which is a slightly more elegant representation than (8.6). By combining (8.6) and
(8.13), we deduce that
1/3
1/3
=9.
(8.14)
We are unable to establish (8.14) directly.
At scattered places in the second notebook, Ramanujan discusses a few addi-
additional class invariants.
Entry 8.3 (p. 263, NB 2). Let t = I/G^v and let x denote the positive real root
of
x6 + 9jc5 + 5jc4 - 2x3 - 5jc2 + 9x - 1 = 0.
Then x = f4, where t > 0. Furthermore,
1-f4
л/29-5
and
Lastly, if
(8.15)
(8.16)
(8.17)
(8.18)
34. Class Invariants 273
where \i > 0, then
M3 + ц = \/2.
(8.19)
Proof. It is not difficult to show that the first claim is equivalent to a result in
Weber's book [2, p. 722].
We now prove (8.16). By straightforward algebra, it is readily verified that (8.15)
is equivalent to the equation
-x
(,20)
Let
Then, from (8.20),
У = -
x \l+x
y2-5y-\=0,
which has the roots j E ± -У29). Since x = 0.119252..., the plus sign must
be taken. Hence,
V29-5
(8.21)
Taking the square root of each side and recalling that x = t4, we complete the
proof of (8.16).
We next prove (8.17). Employing (8.21) and (8.16) and remembering that x =
t4, we see that we are required to prove that
(8.22)
Again, from (8.21),
I 1 -x
x/29-5
i.e.,
Hence, from (8.23),
1 1-х
1 + x
= л/29±2,
-1 +x
X :
1 -X
(8.23)

274 Ramanujan's Notebooks, Part V
and
1 + vWV29 + 2 = X
' + 2x2 - x + 2
1-х2
Thus,
1 ^ + 7^29 - 2 \ _ (-2дс3
:3 + 2л:2 - x + 2J
By (8.22), we want to show that the right side above is equal to A + л)/A - x).
Thus, it suffices to prove that
A -x)(-2x3 -x2 -2x+ lJ = *(l+Jt)(;t3+2*2-;t + 2J.
Expanding both sides and collecting terms, we find that the foregoing equation is
equivalent to the equation
0 = *8 + 9л7 + 6хь + 7х5 + 1хъ - 6jc2 + 9x - 1
= (x1 + 1)U6 + 9л-5 + 5л4 - 2л:3 - 5x2 + 9л- - 1).
Since x2 + 1 Ф 0, it suffices to prove that the second factor above equals 0. But
this is true by (8.15), and so the proof of (8.17) is complete.
Next, we prove (8.19). From (8.18),
= 1+/Л (8.24)
This equality and the symmetry in (8.15) suggest that we set p = jc -x. A brief
calculation shows that
л:6 + 9л5 + 5л4 - 2л:3 - 5л2 + 9л: - 1 = -л3 (р3 - 9р2 + 8р - 16) .
Since л- ф 0, by (8.15),
Р3-9р2 +
From (8.24) and the definition of p,
-16 = 0.
(8.25)
Thus,
Since clearly цг + д > 0, it therefore suffices to prove that
and since p > 0, it is sufficient to prove that
,_ 4 4
34. Class Invariants 275
Squaring both sides and simplifying, we find that it suffices to prove that
p3 - 9p2 + Ър - 16 = 0.
But this is precisely (8.25), and so the proof is complete.
Parts of the proof of Entry 8.3 were taken from the notes of V. R. Thiruvenkat-
achar and K. Venkatachaliengar [1].
Entry 8.4 (pp. 263, 300, NB 2). Let t = 2'/4/С79. Then t is the real root of
t5 - t4 + t3 - It2 + 3r - 1 = 0. (8.26)
Furthermore, if
<1
then
At5 - 2/x4 + \J? + 2/x - 3 = 0.
(8.27)
(8.28)
Proof. The class equation for n = 19 was not computed by Weber [2] and is not
otherwise given by Ramanujan in his paper [3] or notebooks [9]. However, Russell
[2] and Watson [10] determined the class equation, which is easily shown to be
equivalent to that of Ramanujan.
Now (8.28) is valid if and only if
(м5 + ц? + 2iiJ = C + 2M4J , (8.29)
since the square root of each side is positive. Also, (8.29) holds if and only if, by
(8.27),
0 = /г10 - 2м8
- 9
-5
= -Г5 - 2r4 + t - 5 - /"' - 2Г4 + t
= _ (,"> + 2t9 - t6 + 5/5 + t4 + 2f - 1) Г5
= - (ts -14 +13 - It2 + 3t - 1) (r5 + 3r4 + 2r3 +12 + t + 1) Г5.
Since t > 0,/5 + 3f4 + 2f3 + /2 + / + 1 > 0. Hence, (8.28) holds if and only if /
satisfies (8.26), and this is what we wanted to prove.
Entry 8.5 (p. 382, NB 3). Let г = x + l/x, where x = G2,. Then
¦- 0. (8.30)

276 Ramanujan's Notebooks, Part V
Proof. From Weber's book [2, p. 722], as corrected by Brillhart and Morton [1],
\ и)
(8.31)
Since u = G\^, (8.30) and (8.31) are equivalent.
Ramanujan's formulation of Entry 8.5 is, in fact, slightly enigmatic. In particu-
particular, in contrast to the notation used throughout the second notebook [9], Ramanujan
employed the conventional notation in the theory of elliptic functions and wrote
\/x - УШ'.
We now make a few remarks about two entries possibly related to invariants.
On page 294 of the first notebook, Ramanujan claims that
"F
where
1 (9V3+1I'3
x + -
x J Vl + A± л/А}
x+ l/x-2
As intimated in Section 2 prior to the tables, it is not difficult to show that the
indicated formula for Gm is equivalent to the one given in the tables. Parts of the
last two lines of the entry above are difficult to decipher, especially the definition
of A. Moreover, we are uncertain that these two lines pertain to the first two lines,
that is to say, that the value of x in the first two lines is the same as in the second
two lines. Even more enigmatic is that an equality sign seems to be missing in the
last two lines. With the two values of x arising from the second line, we calculated
the expressions in the last two lines and could not account for a missing equality
sign. In conclusion, we are unable to supply any meaningful interpretation to the
incomplete entry offered in the last two lines.
On page 343 in the first notebook, Ramanujan wrote
34. Class Invariants 277
We have no explanation for this mysterious fragment. The definition of у is not
given, but perhaps Ramanujan intended to write у = J (л/П - 3)/2. However,
this value of у is not a solution of either of the indicated polynomial equations.
The top half of the page comprises results discussed in Chapter 37, and underneath
the fragment is Ramanujan's (equivalent) representation for С765; neither topic
appears to be connected with this fragment.
9. Singular Moduli
Recall from the Introduction that the singular modulus к„ is defined by к„ : —
k(e~"^"), where и is a positive integer. It is clear from A.6) that if the value
of Gn (or gn) can be determined, then а„ :— к* can be computed by solving a
quadratic equation. For example, see B.8) or (9.1) below. However, the expression
that one obtains generally is unattractive and does not evince the fact that an can be
expressed in terms of units in certain algebraic number fields. (See Theorem 1.1.)
Thus, formulas for а„ that facilitate their representations via units are desirable.
In his second letter to Hardy, Ramanujan [10, p. xxix] asserted that
- V6L(8 - Зл/7J
*2.o =
- 1LB -
- 3LD -
- л/14JF - л/35J.
This was first proved by Watson [4], who used the following remarkable formula
which he found in Ramanujan's first notebook [9, vol. 1, p. 320] and which enables
one to calculate а„ for even n.
Theorem 9.1. Set
= и»,
и2 + l/u2 = 2U.
W =
v2 + l/i/ = 2V,
v2 - 1,
and
Then
-V - 1}2{VS- W - VS-W-1}2.
Watson's proof of Theorem 9.1 is a verification; it does not shed any light on
how Ramanujan might have discovered the formula. K. G. Ramanatlian [1], [5]
stated Ramanujan's Theorem 9.1 but did not find another proof. Heng Huat Chan
has found a much more motivated proof of Theorem 9.1, and we present his proof
below. Later we show that the algorithm implicit in Theorem 9.1 can be adopted
to determine а„ for odd n as well.

278 Ramanujan's Notebooks, Part V
Proof. From A.6) and the notation above,
1
— - vG = 2u2v2 = 2A/ + Vu2 - \)(V +
- 1)
UV
- 1))
= 2
where we set
and
Jab = f/V + V(l/2 - 1)(V2 -
Squaring each of the last two equalities, we find that, respectively,
ab = 2U2V2 - U1 - V2 + 1 + 2UVy/(U2- 1)(V2 - 1)
and
ab+b-a- 1 =2l/2V2-(/2-V2
These two equalities imply that a -b. Thus,
- 1).
a) .
Solving for .y/a, we find that
-Ja. = (Va + 1 - -Ja)(-Ja - -Ja - 1).
(9.1)
It thus suffices to compute ^a.
From the definition of a and the fact that a —b,
я = l/V + >/((/2 - 1)(V2 - 1)
= 5 (lUV + 2у/игУ2 -U2-
= { BUV + 2-JU2V2 - W2)
34. Class Invariants 279
We write the last equality in two ways as follows:
a —
VJ -U2 - V2-2W
VJ - U2 - V2 + 2W
\-{U-
V2-2W
V2 + 2W
Now,
E - W)(S - 1) = S(S - W - 1) + IV
_(U + V + W+l)(U + V-W-l)
~~ 2 2~
(U + VJ - (W + IJ . „„
+
(9.2)
4
_ (U + VJ -U2-V2-
4
_ (U + VJ-U2- V2 + 2W
= . _ .
Thus, from the first equality in (9.2),
a = (yS(S - W - 1) + /(S - IV)E - I)J
= S(S - W - 1) + (S - W)(S - 1) + 2y/S(S - W - \)(S - W)(S - 1)
= 5E - W) + E - W - 1)E - 1) - 5 + E - 1)
- W - 1)E - W)(S - 1),
i.e.,
a + 1 = S(S - W) + (S - W - \)(S - 1) + 2^/5E - W)(S - W - 1)(S - 1)
Hence,
- W)
a + 1 - v^ = >/5(S - IV) + 7E - W - \)(S - 1)
- W - 1) - V(S - W)(S - 1)
- IV -

280 Ramanujan's Notebooks, Part V
Next,
(S - U - 1)(S - V - 1) = (S - U)(S -V)-W
_ W+\ + V -U W+\ +U - V
2 2
_ (W+\J-(U- VJ
~ 4
t/2 + V2 _ 2W - (U - VJ
- w
Thus, from the second equality of (9.2),
a = (j(S-U- 1)(S-V-l) + y/(S-U)(S- V)J
= (S - U - 1)(S - V - 1) + (S - t/)(S - V)
i.e.,
+ 2j(S-U - 1)(S- V - \)(S-U)(S- V)
= (S-U - 1)(S - V) + (S - l/)(S - V - 1) + 1
+ 2>/(S - I/ - 1)(S - V)(S - U)(S - V - 1),
- V) + V(S - U)(S - V- 1)) .
a - 1 =
Hence,
-U- IMS - V) - 7E - I/MS - V - 1)
= (Vs^V - vs - V - i) (ys^V -y/s-u-
Using our just calculated formulas for y/a + 1 - Va and лА ~ Vfl - 1 in (9.1),
we complete the proof.
Furthermorc,Watson [4] inexplicably claimed,"... this is the sole instance in
which Ramanujan has calculated the value of к for an even integer и." In fact, 20
additional values of kn for even n are found in the first notebook. Theorem 9.2
gives 13 of these values.
On page 82 of his first notebook, Ramanujan offers three additional theorems for
calculating а„ when n is even. The first (Theorem 9.3) expresses a4p as a product
of units involving Gp. The second (Theorem 9.5) expresses a^p as a product of
units involving Gp. The third (Theorem 9.6) enables one to determine щр as a
product of two fourth powers of units, provided that aiP can be expressed as a
product of units of a certain form. We calculate eight examples of Ramanujan as
illustrations.
The calculation of an when и is odd is slightly more difficult. On page 80 in
his first notebook, Ramanujan recorded the values of а21,азз» and «45 in terms
34. Class Invariants 281
of units. This list is repeated, with the addition of a,5, at the bottom of page 262
in his second notebook. On pages 345 and 346 in his first notebook, Ramanujan
recorded units that appear in representations of an when и = 3,5,7,9,13,15,17,
25, and 55. (Inexplicably, the units foror7 and«i5 are recorded twice.) Ramanujan
also indicated that he had intended to calculate a39, but no factors are given.
Of course, the result for n = 15 is superseded by the complete formula given
on page 262 in the second notebook. It is unclear to us why Ramanujan only
listed portions of а„ and not complete formulas. Initially in our investigations we
employed computational "trial and error" to "guess" the complete formulas for
а„, n = 5,9,13,17,25, and 55. We remark that the values for a3 and a7 are easily
determined from B.8), i.e.,
<*n = 5G
-12
(9.3)
For further values of n, however, (9.3) becomes unwieldy, and so better algorithms
were sought.
We adopt the algorithm of Theorem 9.1 and reformulate it in Theorem 9.8 in
terms of Gn to calculate some values of а„ when n is odd. Theorem 9.9 provides
a list of all of Ramanujan's values for odd n. Although Theorem 9.8 yields a
systematic procedure for calculating а„ when n is odd, the calculations are often
cumbersome and the representations that we obtain, although expressed in terms
of units, are frequently more complicated than we would like. Thus, we establish
three simple lemmas, Lemmas 9.10-9.12, that provide an alternative procedure
for calculating all of Ramanujan's singular moduli for odd n.
We conclude Section 9 with two further algorithms of Ramanujan for computing
а„. These were cryptically stated by Ramanujan in his first notebook and are rather
different from his other algorithms. The first provides a method for determining
etin from a certain type of modular equation of degree n. The second also arises
from modular equations and gives a formula for a3n.
Ramanujan likely learned about singular moduli from a brief discussion in A. G.
Grecnhill's book [3, p. 331]. It would be extremely difficult to assess the priority
of each singular modulus that has been determined. Ramanathan [1] and J. M. and
P. B. Borwein [1] previously calculated some of Ramanujan's values for а„, and
we shall cite their specific determinations in the sequel.
Singular Moduli for Even n
We begin with a list of 13 values for а„ found on scattered pages in the first
notebook.
Theorem 9.2 (pp. 214, 288, 289,310,312,313, NB 1). We have
- IJ,

282 Ramanujan's Notebooks, Part V
= B-
- 3JC - 2V2J =
V2-1
7V2-2V6-5
a22 = A0 - Зл/ГТJ(Зл/ГТ -
изо = E - 2>/6JD - \/Т5J(>/б - V5JB - V3J,
a42 = (8 - Зл/7JG - 4V3JC - 2-Jlf(sh - V6J,
Of58 = A3V58 - 99J(99 - 7(k/2J,
a70 = A5 - 4\/l4J(8 - Зл/7JC\/Й - 5n/5JF - л/35J,
a7S = B - >/3NCV3 - л/26J(ч/!з - 2л/3LE - 2л/бJ,
«юг = [ ^Г
«130 = Eл/Т30 -
7 J E - 2x/6L
2 - л/3L,
- 3L(V26 - 5LC - 2V2L,
and
_ 3V2L
Proof. The value of a2 was, in fact, established in Example 1, Section 2 of Chapter
17 in the second notebook (Part III [3, p. 97]). Ramanathan [1] and the Borweins
[ 1, p. 139] also determined a2.
All of the remaining values for an are easily determined from Theorem 9.1. The
required values for gn can be obtained from the tables of Weber [2] or the table
in Section 2. In each instance, we list the values for u, v, U, V, W, and S in the
table below. The reader can easily verify the calculations.
34. Class Invariants 283
n
6
10
18
22
30
42
58
70
78
102
130
190
и
1
1
1
1
2 + x/5
2y/i+-Jl
1
9 + 4^5
18 + 5^13
7 + 5V2
38 + 17л/5
38+ 17V5
V
1+V2
2 + V5
5 + 2ч/б
7 + 5ч/2
3 + vT0
Зл/3 + 2л/7
7O+13V29
7 + 5л/2
5 + >/26
35 + 6n/34
18 + 5vT3
117 + 37л/Т0
и
1
1
1
1
9
15
1
161
649
99
2889
2889
V
3
9
49
99
19
55
9801
99
51
2449
649
27,379
W
3
9
49
99
21
57
9801
189
651
2451
2961
27,531
S
4
10
50
100
25
64
9802
225
676
2500
3250
28,900
The second and third formulas for «& and «ig, and the second formula for «щ
can be easily verified by direct calculations.
The Borweins [1, p. 139) calculated an for 1 < n < 9. Ramanathan [1] also
established «130 by using Theorem 9.1.
Recall the definition of F(x) given in B.3).
Theorem 9.3 (p. 82, NB 1). If p > 0, n > 1, and
then
From B.9), n = Gj,2. Hence, in Theorem 9.3 Ramanujan provides an algorithm
for determining cc4P from the value of ap, or from Gp, namely,
+ 1 - /cj2) (JGJ - jGp* - I). (9.4)
Before proving Theorem 9.3, we verify four examples recorded by Ramanujan.

284 Ramanujan's Notebooks, Part V
Examples 9.4 (p. 82, NB 1). We have
«4 = (V2-1)\
an = (л/3 - V2L(V2 - IL,
a28 = (V2-l)8BV2-V7L,
and
- 3L(V2 - 1L(л/б - л/5L(ч/3 - л/2L.
The value of a4 was also recorded in the second notebook (Part III [3, p. 97]).
Both o<4 and ащ were also determined by Ramanathan [ 1 ], and the Borwein broth-
brothers have determined c*4 and an [1, pp. 139,151].
Proof. Let p = 1, so that trivially G\ = 1. Then, from (9.4),
«4 = (V2 - 1LA - 0L = (V2 - IL.
Let p = 3, so that, from the table in Section 2, G3 = 2I/I2 and n = 2. Thus,
from (9.4),
al2 = (V3-V2L(V2-lL.
Let p = 7, so that, from the table in Section 2, G7 = 2I/4 and n = 8. Thus,
from (9.4),
«28 = C - 2V2LBV2 - V7L = (V2 - 1)8BV2 - V7L.
Let p = 15. From the table in Section 2, d5 = 2/l2(l + V5I/3. Thus,
n = A + V5L/2 = 4G + 3V5). Hence, from (9.4),
/ I I \4 / I I \4
«60 = (v29+12^5-2V7 + 3^5) BV7 + 3V5-\/27+12V5J .
To denest these radicals, we employ the following denesting theorem (Landau
[1]). If a2 - qb2 = d2, a perfect square, then
(9.5)
2 ' -""-'V 2
where we have coяected a misprint. To that end, from (9.5),
_ / /29+11
= ^-2-
*[2\l^~+ V^~ V 2 V 2 j
= (V20 + 3 - Зл/2 - VlOLCV2 + VTO - a/15 - л/ИL
j _ 3)(V2 - 1)}4{(V6 - V5)(V3 - V2)}4.
34. Class Invariants 285
Proof of Theorem 93. From Part III [3, p. 215, eq. B4.21)], we find that
(9.6)
where f) has degree 2 over a. To prove Theorem 9.3, we must show that
(9.7)
P = (>/n-l - VnL(v/nTT - лАL-
From our hypothesis, with n := Cj,2 and a := ap,
a =
и - Vn2 - 1
~2n '
which implies that
Since 4a(l - a) — n~2, we deduce that
1 - a
Hence, from (9.6), (9.8), and (9.9),
(9.8)
(9.9)
/ 1 \4
- I)L 1 , , )
. \ 4
^ Vn + 1 + Vn - 1 \
v« 1
= (\/n2 + n + >/«2 - n — n - y/n2 - IL
= (Vn- 1 - V")\Vn + 1 - VnL,
and so (9.7) has been shown.
Theorem 9.5 (p. 82, NB 1). Under the same hypotheses as Theorem 9.3,
-**Jp = F ((%/п + j + ^s j jjn + l -y2vWn~+T + V2)|
x JV2^- 1 - /zVn^VnTT- V2)| ).
Thus, together Theorems 9.3 and 9.5 yield the formula
a16p = (/GJ2 + 1 +
+ 1 -
- 1 -
+ 1 - V2)| .

286 Ramanujan's Notebooks, Part V
For example, if p = 1, then n = C, = 1, and a simple calculation shows that
Proof. From Theorem 9.3,
(9.П)
Also, by Entry 2(v) of Chapter 17 in the second notebook (Part III [3, p. 93]), for
0 < x < 1,
Thus, from (9.12), (9.10), and (9.11), it suffices to show that
= (ATT + s/n)* I у/Ъг + 1 - JzJH(.JZT\ +
From (9.11), it follows that
1
(9-13)
(9.14)
Let и = yfx. Since и tends to 0 as л tends to oo, the solution of (9.14) that we
seek is
1
к =
1
+ -
2(л/лТТ - VnJ(л/л ~ л/и ~ О2 2
2(>/йТТ - л/йJ(л/« - л/и - IJ
- (VnTT - V^JJ
= (лЛГТТ + V«J (^2/1 + л/л (и - 1) - >/л (я +
-Ул/и(л- 1) + л/л (л + 1) - 1 j
«TT + V«J Bи - 1 + 2л/л(« -1)
34. Class Invariants 287
1)л/л(л + 1) + Bл - 1)л/л(л - 1) - 4л . (9.15)
Comparing the proposed value of и from (9.13) with that of (9.15) above, we see
that it suffices to show that
Bл — 1 )у/п(п — 1) — 4и
(л/2л-
^n+ 1 + л/2).
(9.16)
If we square both sides of (9.16), it is a routine matter to show that (9.16) indeed
is a correct equality. This therefore completes the proof.
The next theorem enables one to determine a8p from the value of a2p.
Theorem 9.6 (p. 82, NB 1). Ifn > 1, p > 0, and
(9.17)
then
•v/w
Jn - 1 + y/n + 1
М8)
Observe that
(9.19)
л/л ± 1 + л/" + 1
Thus, if a-ip can be expressed as a product of units of the form (л/л + 1 - -УйJ
(л/й — л/и — IJ, then agp сап be expressed as a product of two fourth powers
of units. Before proving Theorem 9.6, we present three examples recorded by
Ramanujan.
Examples 9.7 (p. 82, NB 1). We have
a8 =

288 Ramanujan's Notebooks, Part V
and
«40
(
Proof. Let ,p = 1. Then from Theorem 9.2, a2 = (\/2 - IJ. Thus, n = 1, and
from Theorem 9.6,
(9.20)
But from (9.5),
2 V 2
Using this in (9.20), we achieve the desired representation of a8.
Let p = 3. From Theorem 9.2, a6 = B - ч/3J(-ч/3 - V2J. Thus, л = 3, and
from Theorem 9.6,
«24 =
But from (9.5),
and
(9.21)
2 ' V 2 V2
Using these calculations in (9.21), we complete the verification of Ramanujan's
representation for a24.
Let p = 5. From Theorem 9.2, аю = (vTO - 3JC - 2</2J. Thus, л = 9,
and from Theorem 9.6,
a,» = f2a/2 + V5 - ^4C + y/loU
which is what is claimed.
Proof of Theorem 9.6. From (9.17),
Hence, as in (9.14),
(¦s/nTT - y/nX^/n - vV. - 1)
(9.22)
34. Class Invariants 289
and, by (9.18) and (9.12), it suffices to prove that
—T)|
J
Let м = y/x. In view of the form (9.22), it is natural to assume that
where
a]-b\=\=a\- b\.
Then, by (9.22) and (9.24),
— ^__ _ j/ Л _
( л+ +V«)(V"+ « 1}-2V + uJ~
(9.23)
(9.24)
(9.25)
(9.26)
If 5 := -Jn + *Jn + 1, the values of a,, />], a2, andfc2 that satisfy (9.25) and (9.26)
are
- ^J. a - — fc - л/? /JS—1 and b - J~sJ^ft-\
Then, as already observed in (9.19), (9.25) is satisfied. Furthermore,
= (y/n + 1 + -Jn)(Vn + V" - 1),
and so (9.26) is satisfied. Hence, (9.23) has been shown, and the proof of Theorem
9.6 is complete.
Singular Moduli for Odd n
From A.6), we find that, in the notation of Theorem 9.1,
1 "~ (9.27)
By elementary manipulation, we find from the other equality of A.6) that
(9.28)
If we set
and

290 Ramanujan's Notebooks, Part V
then (9.28) takes the form
(9.29)
Comparing (9.27) and (9.29), we deduce the following theorem from Theorem
9.1.
Theorem 9.8. Set
(8'nf = «».
и2 + 1/М2 = 21/, v2+l/v2 = 2V,
and
Then
^/ - л/S-U -IJ
- V -
-W- I}2
The next theorem gives the twelve values of а„, when n is odd, that are found
in Ramanujan's notebooks. In those instances when two representations are given,
the former one is that which is in the notebooks, or that which contains the units
provided by Ramanujan in his notebooks. The Borweins [1, pp. 139, 151] calcu-
calculated an, for n — 3,5,7, 9, and 15, and Ramanathan [1] determined <*„, for и =
3,7,9, and 15.
Theorem 9.9 (pp. 80,345,346, NB 1; p. 262, NB 2). We have
2-V3
4
«5 = Г
a7 =
8-Зч/7
16 '
a9 =
34. Class Invariants 291
2
«13 = Г
= df\2
«15 = 77
(
VTT-3
(/ 1 \ 4
M /iW3\
«45 = i

292 Ramanujan's Notebooks, Part V
and
ass = 1 Л/5 - if A0 - Зл/П)C^5 - 2ч/ТТ)
16 V /
Proof of Theorem 9.9 for n = 3, 5, 7, 9, and 13. These five values arc easily
computed by using Theorem 9.8. The required values for Gn may be found in the
table in Section 2. In each instance, we list the values for м, v, U, V, W, and S in
the table below. The reader should easily be able to verify the calculations.
n
3
5
7
9
13
и
expGn'/4)
ехр(лч/4)
exp(jri/4)
ехр(лч/4)
exp(?ri/4)
i;
¦Л
V2W5
23/2
2 + л/З
V18 + 5V13
U
0
0
0
0
0
V
5
4
V5
65
16
7
5^13
W
3
4
2
63
16
4>/3
18
S
3
2
iC + v/5)
9
2
4 + 2V3
iA9 + 5>/l3)
Except for и = 55, we have also used Theorem 9.8 to calculate the remaining
values in Theorem 9.9. However, the following lemmas lead to simpler calcula-
calculations.
Lemma 9.10. Ifr is any positive real number and t = y/(r + l)/8, then
Proof. The equality (9.30) can be readily verified by elementary algebra.
Lemma 9.11. Ifr and t are as given in Lemma 9.10, then
- Jr1 - 1 =
(9.31)
34. Class Invariants 293
Proof. It is readily verified that
(9.32)
Using Lemma 9.10 in (9.32), we deduce (9.31).
We frequently set G — Gn below, when the value of n is understood.
Proof of Theorem 9.9 for и = 5, 9,13, 15,17, and 25. Let n = 5. From the
(able in Section 2,
= V5 + 2.
(9.33)
Ifr = Gl2 in Lemma 9.10, then
and
(9.34)
Thus, the given value for as follows immediately from (9.3), (9.33), and (9.34).
Let n = 9. From the table in Section 2,
(9.35)
Of. ?+1 =7+4V5.
Applying Lemma 9.11 with r = G^2, we find that
2 '
and
4 V 4
Thus, by (9.3), (9.35), and (9.36), we deduce Ramanujan's value for a9.
(9.36)

294 Ramanujan's Notebooks, Part V
Let n = 13. From the table in Section 2,
(9.37)
Then in Lemma 9.10, set r = G\\ to deduce that
t =
4
and
— 1 =
(9.38)
Hence, the given value for ац follows from (9.3), (9.37), and (9.38).
Let n — 15. From the table in Section 2,
G\] = i(V5 + IL = 28 + 12л/5.
Apply Lemma 9.10 with r - G^-
(9.39)
t —
29 +
and so
С12-
/29+12V5
V 8
29+12V5
2 1
1
2
/27-
/
N
M
2
/29
V
2V5^
+ 12V5
8
)
1
2
= 28 + 12V5 - /B9
= 28+12ч/5-16л/3-7л/15
-B-V3JD-
(9.40)
Hence, by (9.3), (9.39), and (9.40), the desired result follows.
Let n = 17. From the table in Section 2,
12
I
34. Class Invariants 295
after a lengthy calculation. We now apply Lemma 9.10 with r = 20 + 5-/17. Then
21+5VT7 5 + VT7
8
and so
17 =
(9.41)
Next, set r = 20 + 5vT7 + ^B0 + 5л/ПJ - 1 in Lemma 9.11. Then
21 + 5vf7
8
42 + Юл/17 + 4^06 + 50л/17
Гб
3 + л/17 + 2ч/4+л/Г7
and
//+2 =
Thus,
G'2_v/G24_ , =
м+Тп
-V—5—j-
Using (9.41) and (9.42) in (9.3), we complete the proof.
Let n — 25. From the table in Section 2,
Withr = 0^ in Lemma 9.11,
t =
and
3V5
(9.42)
(9.43)

296 Ramanujan's Notebooks, Part V
Hence,
(9.44)
4 V 4
Putting (9.43) and (9.44) in (9.3), we complete the proof.
The next lemma will enable us to calculate ОГ21, азз,««, and а^$.
Lemma 9.12. Let r = uv, where v > и and и is a unit. Set и = U] + «2, where
и i, «2 > 0 and и2 — u\ — 1. Furthermore, let
a2 = \+2vu\+v2 and b2 = 1 - 2vux + v2,
where a,b > 0. Then
- v/r2 - 1 =
ja + b + 2
16
1_
+2
a+b+2 1
~l6 2
la-b + 2
~1б
Proof. The right side of (9.45) equals
1 I ja-b + 2 l\
2"VV 16 2j ¦
(9.45)
a+b-2
a-b + 2 la-b~2
(9.46)
Set
Then, by an elementary calculation,
-b
-b
34. Class Invariants 297
Hence, we see that the right side of (9.46) equals r' - vV'2 - 1, and it therefore
remains to show that r = r'.
In fact, from the definitions of a2 and b2.
r' = vu\ + ^1Jи\ + 1 - A + v2)
= VU\ + VJU2 — 1 = DM] + VU2 = VU = Г,
which completes the proof.
ProofofTheorem9.9for л = 21,33,45, and 55. Letn = 21. From the table in
Section 2,
= (8
. (9.47)
Set mi = 2-У7 and v = 8 + 3^7 in Lemma 9.12. Then
a2 = 1 + 2(8 + Зл/7J>/7 + (8 + Зл/7J = 212 + 80л/7 = A0 + 4л/7J
and
fe2 = 1 - 2(8 + Зл/7J>/7 + (8 + Зл/7J = 44 + 16л/7 = D + 2л/7J.
Hence, в = 10 + 4л/7 and b = 4 + 2л/7. Moreover,
ja+b + 2 /16+ 6-
V 16 ~ V 16~
6ч/7 3 + V7
and
16
a-b + 2 /8+2v/7 1 +
16
Thus, by Lemma 9.12,
On using (9.47) and (9.48) in (9.3), we complete the proof.
Let n — 33. From the table in Section 2,
V3~)
31 \
=B6+15v/3)A0
(9.48)
(9.49)
1 \^
Apply Lemma 9.12 with м, = 10 and v - 26 + 15 V5. Then
a2 = 1 +2B6+ 15>/3I0 + B6+ 15\/3J = 1872+ 1O8OV5 = C0+ 18V5J

298 Ramanujan's Notebooks, Part V
and
b2 = 1 - 2B6 + 15^3I0 + B6 + 15VIJ = 832 + 480^3 = B0 +
Thus, a = 30 + 18-УЗ and b = 20 + 12^3, so that
and
52 + 30 V3 5 + 3V3
16
Ia-b + 2 _ /12 + 6V3 3 + V3
V 16 ~V 16 ~ 4 "
Thus, by Lemma 9.12,
3^
5 + ч/3 1+ч/3
(9.50)
Upon substituting (9.49) and (9.50) in (9.3), we complete the proof.
Let n — 45. From the table in Section 2,
Gg = (V5 + 2K f * ) = C8 + 17n/5)C1 + 8vT5). (9.51)
We apply Lemma9.12 with w, = 31 and v = 38 + 17л/5. Thus,
a2 = 1+ 2C8 +17n/5K1+ C8 + 17n/5J = 5246 + 23464/5 = E1 + 23V5J
and
b2 = 1 - 2C8 + 17n/5K1 + C8 + 17ч/5J = 534 + 238^5 = A7 +
Hence, a = 51 + 23>/5 and fc = 17 + 7-У5, so that
la+b + 2 П
V—Тб— = V"
16
and
la-b + 2 _ /36+К
v 16 ~V i6
Ia-b + 2 /36+16V5 2 + V5
2 '
So, by Lemma 9.12,
G'2_v/C;24_7= j
/з + л/5 /1+V5
(9^52)
The desired evaluation now follows immediately from (9.3), (9.51), and (9.52).
34. Class Invariants 299
Let n = 55. From the table in Section 2,
12
2J
99 +
- 1
(9.53)
Apply Lemma 9.12 with м, = (99 + 45^5)/4 and v = 8(V5 + 2J. Thus,
a2 = 1 + 2 • 8(V5 + 2Ji(99 + 45^5) + 64(V5 + 2)"
= 17469 + 7812V5 = (93 + 42 VB"J
and
b1 = 1 - 2 • 8(V5 + 2J|(99 + 45V5) + 64(V5 + 2L
= 3141 + 1404V5 = C9 + 18V5J.
Thus, a = 93 + 42>/5 and b = 39 + 18>/5, so that
ja+b + 2 I
V 16 "V
67 + 30л/5
and
ja-b + 2 /
V те ~V
7 + 3V5 3 + V5
Thus, by Lemma 9.12,
- 1 =
67 + 30ч/5 1
8 +2
/4+ч/5 /2TV5
67 + 30V5 _ 1
8 ~ 2
(9.54)
Now, by Lemma 9.10,
67 + 30У5
8
+ 2-
67 + 30ч/5 1
8 2
= 66 + ЗОч/5 - /F6 + ЗО>/5J - 1
= 66 + ЗОл/5 - 9V55 - 20VTT
= A0 - 3VTT)CV5 - 2VTT).
(9.55)

300 Ramanujan's Notebooks, Part V
Using (9.55) in (9.54) and then (9.53) and (9.54) in (9.3), we complete the proof.
We close this subsection by showing how different modes of calculating а„ can
lead to interesting identities between radicals.
Entry 9.13 (p. 311, NB 1). We have
(a)
and
(b)
- 1 v^5 ±
1 ± V1 - B - %/3L =
Proof. From the table in Section 2, C25 = (V5 + l)/2, and using this value in
(9.3), we easily find that
B«25I/8 = </l-./l-
On the other hand, from Theorem 9.9,
1 E2!1 _ —\
Combining these two calculations, we deduce (a) with the minus signs chosen on
each side. 1/3
From the table in Section 2, G» = (A + -Я)/ -Л) > and with this value ,
(9.3), we readily find that
in
1/8 = Л 1 _,/i -П-
34. Class Invariants 301
On the other hand, from Theorem 9.9,
(W/8 -
Combining the last two calculations, we obtain (b) with the minus signs chosen
on both sides.
We verified via Mathematica that both (a) and (b) also hold with the plus signs
chosen on each side.
Calculating а-щ and огз„ with the Help of Modular Equations
Ramanujan obscurely described two further methods for calculating ctn in his first
notebook [9].
In the first, Ramanujan indicated that a2n may be calculated by solving a certain
type of modular equation of degree n. For several prime values of л, the desired
form of modular equation exists; many of these modular equations can be found
in Ramanujan's notebooks and are proved in Part III [3]. This very novel method
is the only known method that does not require a priori the value of g2n. Thus, the
method is a new, valuable tool in the computation of аъ,.
In the second, Ramanujan disclosed a method for determining a3n arising from
the definition of a modular equation of degree n. We give a rigorous formulation
of this formula and prove it by using a device introduced in Section 6 to calculate
certain class invariants.
The results in this subsection are due to the author and Chan [3]. In this paper
we also devise a method for calculating «5„ that is similar to Ramanujan's method
for determining a3,,.
In the middle of page 292 in his first notebook, Ramanujan claims that, "Chang-
"Changing 0 to4fi/(l + BJ and a to 1 - B1 we get an equation in 45A - 5)/(l + B)
and the value of B2 is for е"л>*." We now state and prove a rigorous formulation
of this assertion.
Theorem 9.14. Let P have degree n over a, and suppose a and f) are related by
a modular equation of the form
', {A -a)il-P))r)=0,
(9.56)
for some polynomial F and for some positive rational number r. If we replace a
by\~x2 and f} by 4jc/A + xJ, then B.1) becomes an equation of the form
G(x) := g({4x(l - лг)/A + *)}') = 0,
for some polynomial g. Furthermore, x = ^/аъ,. is a root of this equation.

302 Ramanujan's Notebooks, Part V
Proof. Under the designated substitutions,
\+x
= 0.
Hence, the first part of Theorem 9.14 follows.
For brevity, let 2F\(\, \; 1; x) = 2F,(x). Setting 0 = 4x/(l + xJ, we find
that
4x
0-
(9.57)
by a fundamental transformation for F(x) (Part III {3, p. 93]), which actually
arises from a special case of Pfaff's transformation. With a replaced by 1 - x2,
we find from A.4), B.5), and (9.57) that
Therefore,
Now recalling the definition of a singular modulus and A.2), we deduce that
x2 = <*2л, and the proof is complete.
Example 9.15(a). Let n = 3. Then
(9.58)
which is originally due to Legendre and was rediscovered by Ramanujan (Part III
[3, pp. 230,232]). With Ramanujan's substitutions, (9.58) takes the form
«+V = 1. (9.59)
where
и = I Ax
V
(9.60)
Solving (9.59), we find that и = л/3 - 1. Then solving (9.60), we find that
x = 2V5 - 3 - 2\/2 + V6. Using two different modes of calculation, we find
34. Class Invariants 303
that
= x2 = B -
y/l + 3
The former representation for a6 can be found in Theorem 9.2.
Example 9.15(b). Let и = 5. Then we have Ramanujan's modular equation of
degree 5 (Part III [3, p. 280]),
{A - o)(l - p))m + 2{l6afi(\ - a)(l - /3)}1/6 = 1. (9.61)
Using Ramanujan's substitutions, we find that (9.61) can be put in the shape
u + 3tt= 1, (9.62)
where
и = H
l+x
1/2
(9.63)
Solving (9.62), we deduce that и = -6 + 2VTo. Next, solving (9.63), we find
that x — 3vT0 - 9 - 4-У5 + 6\/2. Lastly, by two distinct routes for calculation,
- 3JC -
The former representation is given in Theorem 9.2.
We next derive Ramanujan's formula fora3n.
Let q be given by A.5), and suppose that fi has degree и over a. Thus, A.4)
holds. Now suppose also that f) has degree 3 over 1 — a =: a'. Then, by A.4),
2F,(l-0) ..2F,(l-«) (964)
Hence,
2F,(l-a) 2F,09)
(«) fn
-«) V3'
and, from (9.64) and (9.65),
(9.65)
(9.66)
Next, from Part III [3, p. 237], since 0 has degree 3 over a', we have the
parametrizations
where 0 < p < 1. It follows from (9.67) that
A a)fi~ (i + 2PK p {TTi-p) = VTTTp)
(9.68)

304
and
A -P
Next
Ramanujan's Notebooks,
ч , 0-pXl + pK
)U- (l + 2p)
, set
PartV
It -
+
+
- 1
p
2p
\
i—
h
- p
\-P
P\+2p
+ p)B +
1 +2p
(9.69)
(9.70)
l+2p
and observe that
2A - 0 =
l+2p
It follows from (9.68b(9.71) that
«0 = 0- a')P = 16f3(l - t)
and
' = 1бгA-03-
(9.71)
(9.72)
A«)A/}) (р> @ (9-73)
Now, set
k = 4t(l-t). (9.74)
Observe from (9.72), (9.73), A.6), (9.65), and (9.66) that
к = BG6n/,Glyl . (9.75)
We determine p («з,,) as a function of it. From (9.74), we find that, with no loss
of generality in choosing the minus sign in the first equality below,
i-vT=lt
and 1-/ =
Thus, by (9.72) and (9.73), we find that
and
^T) = fc(l - Vl -
A - 0W = A - Vl -fe)(l + Vl - fcK = Щ + Vl -kJ.
Subtracting (9.76) from (9.77), we find that
a' = 4itVl -k+p.
Substituting (9.78) into (9.77), we deduce that
p2 + рDкл/1 - k - 1) + k(l - Vl - кJ - О.
(9.76)
(9.77)
(9.78)
34. Class Invariants 305
Thus,
«Зл = P =
^ft ±
- it - IJ -
- кJ
^it ± A -
1 - ^/l - 4* A - 2k ±
- 4k)f
(9.79)
This last formulation was that given by Ramanujan.
We now resolve the sign ambiguity in (9.79). It is clear that both a' and P satisfy
the equation
Щ - VT^kJ = 0.
(9.80)
Since a' = 1 - a, it follows from (9.65) and (9.66) that the solutions of (9.80) are
аз„ and а-цп- Thus, it suffices to show that
(9.81)
for и > 1. From the definition of <p in B.4), it is obvious that
From B.5), it follows that
Since 2F\ (jc) is increasing on @, 1), (9.81) follows.
Thus, we have proved the following theorem.
Theorem 9.16. Let q be given by A.5), suppose that P has degree n over a, let
P have the parametrization (9.67), and define t by (9.70). Then, if к is defined by
(9.74), огз„ has the representations given in (9.79), where the minus sign must be
chosen.
Ramanujan's formulation of Theorem 9.16 at the bottom of page 310 in his first
notebook is a bit different. He first gives (9.72) and (9.73), but with the left sides
switched. He then states (9.70), followed by the equality
which is a consequence of (9.66) and (9.67), where F is defined in B.3). He con-
concludes by defining it in (9.74) and by claiming that a more complicated, somewhat
ambiguous version of the right side of (9.79) equals e~"^", i.e., he forgot to write
"F" in front of the right side of (9.79). (In recording specific values of F(a,,),
Ramanujan frequently omitted parentheses about the arguments.)

306 Ramanujan's Notebooks, Part V
Example 9.17(a). Let n = 1. Then, from the table in Section 2, G3 = 21/12 =
G|/3, since Gn = Gx/n (Ramanujan [3], [10, p. 23]). Hence, by (9.75), k = \,
and by (9.79),
2-УЗ
which is given in Theorem 9.9.
Example 9.17(b). Let и = 5. Then, from the table in Section 2, G15 =
2-'/'2(У5 + II'3. It can also be verified that G5/3 = г'12^ - 1 I/3. Hence,
it is easily seen from (9.75) that к = ^. Therefore, from (9.79),
16-7л/3-\/Т5
a,5 =
32
which is simpler than the formula given in Theorem 9.9.
10. A Certain Rational Function of Singular Moduli
Set b2 := b2n :— 0, where 0 has degree In and b > 0. On page 312 in his first
notebook, Ramanujan defines
b(\ - b)
and
и ;= и„ :=
«¦-"•¦- -
l+b
A0.1)
A0.2)
(In fact, Ramanujan uses the notation fi, instead oib above, and he has no notation
for the right side of A0.2).) He then provides the following table.
n
1
3
5
7
9
11
15
23
29
35
71
Vn
1
3
9
(^2+1JA+2^2)
49
99
3E + Л-ЛI
9A + V2LC + Ф/2)
992
63(8^2 + 5V5J
9A + лД)юBу/2 + 1JFл/2 + 1)
34. Class Invariants 307
In this section, these eleven values are established. It is unclear to us why
Ramanujan studied this particular function U,,.
Recall from A.6) that
S ¦- Sb, =
It follows that
A0.3)
(Ю.4)
Hence, from A0.1 Ml0.4),
l-b) ,
i+* г(
2b
4ЬA - b2) 1 - b2 8b2
1-/3
8
A0.5)
Equality A0.5) will now be utilized to calculate Un for eleven values of n. All
values for gn below can be found in the table in Section 2.
First, ?2=1. and so, by A0.5),
Second, since g6 = 2"l/4D + 2л/2)|/6, we find that
Thus, from A0.5),
= 5 (C + 2V2) + C - 2V2)) = 3.
Third, g]0 =
Thus, from A0.5),
+ l)/2, and so
Us = 5 ((9 + 4л/5) + (9 - 4>/5)) = 9.
Fourth, ям = 7^1 + л/2 + \Fl\Fi- \\ /2, and so
^j'i12 =11+ 8V2 ± (8
Thus, from A0.5),
?/7= П + )
Fifth, git = (ч/г + л/3I/3. Thus,
g±n = 49±20v/6,

308 Ramanujan's Notebooks, Part V
and so, from A0.5),
U9 = 49.
Sixth, g22 = vV2+ 1, and so
Hence, from A0.5),
UU=99.
Seventh, «да = B + V5)l/6C + л/ЮI/6. It follows that
g±n = (9 ± 4л/5)A9 ± бУТО).
Hence,
(/15 = 171 + 120V2 = 3E + 4V2J.
Eighth, «46 =
It follows that
I/23 = A8 + 13V2J + E + 3V2JG + 6V2) = 9A47 + 104^2)
= 9A7 + 12V2)C + 4V2) = 9A + V2LC + W5).
Ninth, g58 = V^5+ V29)/2- Thus,
and so
иг9 = 702 + 132 • 29 = 9801 = 992.
Tenth, g10 = у/(? + у/5)О + у/2)/2. It follows that
= (9 ±
Thus, from A0.5),
(/3S = 63B53 + 80л/Ш) = 63(8 V2 + 5л/5J.
Eleventh, gl42 = ^(9 + 5^ + УТ27 + 90V2)/2. So,
= 11026 + 725^2 ± F5 + 45V2)v/l27
34. Class Invariants 309
Thus, from A0.5),
?/7i = A026 + 725 V2J + F5 + 45V2JA27 + 90 *Jl)
= 9D67539 + 330600V2)
= 9C363 + 2378 *Л) E7 + 58^2)
= 9A + V2)IOBV2 + 1JFV2 + 1).
11. The Modular y-invariant
Except for four entries discussed in Section 13 of Chapter 33, the last two pages,
392-393, in Ramanujan's second notebook are devoted to values of the modular j-
invariant. Recall (K. Chandrasekharan [ 1, p. 81 ], Cox [ 1, p. 224]) that the invariants
У(т) and j(x), for т е Н, are denned by
and j(t) = 17287(r),
where, forq = expB7rir),
m.n=—00
(m.n)^(O.O)
m.n=—00
(m.nM4(O.O)
27
and
A1.1)
Here M(q) and N(q)ai:e the Eisenstein series defined at the beginning of Section
4 of Chapter 33. (See also the beginning of Section 7 in this chapter.) Furthermore,
the function уг(т) is defined by (Cox [1, p. 249])
Ыт) = /W), (П.2)
where that branch which is real when г is purely imaginary is chosen.
At the top of page 392, which inexplicably is printed upside down, Ramanujan
defines J := Jn and и := un by
Л =
8DаяA -«„
and
A1.3)
where n is a natural number. (To avoid a conflict of notation later, we have replaced
Ramanujan's tn by и„.) For 15 values of и, я = 3 (mod 4), Ramanujan indicates
the corresponding values for /„, although not all values are explicitly given. In

310 Ramanujan's Notebooks, Part V
each case, the value had been given in the literature or can be readily deduced
from results in the literature. Now, from A.6) and A1.3), we easily find that
Л = |С«A-4С,724). A1.4)
We now identify Jn with y2. From Cox's text [1, p. 257, Theorem 12.17], for
q — ехрBлч'г),
Setting т - C + yf^n)/!, we deduce from A1.5), F.1), and A.3) that
A1.5)
Hence, from A1.4) and A1.2),
i
) = 2-f (—Г" }
There are 13 cases when the class number of the order in an imaginary quadratic
field equals 1 (Cox [1, p. 260]). In such an instance, the value of the y'-invariant is
known to be a rational integer (Cox [ 1, p. 261 ]), and we first turn to Ramanujan's
seven values in these cases. Formula A1.3) can be used to calculate some values,
but in most instances the value of Gn is unavailable.
Entry 11.1. J, = 0.
Proof. From the table in Section 2, G3 = 21/l2. Thus, by A1.4),
y3 = i(G* - 4G^6) = f B2/3 - 4 • 2-4/3) = 0.
Entry 11.2. У27 = 5 • 3'/3.
Proof. From the table in Section 2, G27 = 2|/12B|/3-1Г1/3. Thus, from A1.4),
4 = i
o4
- I)"8) A - B1/3 - I)8K = 375,
with the help of Mathematica.
For the last five cases of degree 1, we simply use the relation A1.6) in conjunc-
conjunction with the table in Cox's book [1, p. 261].
Entry 11.3. -Л, = 1.
Entry 11.4. У19 = 3
Entry 11.5. Лз = 30.
34. Class Invariants 311
Entry 11.6. Ув7 = 165.
Entry 11.7. y,63 = 20,010.
There are a total of 29 cases when the degree of j (C + v^«)/2) equals 2 (W.
E. Berwick [1]). Ramanujan gives the values of Jn in six of these cases. We will
verify Ramanujan's values by simply referring to the tables in Berwick's paper [ I,
pp. 55, 57-59].
Entry 11.8.
Entry 11.9. J5l = 3 {уП + 4
Entry 11.10.
УТ7
Proof. In Berwick's table on page 58, we need the values of e (given on page 55)
and у (given at the top of page 57). Thus,
- 3 ¦ 5"«,9 + 4V5)
=3
Entry 11.11. J9, - |B27 + 63л/ТЗ).
Proof. Proceeding with the same tables as in the preceding proof, we find that
Entry 11.12. 799 = jB3 + 4Л/33J'/3G7 +
Proof. From the work of Berwick [1, pp. 55, 58],
11)(л/зЗ
= iB3 + 4>/33J/3G7 + 15-ЛЗ).
The value of У99 was not given by Ramanujan; he wrote "У99 = • ¦ ¦ ."

312 Ramanujan's Notebooks, Part V
Entry 11.13. Jm = fG85 + 35lv5).
Proof. From pages 58, 55, and 57 in Berwick's paper [1],
/.,5 =
Ramanujan misrecorded the value of Уп? as
Lastly, Ramanujan discusses two values of Jn when j (C + л/3й)/2) has de-
degree 3.
Entry 11.14. Let un be defined by A1.3) and define p > 0 by
A - P%?
9 = g ¦
Then
-1 = 0. A1.7)
Furthermore, к',3 satisfies an irreducible cubic polynomial over <Q>.
Proof. We turn to the work of A. G. Greenhill [1 ] who sets (with notation altered
so as not to conflict with that of Ramanujan)
AL8)
Comparing A1.8) and A1.3), we see that an = 12873.Thus,fromA1.3),wn = an.
Then Greenhill sets (p. 311)
aS9=
A -
and shows that p is a root of A1.7). Greenhill [1] claims that a := a59 satisfies a
cubic equation but does not give it. However, in a subsequent paper [2, p. 404],
Greenhill gives the equation
a - 392 2|/3а2/3 + 1072 • 4!/V/3 - 2816 = 0. A1.9)
Entry 11.15. Let un be defined by A1.3). Define s > 0 by
A - 256л24K
and set
I-2s- 2s2 - 2.?3
2s*
34. Class Invariants 313
Then
+ 4/32 + 20 - 5 = 0.
1/3
Furthermore, w8j satisfies an irreducible cubic polynomial over Q.
Proof. Ramanujan's claim, which is incomplete as stated on page 392, is a quota-
quotation from GreenhilPs paper [1, pp. 312, 313]. In fact, except for ияз, the notation
is the same. Greenhill [1] indicates thatal/3 := u1^ satisfies a cubic polynomial,
which he gives in his second paper [2, p. 405], namely,
a - 1740 • 21/3a2/3 + 2000 • 41/3al/3 - 32000 = 0,
which he claims is due to Russell.
A1.10)
On page 393, Ramanujan considers various polynomials satisfied by Jn. The
,work is divided into three parts. In the first pan, Ramanujan gives the following
table.
Entry 11.16.
n
11
19
43
67
163
Л
1
3
30
165
20010
8У„ + 3
11
27
243 = 27 • 32
1323 = 27-72
160,083 = 27. 772
64У2 - 24У,, + 9
49 = 72
513 =27- 19
56, 889 = 27 43 • 72
1,738, 449 = 27-312-67
25,625, 126, 169 = 27 • 163 • 24132
Note that
Table 11.1
(87,, + 3J = F4У,2 - 24У,, + 9) + 72У„.
Observe that, except for и = 11, 64УП2 - 24./,, + 9 contains n as a factor. Also
note that, except for n = 11, both 87„ + 3 and 64У„2 - 24У„ + 9 contain 27 as a
factor. The factorizations in the last column are presented as Ramanujan recorded
them. Perhaps he failed to observe that 2413 = 19 127.
In the second part of the page, Ramanujan recorded the following remarkable
theorem.
Entry 11.17. For q = exp(-7rVw), define
A1.11)

314 Ramanujan s Notebooks, Part V
Then
/ \'/0
tn = Ыб4^ - lAJn + 9 - A6Л - 3)J .
i/6
A1.12)
Entry 11.18. For the values of n given below, we have the following table of
polynomials p,,(f) satisfied by tn:
n
11
35
59
83
107
Pn(t)
t- 1
tl + t-\
fJ + 2r - 1
t' + 2^ + 21-1
t* -2t2 +4t - 1
Table 11.2
The simplicity of these polynomials is remarkable, since the corresponding
well-known polynomials of the same degrees satisfied by J» are considerably more
complicated, especially in the latter three instances. If и is squarefree, n = 1 l(mod
24), and the class number of the Hilbert class field K0) is odd, then tn and Jn satisfy
irreducible polynomials of the same degree; for a proof, see a paper by the author
and Chan [4].
The form of Entry 11.17 suggests hitherto unknown connections between the j-
invariant and Ramanujan's cubic theory of elliptic functions to alternative bases
developed in Chapter 33. We defer a proof of Entry 11.17, as a similar result
commences the third part of page 393. A proof of the latter claim also depends
upon Ramanujan's cubic theory.
Proof of Entry 11.18. Using Entry 11.3 and Table 11.1, we find that
(,, =B-7-13I/6= 1,
as desired.
Second, from Entry 11.8,
/35 = 2
*•
¦№)'-))
= B/7349 + 3276^5 - 117 - 56^5 j
'/6
34. Class Invariants 315
Now,
73492 - 5 • 32762 = 5892.
Thus, by the denesting theorem (9.5),
/7349 + 3276^ = J
/7349 + 589
7349 - 589
2 " V 2
= V3969 + V3380 = 63 + 26^5.
Hence,
/35 = B (бЗ + 26У5") - 117 - ЬЬ-Щ'* = (9 - 4V5)'/6 =
V5-1
Hence, /35 is a root of t2 + t - 1, and the second result is established.
For n = 59, recall that ai9 = -y}/256 is a root of A1.9), where y2 =
Y2 (C + V-59)/2) . Thus, y2 satisfies
y% + 3136y22 + 686О8У2 + 720896 = 0. A1.13)
Set* :— t6 := f^andi := yn.FromA1.12),weseethatxisarootofaquadratic
polynomialx2+bx+c, withb = 2A67-3). Sinceb2-4c - 16F472-24У+9),
we easily calculate that с = -27. Thus,
x2 + 2A67 -3)*-27 = 0. A1.14)
This suggests that we make the substitution
y2 = x - 6 - 27x"' A1.15)
in A1.13). Upon turning to Mathematica, we find that
x6 + 3118л5 + 31003*4 + 25355jc3 - 837081*2 + 2273022л: - 19683
= (*3 + 13jc2 + 115x - l)(x3 + 31O5x2 - 9477x + 19683) = 0.
Numerically checking the roots, we see that x is a root of the first factor above,
i.e..
We use Mathematica to factor this polynomial and find that
t18 + 13r12 + 115i6 - 1 = (r3 + 2t - \)(P + Ъ + 1)
x (f6 - 2r4 + 2r3 + 4r2 - It + \)(tb - 2t* - 2r3 + 4/2 + 2t + 1).
Again, numerically calculating the roots, we find that
t3 + 2/ - 1 = 0,
as claimed by Ramanujan.
For n = 83, recall that а8з = -y23/256 is a root of A1.10), where y2 =
y2 (C + V-83)/2). Thus, y2 satisfies
y\ + 13920y22 + 128000/2 + 8192000 = 0.

316 Ramanujan's Notebooks, Part V
As in the last example, set x = /6 and уг = * - 6 - 27л "'. Using Mathematica,
we find that
x6 + 13902л:5 + 39013л4 + 7174196л:3 + 1053351л2 + 10134558л - 19683
= (л3 - Зл2 + 515л - 1)(л3 + 139O5jc2 + 2187* + 19683) = 0.
A numerical examination of the roots shows that t is a root of the first factor, i.e.,
2/5 + 2/4 + 6/3 + 6/2 - It
+2t_
Numerically inspecting the roots, we conclude that
/3 + 2/2 + 2/ - 1 = 0,
which is what Ramanujan claimed.
For n = 107, Greenhill [2, p. 405] proved that a := аЮу satisfies
a - 79 ¦ 80 • 21/3a2/3 - 69 • 800 • 41/3al/3 - 17 ¦ 16000 = 0.
As in the previous two examples, setting a = -y23/256, we find that
y\ + 50560y22 - 3532800/2 + 69632000 = 0.
As before, set x = t6 and y2 = x - 6 - 27x"'. Using Mathematica, we find that
хь + 50542л-5 - 4139493л4 + 89919476л3
+ 111766311л2 + 36845118л - 19683
= (л3 - 83л2 + 1875л - 1)(л3 + 50625л2 + 60507л + 19683) = 0.
A numerical examination of the roots indicates that the first factor equals 0, i.e.,
,'8 _ 83,12
1875,* _
41- 1
2f2
At + \)(t6 + 2t5 + 6/3 + 14f2 + 4/
= (Z3 - 2/
x (,A _ 2/5 - 6/3 + 14f2 - 4/ + 1) = 0.
Checking the roots of each polynomial, we conclude that
t3 - 2/2 4- 4/ - 1 = 0,
which is what Ramanujan asserted.
In the third portion of page 393, Ramanujan first sets
A1.16)
and then gives the following table of values for tn.
34. Class Invariants 317
Entry 11.19.
n
19
43
67
91
115
163
7 + 2v/l3
13 + 6n/5
tn
1
3
7
(r2-
o2 -
77
14/ -
26(-
3
11
= 0)
= 0)
Table 11.3
Proof. The values of <„ for n — 19, 43, and 67 follow trivially from Entries
11.4-11.6, respectively.
From Entry 11.11,
252^13.
Now
9092 - 13 ¦ 2522 = 272.
Thus, by the denesting equality (9.5),
/9i = \ (a/468 + v^) = i (бл/13 + 2l) = 7 +
which proves Ramanujan's assertion.
Next, by Entry 11.13 and another application of the denesting theorem (9.5),
1404л/5 = |C9
= 13
which establishes the desired result.
Lastly, the value for /^3 follows very readily from Entry 11.7.
Return to the definition of/„ given in A1.11) and set H(n) - 27/~12. For the
four values 19, 43, 67, and 163 in Entry 11.19, H. H. Chan found that
//A9) = 151 +20V57,
Я D3) = 16855 + 1484л/129,
Я F7) = 515095 + 36332 %/20Т,
and
Я A63) = 7592629975 + 343350596^/489.
Amazingly, these numbers are fundamental units of their respective real quadratic
fields. Essentially, these same observations were also made by H. M. Stark [1].

318 Ramanujan's Notebooks, Part V
Entry 11.20. Let tn be defined by A1.16). Then, as n tends to oo,
Proof. From the definition A.3), as n tends to oo,
Gn ~ 2-u*en^"'2\\ +e
Hence, from A1.4),
Jn ~ |2" V^/3(l + e~"^f (l - 4 • 26e
which implies that
Л = ±е'*/3 + О
Thus,
О {*-**
О
from which Ramanujan's asymptotic formula for /„ follows.
Entry 11.21. Letq — ех.р(—тг*/п) and put
R := Rn :=
Then
3
1/36
/(g)
A1.17)
A1.18)
We now use Ramanujan's cubic theory from Chapter 33 to prove both Entries
11.17 and 11.21.
Proof of Entry 11.17. We shall prove a stronger version of Entry 11.17 by re-
removing the condition q = ехр(-Лу/п), i.e., we interpret A1.12) as a ^-identity
in the following way. The more general definition of t is clear from A1.11). To
extend the definition of У„, note that, fromA.3), G,, = 2~]/tiq-l/24f(q)/f(-<i2),
where q — ехр(-я*/п). Removing this stipulation on q, in view of A1.4), it is
natural to define J by
J - .,-./3
34. Class Invariants 319
To prove A1.12), recall that in proving Entry 11.18 we showed that it suffices to
establish A1.14). Replacing 4 by -q3 in A1.14) and using A1.11) and A1.19),
we find that it suffices to prove that
Setting
h{q) =
/12(-g3)
A1.20)
A1.21)
and using A1.5) with q replaced by q*, we find that A1.20) assumes the more
palatable form
or
Set
21h-\q) + 2*-'fo) (|wCt) + 3) - 1 = 0,
У2Cт) = h{q) - 6 -
A1.22)
s =
fJT^Tv A1-23)
Then upon cubing Entry l(iv) of Chapter 20 of Ramanujan's second notebook
with q replaced by <?3 (Part III [3, p. 345]), we find that
Hq)=s+9+ —.
A1.24)
Substituting A1.24) into A1.22) and utilizing elementary algebra, we now find
mat we are required to prove that
A1.25)
+ 9)E + 3)(s2 + 27)
s(s2 + 9s + 27)
We now recall some basic facts from Ramanujan's cubic theory. From A.7) of
Chapter 33,
i 2*
q -=qi :=exp - —
0 <a < 1.
Returning to A1.1), if we employ Theorems 4.2 and 4.3 of Chapter 33, we deduce
that
+ 8aK
that
Дт) = 27
«A -aK'
A1.26)

320 Ramanujan's Notebooks, Part V
By Lemma 2.9 of Chapter 33, a has the representation
a:=a(q) = L^l, A1.27)
where a(q) and c(q) are defined by B.2) and B.4), respectively, in Chapter 33.
From Lemma 5.1 of Chapter 33,
c(q) =
A1.28)
П-q) '
By combining Entry l(v) of Chapter 20 of Ramanujan's second notebook (Part III
[3, p. 346]) with B.6) of Chapter 33, we find that
A1.29)
Hence, by A1.27H П 29) and A1.23),
З5
or, with the argument <?3 suppressed,
5 = 3-
A1.30)
0,./.-
Upon substituting A1.30) into A1.25) and simplifying, we find that we are required
to prove that
,(l+2a)(l-2« + 4a2/3) 1 + 8a
ЫЗТ) = 3«-/»(l-«-/3)(l+,./» + ,V») = 3^T7^-T- A1.31)
But, by A1.2) and A1.26), since the argument of a is 43,
1 +8a
nOr) = /лзт) = 3-
Thus, A1.31) has been shown, and so the proof of Entry 11.17 is complete.
Proof of Entry 11.21. By A1.17),
3V3 f4qiP)
x :=
From A1.18), we want to show that x is a root of a certain quadratic polynomial
x2 + bx + c, where b - -2-JbJ + 3 and
b2 - 4c = 4 By/64J2 - 24/ + 9 - 87 + )
A simple calculation shows that
с = 16J - 3 -
}
by Entry 11.17. Hence, we want to prove that
Cancelling /V/3)/D1/6/6(<?)) >n A1.32), we find that
34. Class Invariants 321
1.32)
A1.33)
By A1.19),
(П.34)
Squaring both sides of A1.33), substituting A1.34) into the left side, replacing #
by — g3, and simplifying slightly, we find that
- 12
A1.35)
By using A1.5), A1.21), and A1.23), we can rewrite A1.35) in the form
у2(Зг)-.2=^- + ^-_^_
h(q) s4{q) h(q)'
or, by A1.24),
У2Cт)
= (s2 + ^ - 54)
V s2 )
1
s + 9 + 27/s
+ 12
s(s2
A1.36)
after elementary simplification. However, A1.36) is the same as A1.25), estab-
established in the proof of Entry 11.17. This completes the proof of Entry 11.21.
It has been conjectured that Ramanujan had seen Greenhill's book [3] on elliptic
functions. For example, see a letter by K. Ananda Rau and the following com-
commentary in the book by R. A. Raiikin and the author [1, pp. 289,290]. On pages
327-329, Greenhill [3] briefly summarizes much of Russell's work on modular
equations. This is further evidence that Ramanujan had seen both of Russell's
papers [1J, [2], and also both of Greenhill's papers [1], [2], since all four of these
papers appear in volumes 19 and 21 of the Proceedings of the London Mathemat-
Mathematical Society.
We have, for convenience, quoted Berwick's paper [1], published in 1923, for
some of Ramanujan's results. However, the quoted results were originally discov-
discovered much earlier, as Berwick cites Chapter 19 of the third volume of Weber's
Lehrbuch der Algebra [2]. It is interesting that in 1916 Ramanujan wrote Berwick

322 Ramanujan's Notebooks, Part V
inquiring about references for singular moduli and modular equations. Ramanu-
jm's letter is evidently not extant, but Berwick's reply has been preserved (Berndt
and Rankin [1 pp 138-141 ]). Among the several references that Berwick provides
are the four aforementioned papers by Greenhill and Russell.
There is evidence that several entries in the third notebook were recorded while
Ramanujan was at Cambridge. From the remarks above, we conjecture that the
entries on pages 392 and 393, which are the last two pages in the third notebook,
were also recorded during Ramanujan's stay in England.
Following a suggestion by A. O. L. Alkin, Birch [1. p. 291] briefly cons.dered
the function g defined by
+ 6.
g6 - 27g-6 =
This is exactly the function studied by Ramanujan in Entry 11.17. In particular, see
A115) Atkin's suggestion was motivated by certain relations between modular
forms that would have been unknown to Ramanujan
For further recent work on invariants, readers might examine papers by В. Н.
Gro"«dS. B.Zagier[1], E. Kaltofen and N. Yui [1], N. Yui and D. Zagier [1].
and I. Chen and N. Yui [1).
35
Values of Theta-Functions
0. Introduction
For the convenience of the reader we briefly review here some definitions from
earlier chapters.
After Ramanujan, define the theta-functions <p(q) and f(q) by
9(Я) ¦= 2_, «" =
Я = -00
@.1)
and
@.2)
n=0
where the infinite product representations arise from the Jacobi triple product
identity (Part HI [3, pp. 35-37]). The Dedekindeta-function rj(z) and Ramanujan's
function f(-q) are defined by
@.3)
@.4)
= q>/1\q^)oo=:qw24f(-q), q=e2"u, Imг > 0.
It is well known that (Part III [3, p. 102 (with a misprint corrected)])
71/2
dO
2
=:-K(k),
^o v/l -fc2sin20 n
where 2^*1 denotes the ordinary or Gaussian hypergeometric function; k,0 < к <
1, is the modulus; К is the complete elliptic integral of the first kind; and
q =exp(-nK'/K),
@.5)
where K' = K(k'), and k! = Vl - k2 is the complementary modulus. Thus,
an evaluation of any one of the functions (p, 2^1. or К yields an evaluation of
the other two functions. However, such evaluations may not be very explicit. For
example, if К (k) is known for a certain value of k, it may be difficult or impossible
to determine explicitly AT', and so q cannot be explicitly determined. Conversely, it
may be possible to evaluate <p(q) for a certain value of q, but it may be impossible

324 Ramanujan's Notebooks, Part V
to determine the corresponding value of k. (Recall that it — *J\ — (pA(—q)/<p*(q)
(Part III [3, p. 102]).)
In the literature more attention has been devoted to determining jF\ and K(k).
In particular, using the Selberg-Chowla formula, I. J. Zucker [ 1 ] evaluated К
when K'/K = -/к, where к is a positive integer such that 1 < к < 16, к Ф 14.
See also papers of G. S. Joyce and Zucker [1], J. M. Borwein and Zucker [1], and
J. G. Huard, P. Kaplan, and K. S. Williams [1].
In his second notebook, Ramanujan recorded the values of <p(e~"), <p(e~"^),
<р(е~2л), and^(e~5'T)(PartIII[3,pp. 103,104]); no other values of^> are recorded.
The first three values are classical (Whittaker and Watson [1, p. 525]), while the
value of <p(e~5") is new. However, in his first notebook, Ramanujan recorded many
values of <p, as well as values of ф and /.
Ramanujan's modular equations are central to most of our proofs of his evalu-
evaluations. Recall that in the theory of modular equations the multiplier m is defined
by
m \ —
; \;a) = <p2(q)
@.6)
by @.4). This fact is the key to our evaluations below.
For some evaluations of <p in Section 2, we need to recall a few basic facts about
class invariants from Chapter 34. Define, for \q\ < 1,
The class invariant Gn is defined by
@.7)
where q = ехр(-7г,Л»)- Since x(?) = 2'/6{a(l - a)/q}-'/2* (Part III [3, p.
124]), it follows from @.7) that
С„ = {4аA-а)Г1/24.
If p has degree n over a, it follows from @.8) that
@.8)
@.9)
where now q = ехр(-л).
This chapter is divided into three parts. In Section 1, elementary values of
<p(q), $(q), f(-q), and x(q) arc given. These are easy to derive and probably
most of these values can be found somewhere in the classical literature. Section 2
is devoted to values of <p(e~"n), for certain positive integers n, and these values
are mostly new. Section 3 focuses on values for a certain remarkable product of
the functions <p and ij/.
35. Values of Theta-Functions 325
1. Elementary Values
The values given in Entries 1 and 2 are easy consequences of the "catalogue" of
evaluations given by Ramanujan in Chapter 17 (Part III [3, pp. 122-124]). Since
the more general evaluations are given in the second notebook and not in the first,
it appears that Ramanujan first derived the special values and then observed that
his arguments held more generally. Since proofs of the more general results have
already been given in Part III, we follow the reverse tack.
Entry 1 (p. 248). Let
Then
(i)
a =
ГC/4)
A.1)
<p(e-")=a,
(iii)
(iv)
(v)
(vi)
(vii)
(viii)
(ix)
(x)
(xi)
(xii)
(xiii)
(xiv)
(xv)
(xvi)
cpi-e") = a2-1/4(V2 - l)l/2,
= а2A - 2~
= а2-7/16A +
The evaluations in (xv) and (xvi) give an extra factor of 2l/16 in the denominators
that Ramanujan does not have. In our proofs below, all references are to Entries
10 and 11 of Chapter 17 (Part III [3, pp. 122, 123]). Note that the evaluations arise
from taking x = i in these formulas, and so q = exp(-7r).
Proof. Part (i) is the same as Example (i) in Section 6 of Chapter 17 (Part III [3
p. 103]).
Part (ii) follows from Entry 10(ii).

326 Ramanujan's Notebooks, Pan V
Part (iii) is the same as Example (iii) in Section 6 of Chapter 17 (Part III [3, p.
104]).
Part (iv) follows from Entry 10(iii).
Use Entry 10(v) to prove part (v).
From A0.1) of Chapter 17 (Part III [3, p. 123]),
ipi-e-*") = у/<р(е-2л)<р(-е-2»).
Now use parts (iii) and (iv) to complete the proof of (vi).
Parts (vii)-(x) follow from Entries 10(vi)-(ix), respectively.
Parts (xi)-(xvi) follow from Entries 1 l(i), (iii), (iv), (v), (vi), and (viii), respec-
respectively.
Entry 2 (p. 250). Let a be given by A.1). Then
(ii) f(-e~2") = a2~l/ e" ,
(iii) /(-e") = a2~7/V/6,
(iv)
(v)
(vi)
(vii)
(viii)
(ix)
(-«Г8") =a2-21/'6(V2-
X(-e-2*) =
(-<Г4*) = 25/I6B
Proof. We shall make several references to Entry 12 of Chapter 17 (Part III [3, p.
124]).
Parts (i)-(iii) follow readily from Entries 12(ii>-(iv), respectively, and Entry
l(i) above.
From Part III [3, p. 124, eq. A2.3)],
If we now employ Entries l(vi), (xiii) above, we readily deduce (iv).
Parts (v) and (vi) are easy consequences of Entries 12(vi), (vii), respectively.
From Part III [3, p. 124, eq. A2.4)],
Xl ; f(-e-4")
If we now employ Entry l(vi) and Entry 2(iii), we easily complete the proof of
(vii).
Part (viii) follows readily from Entry 12(v).
35. Values of Thela-Functions 327
From Entry 24(i) of Chapter 16 (Part III [3, p. 39]),
Substituting the values from Entries I (iii). (iv) and Entry 2(vi), we readily deduce
(ix).
By repeated applications of Entries 10-12 of Chapter 17 along with identities
from Entry 24 of Chapter 16, it is clear that we can explicitly determine F(±e~2""),
where n is any integer and F is any of the functions <p, ф, /, or x •
2. Nonelementary Values of <p(e "*)
Ramanujan recorded the values of (p{e~"n) when n — 3,5,9, 45, and 7 in his
first notebook. We give here the proofs that appeared in our paper with Chan
[2]. These proofs are very natural and employ some of Ramanujan's modular
equations and class invariants. As corollaries, we are able to obtain three new
explicit determinations of 2Ft. We also explicitly determine a(e~2n), where a(q)
is the Borweins' cubic theta-function studied in Chapter 33; no other value of a (q)
had been previously determined. Other new values for <р(е~пл) are also obtained
in our paper with Chan [2]. Different proofs for the cases n — 3, 5 have been
given elsewhere in the literature; see Zucker's paper [1] for и = 3, and the paper
of Joyce and Zucker [1] and our book [3, p. 210] for n = 5.
Ramanujan recorded most of his values for tp(e~nrr) in terms of <p(e~"), but in
view of A.1), (pie'"") is therefore determined explicitly.
Entry 3 (p. 285).
Entry 4 (p. 284).
Corollary 1. //
then
P =

328 Ramanujan's Notebooks, Part V
Corollary 2.
Corollary 3. Let a(q) be defined by B.2) in Chapter 33. Then
а(е2л) 1
Entry 5 (p. 287).
Corollary 4. //
t -
BG5-1)) -1
BG1-1)) +2
then
Entry 6 (p. 312).
;64f-
1 - f3 \ _ (l+
1+8*V~
Vie-')
Entry 7 (p. 297).
14
B8)l
We now provide proofs for the entries and corollaries given above.
Proof of Entry 3. If P has degree 5 and m is the multiplier for degree 5, then
from Chapter 19 of Ramanujan's second notebook (Part III [3, Entry I3(xii), pp.
281,282])
35. Values of Thcta-Functions 329
and
1/4
.1-0/ V0O-0),
Set a = i, so that, by @.5), q = «r*. From C.1) we find that
B?) + BA - 0)I/4 = m + D/3A - /3))l/4 ,
and, from C.2) and C.3), we find that
_5 _ B(l-l))l/4 + B/3I/4~ 1
m ~ D0A - 0)I/4
()I/4 - 1 m + G«6 - 1
C.2)
C.3)
D0A -
G
-6
25
C.4)
by @.9), with n = 5. From Lemma 7.1 of Chapter 32, or the table in Section 2 of
Chapter 34,
I+n/5
C.5)
G := G2s =
Hence, from C.4), since G3 = 2 + 75,
Сзт _ 5 = з _ , =
C3m
from which we deduce that G*m = 5, or m = 5G5 - 2). Entry 3 now follows
from @.6).
First Proof of Entry 4. Our first proof is similar to that for Entry 3.
From Entry 5(vii) of Chapter 19 of Ramanujan's second notebook (Part III [3,
p. 230J),
jLgV/2 /in-m'/2
-a) \a(\-a))
and
D.1)
D.2)
where P has degree 3. Setting a = \ in D.1) and D.2) and eliminating the terms
af})in + BA - /9))l/2, we deduce that
by @.9) with n = 3, where, from the table in Section 2 of Chapter 34,
D.3)
D.4)

330 Ramanujan's Notebooks, Part V
Rewriting D.3) and employing D.4), we arrive at
(G'mJ -
= Gb - G = 2V3.
Hence, (G3mJ = Зч/З, or, by D.4), m2 = 6V5 - 9. Appealing to @.6), we
complete the first proof of Entry 4.
Second Proof of Entry 4. From the Borwein brothers' book [1, p. 145],
where n is any positive rational number. We provide here a proof somewhat dif-
different from that in the Borweins' book [1].
From Part III [3, p. 347],
D.6)
Recall the transformation formula for <p (Part III [3, p. 43]). If a, b > 0 with
ab = 7Г, then
) = </b<p(,e-h2). D.7)
Using D.7) twice, we easily find that, for Re(z) > 0,
Recall also the transformation formula for x (Part III [3, p. 43]). If a, b > 0 with
ab = л2, then
е<4х(е-а)=еь>24Х(е-ь). D.9)
Hence, from D.9), we deduce that
ХЪ(е~Я1) еЪлг/глхЪ(.е)
Utilizing D.8) and D.10) in D.6), we deduce that
D.10)
D.П)
where qx = е-0. И we now set q\ = e""" in D.11). we deduce D.5).
Setting и = 1 in D.5), employing D.4), and noting that G, = 1, we find that
= 3 + 2-Л.
The desired formula now follows by elementary algebra.
35. Values of Theta-Functions 331
Third Proof of Entry 4. From Entry 6a of Chapter 19 in Ramanujan's second
notebook (Part III [3, p. 238]),
<p2(q)
= l+2p,
D.12)
where 0 < p < 1. Seta - 5, so that, by @.6) and D.12),
Hence,
2ръ + 6p2 + 5p - \ = 0.
To solve this quartic equation, we use Ferrari's method, as found, for example, in
H. S. Hall and S. R. Knight's text [1, pp. 483, 484]. Thus, adding (ap + bJ to
both sides above, writing the left side as (p2 + p + kJ, and equating coefficients
of like powers of p, we are led to the equations
Hence,
k=\+ab.
(k - \J = a2b2 = Bk - 5)(k2
Obviously, it = 5 is the real root. It follows that a = 0 and b = \^/b. Hence,
Since p > 0,
P =
-1 + v/бл/З - 9
Using D.13) in D.12), we complete the proof.
Proof of Corollary 1. From @.4) and A.1),
Hence, the desired result follows from D.12) and D.13).
Proof of Corollary 2. By Theorem 5.6 in Chapter 33, if
a_ 21p2O+pJ
3
a =¦¦ p
1 +2p
and
4A
D.13)
D.14)
D.15)

332 Ramanujan's Notebooks, Part V
where 0 < p < 1, then
A + p + P2) iF^, i; 1;«) = vTT2p 2Ж5, §; 1; P). D.16)
If p is given by D.13), then, by D.15) and a straightforward calculation, we find
that
„ Зч/3-5
Another elementary calculation shows that
l+p + p2
D.17)
D.18)
%/бл/З - 9V1 + 2p A2)'/8v/V3-1
Using Corollary 1, D.17), and D.18) in D.16), we readily complete the proof.
Proof of Corollary 3. From Theorem 2.12 of Chapter 33, if
V V3 2^, E, 5;1,jc) /
then
e(«)= 2F,(i, |;l;x). D.20)
We also need Ramanujan's cubic transformation, Corollary 2.4 of Chapter 33,
Now let* = (-/3-0/2. Then a simple calculation shows that A
(\/3 - l)/2. Using these values in D.21), we find that
iFi
D.22)
Substituting D.22) into D.19), we see that q = e~2". Using this value of <? in
D.20) and noting that x3 = C^3 - 5)/4, we then find that
by Corollary 2 and A.1). This completes the proof.
Another short proof of Corollary 3 can be effected by employing a formula of
Ramanujan for a(q2) (Pan III [3, p. 460]) and Entry 4.
35. Values of Theta-Functions 333
First Proof of Entry 5. If fi and у have degrees 3 and 9, respectively, over a,
then from Entries 3(vi), (x) of Chapter 20 in Ramanujan's second notebook (Part
III [3, p. 352])
- fi))
l/24
and
= y/mm ,
E.1)
E.2)
respectively, where m is the multiplier connecting a and f) and m' is the multiplier
relating fi and y. Setting a — \ in E.1) and E.2), we find that, respectively,
BA-y))l/8 + BK)l/8 = v/2G9-1 E.3)
and
¦ X)I/8 - GK-,3 = ^S. E.4)
From the table in Section 2 of Chapter 34,
1/3
E.5)
which was first proved in print by Watson [12]. Thus, from E.3), E.4), D.4), and
E.5), we conclude that
Vie'")
(р(е-9л)
2-
E.6)
and so the proof is complete.
Another proof can be constructed by combining E.2) and the "reciprocal" mod-
modular equation (Part III [3, p. 352, Entry 3(xi)])
«A -
yd
-a)\"* =
-У))
when a = \. However, the resulting radicals are more difficult to simplify.

334 Ramanujan's Notebooks, Part V
Second Proof of Entry 5. From the Borweins' book [1, p. 145], for any positive
integer n,
= 1+V5??.
E.7)
The proof of E.7) is very similar to that for D.5). We begin with Ramanujan's
identity (Part III C, p. 345, Entry l(ii)])
—L = \+2q">-^- E.8)
<p(q*) x3(<?3)
and set q - e~"z. After applying the transformation formulas D.7) and D.9) to
E.8), we obtain the identity
1 л —
E.9)
where <?i = e "/г. If we now set q\ = e 3л1Г in E.9), we easily deduce E.7).
Setting n = 1 in E.7), we deduce that
3y(* 7 = 1 + V2G9 = 1 + BA + 73))'/3,
by D.4). This completes the second proof.
Third Proof of Entry 5. From Part III [3, p. 354, eqs. C.10), C.11)],
Setting q = e~" and comparing E.10) with Entry 5, we see that it remains to
show that
1+2/ = F.
1 + B(л/3 + I))'/3
But from E.10) and Entry 4, we know that
E.11)
After some elementary algebra, we find that
I'2 + 5' + i?0 -
It is easily checked that / = -i is a root, and so upon dividing by t + j, we
deduce that
,ч + F - 3y/b)t2 - f A - y/b)t + i(l - y/bf = 0.
This equation has two complex roots and a real root given by
t =
E.12)
35. Values of Thcta-Functions 335
a fact easily verified via Mathematica. By elementary algebra, similar to that used
in E.6), it may be verified that E.11) and E.12) are equivalent, and so the third
proof is complete.
Proof of Corollary 4. From Entry 41(iii) and D1.3) in Chapter 25 of Part IV [4,
pp. 193,194],
Set a = j and use D.14), E.11), and E.12) to complete the proof.
Proof of Entry 6. Setting n — 5 in E.7), we find that
Now from the table in Section 2 of Chapter 34,
G225 =
B
F.1)
F.2)
which was first proved in print by Watson [7]. Hence, by F.1), F.2), C.5), and
Entry 3,
Vie-') (pie-**) V(e~')
x B + V3I'3 (У4 + VT5 + A5)l/4) + Л
3C +
F.3)
By (9.5) of Chapter 34,
Using this in F.3), we find that
1
= (B + V3) G5 +
+10v v
+ 3 +

336 Ramanujan's Notebooks, Part V
and so the proof of Entry 6 is complete.
The following proof of Entry 7 due to H. H. Chan supplants the more lengthy
proof given in the author's paper with Chan [2].
Proof of Entry 7. We employ the two modular equations of degree 7,
and
49
G.1)
G2)
recorded as Entries 19(i), (v), respectively, in Ramanujan's second notebook (Part
III [3, p. 314]). Setting a = 5 in G.1) and using @.9), we find that
G.3)
Next, set a = \ in G.2), use @.9) again, and substitute G.3) into G.2). Accord-
Accordingly,
-l\-Gll- 8G* . G.4)
Since, by (9.5) of Chapter 34,
from the table in Section 2 of Chapter 34,
+ y/T/2,
±,
-Л
v 49 — л л 1
** 4 4 2
Substituting G.5) into G.4) and turning to Mathematica, we deduce that
or, upon taking the square root of each side,
I = B8)'/8 (г-Л ¦ 73/4 + 20 + 8V2 71/4
m 14 V
G.5)
G.6)
G.7)
Since
and
2V2 ¦ 73/4 + 8V2 ¦ 71/4 = 2^/A3 + ч/7)G + Зл/7)
20 + 4V7 = A3 + л/7) + G + Зл/7),
35. Values of Theta-Funclions 337
we find that G.7) can be put in the form given in Entry 7.
Values for ^(e3"), <р(е~11л), and <р(е~вУл) can be found in the paper by the
author and Chan [2].
It is easily noticed that all of the values for (р(е~"пI<р(е~л) that we have
calculated are algebraic. Indeed, this holds in general, and in a paper with Chan
and Zhang [4], a stronger theorem is proved: If n is an odd positive integer, then
y/2n<p(e~1"')/<p(e~") is an algebraic integer dividing 2л/п, while if и is an even
positive integer, then 2<Jn<p(e~111')/<p{e~n) is an algebraic integer dividing 4л/п.
3. A Remarkable Product of Theta-Functions
On page 338 of his first notebook Ramanujan defines
v " ;
where evidently m and n are positive integers. He then, on pages 338 and 339,
offers a list of 18 particular values, which we present in the following table.
Entry 8 (pp. 338,339).
m, n
m,n
3,3
1
7!
3,13
л/Тз /л/Тз-з
3,9
1
+ IJ
3,23
4л/3 /5 + 4V3
3,15
2-V3
3,71
f175+l00v/3 /173+100л/3
3,5
3-V5
5,9
B - V3J
3,7
2-V3
5,11
3,11
2л/3-л/ТТ
5,13

338 Ramanujan's Notebooks, Part V
3,19
3,31
3,59
2vA
B
102 ч/
9-5>/3
-V3K
5-23л/59
5,
5,
7
17
29
,9
(л/17 - 4J
(\/49 + 4Vl45-V48
f /5 + V2T /У
^V 8 V
+ 4V145)
21-5^
8 )
Our goal in this section is to establish each of these 18 values, and the tools
we shall use were entirely known to Ramanujan. The methods we develop can
be easily applied to determine further values of am „, and the reader is referred to
our paper with Chan and Zhang [4] for these. Upon examining the table, we see
that when (m,n) = 1, each value is a unit in some algebraic number field. In the
paper mentioned above [4] we prove that amM is a unit for large classes of pairs
m, n. We also establish in [4] some general formulas for evaluating am „, but we
use ideal class theory, with which Ramanujan was likely unfamiliar.
Ramanujan's class invariants are at the heart of our evaluations. Replacing и by
m/n in @.8) and @.9), we find that if q = ехр(—Лч/т/п) and fi has degree n
over a, then
Gm/n = f4a(l-a)}-1/24 and Gmn = D0A - 0)Г1/24. (8.2)
Next, we derive some formulas foram „ that will be useful in our calculations.
Set $ = ехр(-д Jmjn). Then, by @.1), @.2), and (8.1),
атя - "^''^(qn.^j:^:^^. q2)lo(qi. q7)lo-
Since (-<?; -q)^ = (q2; q2)oc(-q\ ?2)oo.
(8.3)
(-q; -Я),
(я;-
Using Euler's identity,
(8.4)
(8.5)
in (8.4) and then using (8.4) in (8.3) twice (once with q replaced by q"), we find
that
(8.6)
35. Values of Theta-Functions 339
by @.1) and @.7). Using (8.5) once again in (8.4), but with the reverse substitution,
and then using (8.4) in (8.3) twice, we also find that
— и-
~^~ )
G2mln
G2mn '
ат, -п-
(8.7)
by @.3) and @.7). From @.3) and @.7), it is easy to see that, if г = У=г,
г ч-1/4''\ 2 / /Я йч
r/(r)
and so, using (8.8) in (8.7), we find that
(8.9)
К. G. Ramanathan [5, p. 88] very briefly discussed the numbers am „ and derived
an equivalent formulation of (8.7) (with a misprint). He wrote that he planned to
return to am „ in a future paper, but unfortunately he died before he was able to do
so.
To help determine Gm/n, we shall employ three of Ramanujan's modular equa-
equations, Lemmas 4.3-4.5 of Chapter 34.
We also need three additional facts about <p(q) and invariants. The first is a
restatement of D.5), but with n replaced by yfn. If n is any positive rational
number.
(8.10)
Second, if/9 has degree 5 over a (Part III [3, p. 281, Entry 13(iv)]),
Vo?)
fni-p)
= 1+2
Gbn
/"¦5
Gn/5
(8.11)
by (8.2). Third, we need the standard mcta-transformation formula, D.7). In par-
particular, if a2 = л/s/r, then, from D.7),
(8.12)
Calculation ofamn
m = 3

340 Ramanujan's Notebooks, Part V
We first calculate a3.3. By (8.6),
(8.13)
The value for <р4(е~3д )/<р*(е~л) is given in Entry 4 of this chapter, and the values
I Г \ |/3
G\ = 1 and G<) = I —т=— I
are found in the table in Section 2 of Chapter 34. Thus, putting these values in
(8.13). we find that
_L_(V-
6ч/3-9\ч/3+1/ \/3
By (8.6) and (8.12).
a-s.n = n 4 ^„гщ
%ln
= 3-
(8.14)
since Gn = Gl/n (Ramanujan[31,I10,p.23]).Using(8.10)in(8.14), we find that
(8.15)
Provided that the invariants С„/з and C3n are known, (8.15) can be utilized to
compute several values of a3 „.
First, let n — 9. Then from the table in Chapter 34,
3 and
Putting these values in (8.15), we find that, upon simplification,
«3.9 =
Second, let n = 15. Then, from the table in Chapter 34,
1/4
and
.(8.16)
Using these values in (8.15), we find, after a straightforward, lengthy calculation,
that
2V3
But Уз + л/5 = V572 + лД72. Thus, after further simplification, we find that
2A + 2yfb + 2V5) _ 2-Уз
a'15 = 3(V5 + V3JB+V5) " 3 '
35. Values of Theta-Functions 341
by a direct calculation. (It is curious that the middle expression above is in Q(V3)
and not in 4}(л/3, л/5), as we would expect.)
In the remaining calculations, we need to calculate Gn/3 when n/3 is nonintegral.
First, we could use the methods developed in our paper with Chan and Zhang
[2], but these are nonelementary and depend upon the Kronecker limit formula.
Second, with the use of (8.2), we can employ Lemma 4.3 of Chapter 34 with
P = (Gn/3C3n)-3 and Q = (Gn/3/G3nN to deduce that
<817)
For specific values of n and G3n, we could solve (8.17) for G,,/3, but this procedure
is normally very laborious. It is better to "guess" the solution and then to verify
that our "guess" is indeed correct. That we have guessed the correct solution and
not another solution can be simply verified by numerically checking all the roots
of the polynomial equation.
Now set n = 5 in (8.15). From the table in Chapter 34,
(8.18)
From (8.17) we easily verify that
Using this value and (8.18) in (8.15), with n — 5, we easily find that
З-л/5
аз.5=
Next, let n = 7. From the table in Chapter 34,
Just as in the previous proof, we substitute this value in (8.17), when n = 7, and
verify that
G7/3 = 2-1/3(ч/7 - л/3I/4C + VV6.
Using the foregoing values in (8.15) when n — 7, we readily find that
дз.7 = 2 - >/3.
Next, let n = 11. From the table in Chapter 34,
Letting n = 11 in (8.17), we check that
Upon putting these values in (8.15) when n = 11, we readily deduce that

342 Ramanujan 's Notebooks, Part V
Let n = 19. From the table in Chapter 34,
1/6
Putting n = 19 in (8.17), we verify that
/зУ!9+1з\
Gl9/3 = \~7^J
'/6
B -
Employing these values in (8.15) with n = 19, we easily deduce that
аз.,9 = 2-УТ9 - 5>/3.
Let и = 31. From the table in Chapter 34,
Using (8.17) with n = 31, we check that
"
Using these values in (8.15) when я = 31, we easily find that
аз.31 = B - n/3K.
Let n - 59. From the table in Chapter 34,
By using (8.17) with n = 59, we find that
\ V2
3/2
With these values in (8.15) with n = 59, it is now an easy task to show that
аз.59 =
Let n = 39. From the table in Chapter 34,
From (8.17), we can verify that
35. Values of Theta-Funciions 343
Using these values in (8.15) with n — 39, after a calculation with the help of
Mathematica, we find that
a, ,3= X-(l + Зл/ТЗ - D +
_ / /s + УГз /УГз-з
where we established the last equality by applying the binomial theorem on the
right side.
Let n =23. From the table in Chapter 34,
1/2
Next,
1/2
Once again, we can use (8.17) to verify this value, but the calculation is quite
laborious. Alternatively, and preferably, we can use the method that we used in
our joint paper with Chan and Zhang [2] to calculate G^ to also calculate G23/3.
We use the values above in (8.15), when n — 23, and find that, after a very lengthy
calculation,
а3.2з = 6 + 4a/3 - i
= 6 + 4>/3 -
- 2
4V3
26)Cл/3 - 2)
Let n =71. From the table in Chapter 34,
C213 =
1/12
/ /21 + 12л/3
x W ; +
1/2

344 Ramanujan's Notebooks, Part V
By using the same method that we employed in the aforementioned paper [2),
which also utilized Lemma 4.3 of Chapter 34, or (8.17), we can deduce that
"¦¦"¦H—) НН
1/2
Putting these two values in (8.15) when n = 71, we find that, after a lengthy
calculation,
L
a3 7, = 2(87 + 5(K/3) - -L(95
V2
= 2(87 + 50V3) - 5^C62 + 209%/3)Eл/3 - 2)
l(XK/3
/l73+100V3~\
V 2 )
We close this subsection by proving two additional formulas for а3п. Theorem
8.1 below is an analogue of Corollary 8.4 (for m = 5) and Theorem 8.5 (for
m = 7), and provides an optional method for calculating a3 „. For calculational
purposes, the formulas for m = 5, 7 are more advantageous than the ones below
for m = 3.
Theorem 8.1. Ifn is a positive integer and
then
\ (Vn6
3
(8.19)
Proof. If p has degree 3 over a, then, from Part III [3, p. 230, Entry 5(vii)J,
т = I —
and
where m =
(8.21)
3). By combining (8.20) and (8.21), we deduce that
35. Values of Thela-Functions 345
Let 9 = exp(-TrVnT^). so that 93 = ехр(-7Гл/Зй). Then, by (8.2), Gn/3 =
D«A - a)}-|/24 and GJn = {4(9A - 0)Г1/24. Thus, (8.22) may be rewritten in
the form
(?)"(-¦(?)>*¦(?)"¦ «>
Rearranging (8.23) and using (8.6), we readily deduce (8.19).
By combining Theorem 8.4 with the modular equation (8.17), it is not difficult
to deduce the following corollary.
Corollary 8.2. Ifn is a positive integer and Un = GnpG^n, then
аз..-
First, let n = 9. By (8.6) and two applications of (8.10),
«5.9 = 9-
= 9-
Next, by (8.2) and Lemma 4.4 of Chapter 34 with P = (Gn/5C5n) and Q -
3
Set л = 9 in (8.25) and substitute the value of G45 given in (8.16). We then verify
that, as in our applications of (8.17),
Putting this value and the values for G5 and G45, given by (8.16), into (8.24), we
find, after a long, but straightforward, calculation, that
a5,9 = B -

346 Ramanujan's Notebooks, Part V
From (8.6) and (8.11),
(8-26)
We shall use (8.26) in further computations and in our proof of Corollary 8.4
below.
We next derive a formula by which the computation of a^n generally will be
simpler than that by using an analogue of (8.15). The following theorem is proved
in a paper with Chan and Zhang [3].
Theorem 8.3. Let f(-q) be defined by @.3), and let к be a positive rational
number. Put
\е-*Л) G25k
— and V = —^—.
Then
Corollary 8.4. Ifn is a positive integer and
then
= ± (v,,3 - v-) = ^ (v,, - v
-'
Proof. From (8.26) and (8.12),
and from @.1), @.3), and @.7), with * = l/En) in the definition of A\,
A' _,*
1
Therefore,
e-"lJ?") G\,n
_ /
Thus, by Theorem 8.3,
L
= — -7=
5^ v5
35. Values of Theta-Functions 347
since V = V,,. This completes the proof.
Now let n = 11. From the table in Chapter 34,
Setting n = 11 in (8.25), we verify that
Gll/S=
h + Vs U5-1
Thus,
V,, - v.71 = -
2
and so from Corollary 8.4,
Va5.11 V5 V 2
Solving for уогГп", we find that
4
19 + 9V5
2 / / I ч 8
I / h + Vs /Vs-i
8 V 8 I ~ I V 8 V 8
Let n = 13. From F.15) and the paragraph following F.15) of Chapter 34,
V65
8
У65-1
8
Thus,
V,3
Therefore, by Corollary 8.4,
1
ч/65-
+ V65 L/6S-1 1л r-
t-\\ г = -V14+2V65.
2^5 V 2 v
Hence,
/as. 13 ¦
2л/б5.

348 Ramanujan's Notebooks, Part V
Solving for asn, we deduce that
л/65 /7 + л/65"
— i у 2 V 2
Let и = 17. From the table in Chapter 34,
_/У5+1\Л/85Ч9у/а
With n = 17 in (8.25), it is quite easy to verify that
'17/5
= (л- Л (
\ 2 A
1/4
Putting these values in (8.26), with n — 17, we readily find that
Next, let n = 29. In the course of establishing the value of G145 in the proof of
Theorem 5.5, either in Section 5 or Section 6 of Chapter 34, we proved that
V29-
17 + VT45 /9+ л/145
Thus,
v29 -
9+vf45
and so, by Corollary 8.4,
1 15+vf45 J9 + VT45
5.29 -
Hence,
VOS19 + = Vl96 + 16VT45.
v/O5l9
Solving for 05,29, we conclude that
«5.29 = Ы49 + 4л/145 - У48 + 4n/145 j .
Similarly, by using Theorems 5.6, 5.8, 5.10, and 5.11 of Chapter 34, we can
deduce the values
ч 4
Я5.41 =
Зч/4Г
35. Values of Theta-Functions 349
«5.53 =
Я5.89 =
/85 + 9^89
83 + 9V89V
2 J '
and
First, let и = 9. By (8.6) and two applications of (8.10),
07,9 = 9?
G6
°63
<827)
Using (8.2) and setting P = (CpGjn)^ and Q = (Gn/7/G7nL, we find that
Lemma 4.5 of Chapter 34 assumes the form
7 -
From the table in Chapter 34,
G6i =
1/6
8
+
V2T /V2T-3'
8
Using (8.28) with n = 9, we easily verify that
i/л / /
V2T-3'
Substituting in (8.27) and using the denestings
= v/772 ± УЗ/2.

350 Ramanujan's Notebooks, Part V
we find that
a7.9 = j ^62 + 14V2I - B1ч/б
upon expanding the right side and simplifying.
Of course, the method of calculation used above cannot be generalized to n ф 9.
Thus, generally, it seems best to prove an analogue of Theorem 8.1 and Corollary
8.4.
Theorem 8.5. Ifn is any positive integer and
i/ _ G"P
Gln'
then
The proof of Theorem 8.5 follows along exactly the same lines as that for
Theorem 8.1. The relevant modular equations of degree 7 are given by (Part III
[3,p. 314, Entry 9(v)])
.'/2 /1_Д\1/2
-81
- (ЁЛ
~\а)
and
where m = <
As an example, let и = 11. Then, from E.12) and (9.5) of Chapter 34, we find
that
Hence,
and, from Theorem 8.5,
1
v,2, - v,72 = -/19 +8л/П,
ai „ -
fl7.ll
35. Values of Theta-Functions 351
Solving for 07,11, we deduce that
a1M =^372+ 112VTT- ^371 + П2л/ТТ.
In the author's paper with Chan and Zhang [4], the following two theorems,
guaranteeing that amM is a unit, are proved.
Theorem 8.6. Let m and n be odd positive integers such that mn is squarefree
and -mn = 3 (mod 4). Then атл is a unit.
Theorem 8.7. Let m and n be odd positive integers such that mn is squarefree
and —mn = 1 (mod 8). Then amn is a unit.
In this same paper, we also prove two theorems (Theorems 3.3 and 3.4) which
give formulas for am „ in terms of various parameters connected with the theory
of quadratic number fields.

36
Modular Equations and Theta-Function
Identities in Notebook 1
Chapters 19-21 in Ramanujan's second notebook are devoted almost exclusively
to modular equations (Part III [3, pp. 220-488]). Ramanujan clearly loved modular
equations, and, as the content of Chapters 34 and 35 makes manifest, he found
many applications of these equations. Thus, it is surprising that the first notebook
contains several dozen modular equations that he failed to record in his second
notebook. Some of these are easy to prove with the help of modular equations in the
second notebook, and so Ramanujan might have considered them less important
and not worthy of repeating in his second notebook. However, many of them are
apparently not so easy to prove. Some have degrees not examined in the second
notebook. For example, on page 298 Ramanujan records a modular equation of
degree 49. Not only does Ramanujan not consider modular equations of this degree
in his second notebook, but apparently no one else had found a modular equation
of degree 49 up until that time. Even at this writing, we know of no other modular
equation of degree 49.
It is regrettable that we are unable to prove some of Ramanujan's modular
equations by methods familiar to Ramanujan. As in Part III [3], we have had
to use the theory of modular forms in some instances. In particular, for several
modular equations on pages 86 and 88 in the spirit of L. Schlafli [1 ], we have had
to resort to modular forms.
As with much of Ramanujan's work on modular equations and class invariants,
one continually wonders about Ramanujan's methods. Several modular equations
are in the forms of Schlafli [1], H. Weber [1], or R. Russell [1], [2]. Ramanujan
employs the same functions of the moduli к and I that these authors considered. He
uses similar, but not exactly the same, notation as these predecessors. It would then
seem that Ramanujan had read these papers, but it is doubtful that Ramanujan had
access to them in Madras. As we pointed out at the close of Section 11 in Chapter
34, Ramanujan most likely had studied Greenhill's book [3], and, if so, he would
have learned about Russell's discoveries on pages 327 and 328 of this book, but
not his methods. For those modular equations of Schlafli-type, Weber-type, and
Russell-type, we could have employed, respectively, the methods of these three
men. However, since it seems unlikely that Ramanujan would have been acquainted

354 Ramanujan's Notebooks, Part V
with these methods, we have not proceeded by these means. Nonetheless, we feel
that a study of Ramanujan's work along with that of Schlafli and Weber would be
profitable.
Many deep modular equations are found on page 309, but two of them, of
degrees 25 and 27, are incorrect. As we show in the sequel, if these modular
equations are set in the form F(q) = 0, then the ^-expansions of F have coef-
coefficients equal to 0 up through powers of 7 and 18, respectively. This is evidence
that Ramanujan possibly used a method of comparing coefficients. Of course, this
is also the crux of the method employing modular forms. It is also the underlying
principle in Russell's method. Thus, one might conjecture that Ramanujan was
acquainted with Russell's ideas, which were developed for certain modular equa-
equations of prime degree. Possibly Ramanujan tried to adopt such methods in these
cases and either miscalculated some coefficients or did not calculate a sufficiently
large number of them. From the point of view of modular forms, the subgroups
Г0(и) and ГB) П Г0(л), on which the theta-functions act, have more cusps when
n is composite than when n is prime. This generally requires the calculation of
more coefficients of the q-series in order to rigorously establish modular equa-
equations via the theory of modular forms. Also, if one examines the additive order
of terms in Ramanujan's formulations of his modular equations of this type, one
sees that the ^-expansions begin with successively higher powers of q. This gives
further evidence that Ramanujan's methods hinged upon comparing coefficients
in ^-series.
K. G. Ramanathan [8] illuminatingly examined Ramanujan's work on modular
equations pertaining to those of Schlafli, Weber, and Russell. Near the end of the
paper, Ramanathan [8, p. 419] remarked, "We will defer a detailed consideration
of these, in the light of Weber's method, to a succeeding paper." Unfortunately,
Ramanathan died shortly thereafter, and no manuscript of the promised paper is
extant.
This chapter contains other material besides modular equations, but all of the
entries pertain to theta-functions in some way.
1. Modular Equations of Degree 3 and Related Theta-Function
Identities
Ramanujan states that Entry 1 is valid provided that
A.1)
but the degree of this modular equation is not given. Although A.1) could be a
modular equation of another degree, we are confident that it is a modular equation
of degree 3 (Part III [3, p. 230, Entry 5(ii)]).
36. Modular Equations 355
Entry 1 (p. 176). ///6 has degree 3 over a, then
i;
VI - a sin2 в
Proof. We shall apply Entry 6(v) of Chapter 19 of the second notebook (Part III
[3, pp. 238-239]) with В = sin"'(p7a3)l/8 and С = л/2. Hence, if
tan - - + sin I — 1
2 ^2 \a3/
2tan(sin-'(^/a:!)l/8)+2(l -a)tan3i
~ 1 - A - a) tan4 (sin (fi/a
then (Part III [3, p. 102], with a misprint corrected)
A.2)
[
/l -asin20 * Jo Vl-«sinz6»
It remains to show that A.2) is a modular equation of degree 3.
By the addition and half-angle formulas for the tangent function, A.2) is equiv-
equivalent to the formula
(p7«3)"8
tanH?+sin ' ^
1 +
;)'/8 2A - a)(/6/a3K/8
-(р7а3)'/4 + {1-(р7<*3)|/4}3/2
1-
A.3)
In proving some of Ramanujan's modular equations of degree 3 (Part HI [3, p.
237]), we found it convenient to use the parametrizations
and /3 = p3 ( -r-rZ- I, A.4)
i^V
1+2рУ
whereO < p < 1. (It is easy to verify thatA.4) is compatible with A.1).) Hence,
/3
and
1 -« = (!+p) I
A.5)
<*J V 2 + p
Substituting A.5) into A.3) and simplifying each side, we see that we are now
required to show that
3 + 3p + y/3 - 3p2 _ л/3 - 3p2 {2A + 2p)C - 3p2) + 2A + p)(l - pK}
1 - p + уз - 3p2 C - 3p2J - A + p)(l - pK(l+2p)
A.6)

356 Ramanujan's Notebooks, Part V
After much simplification, each side of A.6) reduces to ^3 - 3p2/(l - p), and
so the proof is complete.
On page 172 Ramanujan records two further results on elliptic integrals. We
have placed these results at the beginning of Section 10 in this chapter, because
their statements require no information about modular equations, although one of
the entries has a connection with modular equations of degree 3.
Entry 2 (p. 230). We have
B.1)
Proof. In C6.8) of Chapter 16 of our book [3, p. 69], set д = 2 and v = 1 to
deduce that
l2)<p(q2)-
Replacing q by —q, we find that
Subtracting the latter equality from the former, we deduce B.1).
Entry 3 (p. 254). We have
Proof. From B2.4) and Entry 24(iii) of Chapter 16 (Part III [3, pp. 37, 39]),
On the other hand, from Corollary (ii) in Section 31 of Chapter 16 and the Jacobi
triple product identity (Part III [3, pp. 35, 49]),
3 2)
Comparing C.1) and C.2), we see that we have completed the proof.
36. Modular Equations 357
Entry 4 (p. 254). We have
D.1)
Proof. By Entry l(iii) of Chapter 20 (Part III [3, p. 345]),
<p(-q) <p(-q3)) '
Comparing D.1) and D.2), we see that we must show that
9 — = fi
<p(—q) <p(—qi)
D.3)
We now translate D.3) into a modular equation of degree 3 by means of Entries
10(ii) and 1 l(i) of Chapter 17 (Part III [3, pp. 122-123]). Thus, if/3 is of degree
3 over a and m = zi/zj, D.3) is equivalent to the equality
or
This modular equation is not found among Ramanujan's modular equations of
degree 3 given in Entry 5 of Chapter 19 (Part III [3, pp. 230-231]). However, it
can easily be verified by using the parametrizations (Part III [3, p. 232, eq. E.1)])
1-cr ) 2
m-\
L — p f
(тГ-
3-m
2m '
3 + m
2m "
D.5)
D.6)
Thus, by D.5) and D.6), D.4) is equivalent to the equality
9 (m+ \\2 /3-i
I ^ I — '"I
Wt у 2. j у J,tTl
which is easy to verify, and so the proof is complete.
Entry 5 (p. 266). We have
3 + w
!~2^T'
. E.1)

358 Ramanujan's Notebooks, Part V
This result is not correct. However, by Entry 24(ii) of Chapter 16 (Part III 13,
P- 39]),
(-<?9) = f\q)
t=o E.2)
The right side of E.1) comprises those terms on the right side of E.2) when the
power of q is a multiple of 3, except there is a discrepancy in sign in the last
displayed term. Of course, the second series on the right side of E.2) provides no
contributions to the right side of E.1), and so this entry is very puzzling indeed.
Entry 6 (p. 282). Iff! has degree 3 over a, then
Ы
Proof. By D.5) and D.6), the left side of F.1) equals
3 +m 3-m 3
1m Itn m
Thus, it suffices to prove that
m =
However, this last equality is Entry 5(iv) of Chapter 19 (Part HI [3, p. 230]), and
so the proof is complete.
Entry 7 (p. 283). Let m and n be positive numbers such that m - n — 1. Define
в and (p, 0 < в, <р < 7Г/2, by
sin в = m s\n<p
and
Then
cos в = n cos<p.
m — sin в m + n
j; l;
G.1)
G.2)
G.3)
36. Modular Equations 359
Proof. The far right side of G.3) suggests that we set« = sin2 в and /6 = sin2 <p,
so that 1 -a = cos2 в and 1 - /6 = cos2 <p. Thus, from G.1) and G.2), respectively,
Thus,
G.4)
G.5)
In fact, by Entry 5(i) of Chapter 19 (Part III [3, p. 230]), G.5) is a modular equation
of degree 3. This confirms our definitions of a and p. Also, the far right side of
G.3) is then the reciprocal of the multiplier of degree 3, which, to avoid a conflict
of notation, we denote here by M. By G.4), the last equality in G.3) then takes
the form
= м- G)
However, G.6) is a modular equ ation of degree 3 that can be deduced by combining
parts of Entries 5(i), (iii) of Chapter 19 (Part III [3, p. 230]).
Lastly, by G.4), the extremal equality proposed in G.3) takes the shape
m — sin2 в
However, this equality is precisely Entry 5(iv) of Chapter 19 (Part III [3, p. 230]).
This completes the proof.
Entry Я (p. 283). Let 0 < n < 1 and set
P
Then, if p > 0,
-П + VI ~ n) + V2 + n + 2V1 + n + n2
(8.1)
(8.2)
Proof. Let f(n) denote the right side of (8.1). Thus,
PA + 1РЪ - 2pf(n) - f(n) = 0. (8.3)
To solve this quartic equation, we follow a standard procedure, as set forth, for
example, in Hall and Knight's text [1, pp. 483-484]. Write (8.3) in the form
рл + 2ръ + а2р2 + 2(ab - f(n))p - f(n) + b2 = (ap + bJ.
We seek constants a, b, and к such that
(8.4)

360 Ramanujan's Notebooks, Part V
By equating coefficients, we deduce that
1 +2k=a\
к = -f(n) + ab,
and
Thus,
fe2 = -/(n) + b2.
(8.5)
(8.6)
(8.7)
(* + f(n)f = Bk + \)(k2 + /(n)),
from which it follows that
2JkJ = /2(n)-/(«) = -i«J.
Let к be the real root -n/2. So, from (8.5),
a = \/l — n.
Then, from (8.7),
2 (" ~ Vl -n3).
That we have taken the correct square root can be verified by letting n tend to 1
above and using (8.6). Thus, from (8.4),
-Vl-и'П
Letting n tend to 1 and recalling that p > 0, we determine the proper square root
on the right side above to be
p2 + p - ln = - (Vl-n p - Un2 +2(l- ч/1-и3)) .
In solving this quadratic equation, we again let n tend to 1 and use the fact that
p > 0 to determine the correct square root. To that end,
-A
^n) +J2
1
^n + 2Jn2 + 2A -
1
)
p = 1 1 . (8.8)
We shall use (9.5) of Chapter 34 to denest the second rightmost inner radical
above. Here, d2 = n2(n + 2J, and so
y/n2+2(\-
„2 + 2 + n(n + 2) /и2 + 2-и(п + 2)
2
= ч/l +я +n2 - Vl - n.
(8.9)
36. Modular Equations 361
Using (8.9) in (8.8) and then simplifying, we deduce (8.2) to complete the proof.
The quotient />3B + p)/(l 4- 2p) occurs in the theory of modular equations
of degree 3 (Part III [3, pp. 237, 238]). Taking the parametrizations of p and
1 - p* in terms of p on page 237, by an elementary calculation, we can show that
n1 = 4/3A - /6), where 0 has degree 3. Thus, by A.6) of Chapter 34, n2 is related
to class invariants.
Entry 9 (p. 283). Let & have degree 3 over a. If m =а1/я and n =/31/8, then
mA + 2mV - 2mn - n4 = 0. (9.1)
In fact, Ramanujan does not indicate that /3 has degree 3, i.e., that (9.1) is a
modular equation of degree 3.
Proof. Ramanujan's equation (9.1) is equivalent to the equation
Using D.5) and D.6), we find that (9.2) takes the form
3+m 3+m
which is trivial to verify.
The modular equations in Entry 10 do not seem to be easily deducible from Ra-
Ramanujan's modular equations of degree 3 listed in Entry 5 of Chapter 19 (Part П1
[3, pp. 230-231]). However, they can be easily verified by using the parametriza-
parametrizations given in D.5) and D.6). Since the details are simple and straightforward, we
do not provide them.
Entry 10 (p. 290). We have
1/4
and
1 +

362 Ramanujan's Notebooks, Part V
Entry 11 (p. 295). Let f) have degree 3 over a, and define x and у by
a =
Then
'('
and
p —
i-n/'-у8
+ x2
-2хП.
(П.2)
The following proof by H. H. Chan supplants the author's more ad hoc proof.
Proof. Routinely solving A1.1) and A1.2) for x and y, we find that
x = Da(l -a)I/3 and у = D/3A - /3))l/8.
Replacing Gn by Da(l - а)Г1/24 = x~1/8 and G9n by D0A - /3))-'/24 = у-'/з
in Theorem 3.1 of Chapter 34, we find that
where
+
—= and и - \
*JX
i/э
After straightforward elementary algebra, we find that
-/A -
+x
-7A - дс) (-1 -
2дс
2) )
- л - J(\-x)BjV+x+x2- 1 -
since
Entry 12 (p. 297). A//3
. A2.1)
36. Modular Equations 363
Proof. Comparing A2.1) with Entry 5(viii) of Chapter 19 (Part III [3, p. 231]),
we find that we must show that
or
A2.2)
ButA2.2) is identical to Entry 5(ii) of Chapter 19 (Part III [3, p. 230]), and so the
proof is complete.
Entry 13 (p. 297). Ifp has degree 3, then
1 +
A3.1)
Proof. We are unable to simply deduce A3.1) from Ramanujan's modular equa-
equations of degree 3 found in Entry 5 of Chapter 19 (Part HI [3, pp. 230-231 ]). Thus,
we use D.5) and D.6) to verify A3.1). Thus, A3.1) is equivalent to the equation
, m2 - 1 f (m2-l)(9-m2)
1 H :— = ntjl
16/n2
which is easily verified.
2. Modular Equations of Degree 5 and Related Theta-Function
Identities
Entry 14 (p. 222). We have
+ 4
<P2(q)
A4.1)
Proof. Let /3 have degree 5 over a, and let m denote the multiplier of degree
5. By Entries 10(i) and 1 l(i) of Chapter 17 (Part III [3, pp. 122-123]), A4.1) is
equivalent to the modular equation
z\
\/2
1/8
z>
or
A4.2)

364 Ramanujan's Notebooks, Part V
However, A4.2) is one of Ramanujan's modular equations of degree 5 (Part III [3,
p. 281, Entry 13(vi)]), and so the proof is complete.
Entry 15 (p. 284). We have
/(-я4)
П5 1
Proof. By using Entries 1 l(i)-("i) and 12(iv) of Chapter 17 (Part III [3, pp. 123-
124]), we can easily show that A5.1) can be translated into the modular equation
of degree 5,
which is Entry 13(iii) of Chapter 19 (Part III [3, p. 280]). This completes the proof.
Entry 16 (p. 285). We have
^(a) МгЧ-д) /VV) = 4^V) {16Л)
Proof. With the use of Entries 1 l(i)-(iii) and 12(iv) of Chapter 17 (Part III [3, pp.
123-124]), it is easily shown that A6.1) is equivalent to the modular equation of
degree 5,
which is Entry 13(ii) of Chapter 19 (Part III [3, p. 280]), and so the proof is
complete.
Entry 17 (p. 286). We have
Proof. Replacing q by -q, we are required to show that
Now by Entry 9(ii) of Chapter 19 (Part III [3, p. 258]),
A7.1)
(П.2)
36. Modular Equations 365
We shall apply transformation formulas to the functions in A7.2) to deduce A7.1).
If a, p > 0 and ap = n2, then, by Entry 27(iv) of Chapter 16 (Part III [3, p. 43]),
and
Thus,
c-af4e~Sa)
A7.3)
Next, from Entry 27(i) of Chapter 16 (Part III [3, p. 43]), if a, /6 > 0 anda/6 = л2,
and
Hence,
cps(e-5a) = J_ /fi W
and
Therefore, using A7.3НП.5) in A7.2), we deduce that
4_L (?\ Z5^5) ' (Р\<РЧе-^) 1
455/2
A7.4)
A7.5)
Cancel the common expression 5~5/2p/a and set e~?/5
simplifies to the equality
A7.6)
q\. Hence, A7.6)
which is A7.1) with <? replaced by <?,.
Entry 18 (p. 295). We have
Proof. Replacing q by — q, we find that it suffices to prove that
X(-q)
A8.1)

366 Ramanujan's Notebooks, Part V
by B2.4) and Entries 22(iii), (iv) of Chapter 16 of Part III [3, pp. 36,37].
Now by Entry 10(v) of Chapter 19 and the Jacobi triple product identity (Part
III [3, pp. 262,35]),
= (-«7;
A82)
by Euler's identity, B2.3) of Chapter 16 (Part III [3, p. 37]). The desired result
now follows from A8.1) and A8.2).
Entry 19 (p. 295). We have
<p2(q)
A9.1)
Proof. After replacing q by — q and employing Entries 1 l(i), 10(ii), and 12(vi)
in Chapter 17 (Part III [3, pp. 122-124]), we can translate A9.1) into the modular
equation of degree 5,
Г
On the other hand, by Entry 13(iv) of Chapter 19 (Part III [3, p. 281]),
Squaring A9.3) and subtracting the result from A9.2), we see that it suffices to
prove that
Recall from Part III [3, p. 284, eq. A3.3)] the notation
/0 = (m3-2m2+5mI/2.
By Part III [3, p. 284, eq. A3.4); p. 285, eq. A3.10)],
ЛЛ"* 5/ Jexi/8 5p+m2+5m 5(p + 3m - 5) 1/5
\J) 'm{<Xfi) = 4^~ 4^ =4U"
Thus, A9.4) has been proved, and the proof of Entry 19 is complete.
Entry 20 (p. 297). Iff) has degree 5 over a, then
(a03I/8 + {A - a)(l - /9K)'/x = /I-U6a/3A _«)(i_/})}!/*. B0.1)
36. Modular Equations 367
Proof. Comparing B0.1) with Entry 13(vii) of Chapter 19 (Part 111 [3, p. 281]),
we find that it suffices to show that
or
l-2{16a/3(l -
¦ /3). B0.2)
But B0.2) is the same as Entry 13(i) of Chapter 19 (Part III [3, p. 280]), and so
the proof is complete.
Entry 21 (p. 325). We have
and
Proof. Both of these identities are readily established by employing the Jacobi
triple product identity.
3. Other Modular Equations and Related Theta-Function
Identities
Entry 22 (p. 246). We have
and
_
<р\д)
<p(q)
B2.1)
B2.2)
Proof. By using the product representations for ф(q), yj/iq1), <p(q), <p(—q), and
<p(-q2) (Part HI [3, p. 36]), we can easily show that
B2.3)
Taking the logarithmic derivatives of both equalities in B2.3), we deduce B2.1)
and B2.2) at once.
The following two entries can be found in Section 24of Chapter 18. Regrettably,
in Part III [3, p. 216], we claimed that the two results are false.

368 Ramanujan's Notebooks, Part V
Entry 23 (p. 300). The equation
B3.1)
is a modular equation of degree 8.
Proof. By using Entries 10(iii) and 1 l(iii) of Chapter 17 (Part III [3, pp. 122-
123]), we find that B3.1) is equivalent to the theta-function identity
By Entries 25(ii), (i) of Chapter 16 (Part HI [3, p. 40]),
<p{-q2) + 292VV6) = <р(~Яг) + \ЫЧ2) ~ Ч>
= i2W(-q2)+<p(q2)) = <
and the proof is complete.
Entry 24 (p. 300). ///3 has degree 16 over a, then
^ = 2^7
B4.1)
Proof. By Entries 10(i), (ii), and 1 l(iii) of Chapter 17 (Part III [3, pp. 122-123]),
B4.1) is equivalent to the theta-function identity
By Entries 25(i), (ii) of Chapter 16 (Part III [3, p. 40]),
2<p(q]b) + 32 4 A)
= <p(q) + tp(-q),
and so the proof is complete.
Entry 25 (p. 297). ///3 has degree 7 over a, then
Proof. The result follows by combining the first part of Entry 19(iii) with the
second part of Entry 19(i) of Chapter 19 (Part III [3, p. 314]).
36. Modular Equations 369
Entry 26 (p. 296). Ifp has degree 7 over a, then
-3
a(\-a)
1/8
B6.1)
Proof. Let
and
//37(l-/3O\'
' Ud-e)j
B6.2)
R := {B - 3/ + 2r2)B - / +12)A - t + 2t2)\m,
where a/3 = t\t > 0. From Part III [3, pp. 316,318, eqs. A9.2), A9.3), A9.15)],
A = | B - It + 1 If2 - 8/3 + 4/4 - A - 2t)R). B6.3)
From Entry 19(vii) of Chapter 19 in Ramanujan's second notebook (Part III [3, p.
314]),
У)
v
Thus, using B6.2) and B6.4), we may recast B6.1) in the form
or
(m2 - IJ - 20Mw2 - 12Л + 48Л2 - 64M3 = 0.
We also know that (Part III [3, p. 319, eq. A9.20)])
m = -3 + 8/ - 6/2 + 4/3 + 2R.
B6.5)
B6.6)
Putting B6.3) and B6.6) in the left side of B6.5) and simplifying with the aid of
Mathematica, we verify that indeed B6.5) holds to complete the proof.
We have altered Ramanujan's notation in the next entry so that it is consistent
with that in Entry 3 of Chapter 20 (Part III [3, pp. 352-353]).

370 Ramanujan's Notebooks, Part V
Entry 27 (p. 286). Let у have degree 9, and put m - zi/zj and rri = Zi/Z9-
Then
yj \i-y/ vW \y
Proof. Using Entries 3(x), (xi) of Chapter 20 (Part III [3, p. 352]), we find that
./4
/mm' +
mm
8
'/4
which completes the proof.
Entry 28 (p. 296). Ifm denotes the multiplier of degree 9 and у has degree 9,
then
a(l-a)
1/4
\-a
1/4
= m
Disproof. Write
1/2
1/4
1/8
a(l-cx)
1/4
-a
36. Modular Equations 371
. i/2
a(l-ot)
1/8
_ 4
iz) V
/8
/4
1/8
B8.1)
From our study of Ramanujan's modular equations of degree 9 in Part III [3, pp.
352-356], each of the expressions above can be expressed in terms of a parameter
/. In particular, from page 356,
Vor/
\-у\иЯ l+2t
and from equations C.7) and C.9) on page 354,
Thus, from B8.1),
_
+8/3(l + 2r)(l - 0 - 4(/A + 20A - 0 + *2A + 2rJ)}
= -7i Г4- {l - 6t2 + 4f3 - 9/4 - 12f5 + 4r*j.
Now from equations C.10) and C.11) on page 354 of Part III [3],
m2 = A +2/L.
B8.2)

372 Ramanujan*s Notebooks, Pan V
Thus, if Ramanujan were correct, the expression in curly braces on the far right
side of B8.2) should equal A + 2/J(l - t)\ But
A + 2/J(l - tL = 1 - 6r2 + 4/3 4- 9t4 - \2t5 + 4t6.
B8.3)
Thus, there is a discrepancy between B8.2) and B8.3) in the terms -9/4 and
+9f4, respectively. It seems therefore that Ramanujan made a sign error in his
calculations. This analysis is evidence that Ramanujan also used parametric rep-
representations.
Entry 29 (p. 230). We have
Proof. In C6.8) of Chapter 16 (Part Ш [3, p. 69]), we set д - 6 and v = 5 to
find that
Replacing q by — q and subtracting the resulting equality from that above, we find
that
But by Entry 18(iv) of Chapter 16 with и = 2 (Part III [3, p. 34]),
Using this above, we deduce B9.1).
Entry 30 (p. 298). Iffi has degree 11 over a, then
ll/i _ й\П '/24
!
i+2'0/3(
+
B + 2^
30.1
}
C0.1)
Proof. By adding Entries 7(vi), (vii) of Chapter 20 (Part III [3, p. 364]), we find
that
¦
C0.2)
36. Modular Equations 373
Comparing C0.1) and C0.2), we find that the proof is complete.
4. Identities Involving Lambert Series
Entry 31 (p. 266).
Ж<?'")- Proof. Using successively Entries 25(vi) and 25(iii) of Chapter 16 and Entry 8(ii)
B9-1) of Chapter 17 (Part III [3, pp. 40, 114]). we find that
{<p\q) + <p2(~q)) = | {<p\q) + <p\-q2))
= - 4-
8
( 2ч
2ч*
and the proof is complete.
Entry 32 (p. 267).
Proof. By Entries 25(i), (iii) of Chapter 16 (Part III [3, p. 40]),
<p(q)<p(q4) = \<p(q) (<f(q) + ?(-*)) = { {<p2(q) + <p4~
Recall from Entry 8(i) of Chapter 17 (Part III [3, p. 114]) that
C2.1)
C2.2)
at once.
Using this twice in C2.2), we deduce C2.1)
Note that Eniry 32 is a companion to Entries 8(iii), (iv) of Chapter 17 (Part
13, p. 114]), and, in fact, is placed after these entries in the first notebook.
Entry 33 (p. 274). Ifn is real, then
III
+9»

374 Ramanujan's Notebooks, Part V
Proof. Comparing C3.1) with Entry 33(iii) of Chapter 16 (Part HI [3, p. 53]), we
find that it suffices to prove that, with г = e'",
or
f(zq,q/z) =
f4-zq,-q/z)
<p(-q2)n-Z2q2, -q2lz2)
C3.2)
f(-zq, ~(i
By the Jacobi triple product identity and B2.4) of Chapter 16 (Part III [3, pp. 35,
37]), the right side of C3.2) equals
(q2; q2)oo(q2; <?4)oo(zV; g4)oo(g2/z2;g4)oo(g4;94)oc
= @2; q2)oo(-zq; q2)oo(-q/z; q2)^ = f{zq, q/z),
again, by the Jacobi triple product identity. This proves C3.2), and so the proof is
complete.
Entry 34 (p. 284). If ('j)denotes the Legendre symbol, then
1 -
Proof. By Entry 4(iii) of Chapter 19 (Part HI [3, p. 226])
and by Entry 3(i) of Chapter 21 (Part III [3, p. 460]),
Thus, from C4.1) and C4.2),
_2_^CL + 2 '
C4.1)
C4.2)
,(m + 5 / '
We now use the elementary equality
2<?2m
- qb" 1 - qm 1
C4.3)
36. Modular Equations 375
with m = Ъп + 1, Ъп + 2. Hence, after some cancellation,
q^2 _ q^ _ g6"+4 \
-?6"+2 l+q3"-1-2 1-?6я+4/
3/1+1
„=ov-46"+2 •"*
where we used the elementary equality
,2m
+
\+qm l-q2 l-q1™
with m = Ъп + 1, Ъп + 2. This completes the proof.
Entry 35 (p. 284). lf('j) denotes the Legendre symbol, then
<p(q)
Proof. From Entries 3(i), (ii) of Chapter 21 (Part III [3, p. 460])
= 4|l+6
ч>\д)
and from Entry 4(iv) of Chapter 19 (Part III [3, p. 227]),
^ = l+6f
Hence, using C4.3), we find that
,l2n+4
-2-
„6/1+4
+ 2-
_ л6/1+5
C5.1)
= 1+2
,I2/H
+ 3-
12n+5
9l2n+7 +_q 3
1 _ ^ 12/i+2 j _ ^12/1+4 T j _ ^12/.+5
12/1+8 -12/1 + 10 „I2n+ll
1 _^!2/i+8
1 _ „12/i + H I-
C5.2)

376 Ramanujan s Notebooks, Part V
On the other hand, the right side of C5.1) equals, by C4.3),
~ / qbn+\ ^6л+2 q6n+4 ?6«+5 ч
l~2f^\\ +q6n+] ~ 1 - q(*"+2 + 1 - qb"+* ~ l+^+sj
_„бП+1
-2-
^12л+2 q
~6n+2
?6и+4 q(m+S
+ 2
^12*4-10 Ч
_ д 12/1+10 J '
C5.3)
If we now compare C5.2) and C5.3), we find that they are equal, and so the
proof is complete.
Entry 36 (p. 284). We have
Proof. From Entry 34,
2я
= ~
since (§) = -1.
Entry 37 (p. 285). We /iave
Proof. This result easily follows from C4.1) and C4.3).
5. Identities Involving Eisenstein Series
Recall that
oo к к
^:=^):=1-24^T^-I, \q\<L
From Entry I2(ix) of Chapter 15 (Part II [2, p. 326]), or from the fourth entry on
page 264 of the first notebook,
C8.1)
2Xo(
2X
C8.2)
36. Modular Equations 377
Ramanujan, in fact, expresses the next entry in terms of the right side of C8.2).
Entry 38 (p. 264). We have
4ZV) - Щ) = ~S<p\q)- C8.3)
Proof. From Entries 13(viii), (ix) of Chapter 17 (Part HI [3, p. 127]),
4L(q4) - L{q) = DZV) - 2L(q2)) + BL(q2) - L(q))
= 2г2A - ±x) + z2d + x) = Зг2 = 3<p4(q),
by Entry 6 of Chapter 17 (Part III [3, p. 101]).
Suppose that, in the summands on the left side of C8.3), we expand 1 /(I — <?*)
and 1/A — «y4*) into geometric series. Collecting coefficients of like powers of
q",n > 0, we find that
- L(q) = 3 - 96
24
л=1
where a(n) is the sum of the positive divisors of n. Thus, equating coefficients of
q", и > 1, on both sides of C8.3), we find that, if rA(n) denotes the number of
representations of n as a sum of four squares,
( 8er(n), ifn#0(mod4),
n(n) - {
[ 8a(n) - 32<r(n/4), if n = 0 (mod 4),
d\n
i\n
Thus, Entry 38 yields a very short proof of a famous result of С G. J. Jacobi [ 1 ],
[2].
Recall that
M := M(q) :=
Entry 39 (p. 264). We have
-q«'
\q\ < I.
C9.1)
Proof. By Entry 12(v) of Chapter 15 (Part II [2, p. 326]),
^ к V _ M - L2
2-^A _9*J - 288 '

378 Ramanujan's Notebooks, Part V
where L is denned in C8.1). By Example (ii) in Section 35 of Chapter 16 (Part
III [3, p. 65]),
5L1 - 2M
Entry 39 thus easily follows from the last two equalities.
Entry 40 (p. 271). IfM(q) is defined by C9.1), then
Proof. From Entry 13(iii) of Chapter 17 (Part III [3, p. 127]),
M{q) = z4\ + \4x+x2). D0.1)
On the other hand, by Entries 10(ii) and ll(i) of Chapter 17 (Part III [3, pp.
122-123]),
<pb(-q) + 256q\ffs(q) = г4A - xJ 4- 16г4л = z4(l + 14x + x2). D0.2)
Combining D0.1) and D0.2), we complete the proof.
6. Modular Equations in the Form of Schlafli
This section contains some of the deepest results in the chapter. As indicated at the
beginning of the chapter, Ramanujan's methods for some entries have remained
hidden from us.
Entry 41 (p. 90). Let
P := 2"<WA -
and Q :=
Then, iffi has degrees 11,13,17, and 19, respectively, over a,
_ f j* _ J_^ _ 340 - 136/>4 + 16P8) = 0,
36. Modular Equations 379
and
These modular equations were established by Schlafli [1] in 1870. Schlafli also
discovered similar modular equations of degrees 3,5, and 7, which also appear on
page 90 of the first notebook but are not given here, since they also appear in the
second notebook and so were proved by us in Part III [3, pp. 231, 282, 315]. In
one sentence, Ramanathan [8, p. 411 ] indicated a proof of the equation of degree
11 above, but we have been unable to complete the proof along the lines that he
indicated. Watson [7] gave a proof of Schlafli's modular equation of degree 13
but erroneously remarked that Ramanujan did not discover it. Schlafli's modular
equations were also examined in another paper by Watson [8].
In Entries 42-49 we set
P := B56apy8{\ - cr)(l -
and
/
\fiY(l-fi)(\-y)J
D2.1)
D2.2)
where, say, a, /3, y, and 8 have degrees a, b,c, and d, respectively. By using
Entries 10(i) and 12(i) of Chapter 17 (Part III [3, pp. 122,124]), we may transform
D2.1) and D2.2) into the representations,
P = V2q<a+b+c
and
Q-q
in terms of theta-functions.
<p(qa)(p{qb)<p(q')<p(qJ)
/(<?*)/(<?'
D2.3)
D2.4)
Entry 42 (p. 86). Let P and Q be defined by D2.1) and D2.2), respectively.
Suppose that a, /3, y, and 8 have degrees 1,5,7, and 35, respectively. Then
Proof. The following proof was given by Ramanathan [8]. Observe that
PQ1 = 2|/6 08y(l - /})A - y))
)>/24
D2.5)

380 Ramanujan's Notebooks, Part V
and
PB = 2l/6(a«O -a)C -<5))l/24. D2.6)
Now from Entry 18(v) of Chapter 20 of Ramanujan's second notebook (Part III
[З.р.423]),
- у)I/2д -
A6/3y(l - /3)A - y))
I/24
- <S))
I/24
~ A6/3y(l - /3)A - y)I/8 - A6a<S(l -a)(l - й)I/24
Using D2.5) and D2.6), we can rewrite the latter equation in the form,
Clearing fractions above, we easily deduce the desired result.
Entry 43 (p. 86). //a, P, y, andS have degrees 3,1,5, and 15, respectively, then
Entry 44 (p. 86). If a, p, y, and S have degrees 5,1,3, and 15, respectively, then
Entry 45 (p. 86). Ifa,fi, y.andS have degrees 1,3, 7, and 21, respectively, then
Entry 46 (p. 86). If a, /3, y, and 8 have degrees 7,1,3, andll, respectively, then
D6.1)
Entry 47 (p. 86). If a, fi, y, and S have degrees 3, 1, 11, and 33, respectively,
then
^ (^)(^H. D7.1)
^)+42 = 0.
36. Modular Equations 381
Entry 48 (p. 86). [fa, /5, у, and S have degrees 5,1,7, and 35, respectively, then
Entry 49 (p. 86). If a, ft, y, and S have degrees 5, 1, 11, and 55, respectively,
then
D9.1)
E0.1)
E0.2)
For the next three entries, we need the definitions
R :=
and
W(l-
/0ЛA-1)A-Д)\'/48
W-у)/ '
By Entries 10(i) and 12(i) of Chapter 17 (Part III [3, pp. 122,124]), Л and Г may
be expressed in the forms
= ,cW.a.W/4. jf(qc)f(qd)<P(qa)<p(gb)
and
=
E0.4)
Entry 50 (p. 86). If a, /3, y, and S have degrees 1,5,7, and 35, respectively, then
2 + ^) + 15 = 0.E0.5)
Entry 51 (p. 88). If a, 0, y, and S have degrees 1, 13, 3, and 39, respectively,
then
E1.1)
Entry 52 (p. 88). If a, p, y, and S have degrees 1, 13, 5, and 65, respectively,
then

382 Ramanujan's Notebooks, Part V
Each of the previous eleven modular equations is of Schlafli-type. They are
equations of composite degree that are analogues of the Schlafli modular equations
of degrees 11, 13, 17, and 19 given in Entry 41. We have been unable to prove
Entries 43-52 by employing ideas known to Ramanujan and so we have resorted
to the theory of modular forms. By using D2.3), D2.4), E0.3), and E0.4), we may
convert each of Entries 43-52 into a proposed identity involving modular forms.
Let ГA) denote the full modular group, and let Г B) and Г0(л) be the subgroups
usually so denoted (Part III [3, p. 327]). If q — exp(;rir), where г е Ш - [t :
Im r > 0), /i(t) := <?1/24/(<?)> and gi(r) := <p(q), then f\ and g\ are modular
forms of weight ± on ГB) (Part III 13, pp. 330, 331]). Their multiplier systems
may be found on page 331 of Part III [3]. In each of Entries 43-52, the degree
n = p\P2, where p\ and p2 are distinct odd primes. Each of the modular forms
f\{mx) and gi(wr), m = 1, pit p2, n, appearing in these entries is a modular
form of weight \ on Г := ГB) П Г0(п) (Part III [3, p. 332]). If n is any odd
positive integer, then (Part III [3, p. 332])
(E2.2)
: ГB) П Г0(в)) = 6п П A + - ),
where the product is over all odd primes p dividing n. It is easily checked that
each of the products of powers of modular forms appearing in the translations of
Entries 43-52 into modular forms is a modular form of weight 0 and multiplier
system identically equal to 1.
As in Part III, the principal theorem to be utilized is the valence formula (Rankin
[1, p. 98, Theorem 4.1.4], Part III [3, p. 329]). If F is a modular form of weight r
and multiplier system v on Г, a subgroup of finite index in ГA), and if F is any
fundamental set for Г, then, provided that F is not constant,
F;z) = >-pr, E2.3)
where Ordp(F; z) denotes the order of F at z with respect to Г and
рг = ^(ГA):Г). E2.4)
In Entries 43-52, as already observed, r = 0, and so E2.3) reduces to the equality
F;z) = 0. E2.5)
Since, in our applications, F is analytic on H, then, from E2.5),
0>?Ord,-(F;z), E2.6)
where С is a complete set of inequivalent cusps for Г.
When Г = ГB) П Г0(л), and n = p\p2, where p\ and p2 are distinct odd
primes, then a complete set of inequivalent cusps is given by (Part III [3, p. 403])
Ilili-iliii?! <*«*
0' n' n' 2p,' pi' pi' 2p2' pi p{ 21 Г Г
36. Modular Equations 383
If С* := С - 1/0, we rewrite E2.6) as
0 > Ordr(F; 00) +
; г).
E2.8)
For a proposed identity
F(r):=
E2.9)
involving modular forms Fj(r), 1 < j < m, of weight 0, we calculate a lower
bound for the order of F(r) at each cusp in C*. Thus, from E2.8), we obtain
an upper bound for Ordr(F; 00). If we can show by a direct calculation of the
Fourier expansion of F about г — /oo, i.e., the power series of F about q = 0,
that Ordr(F; 00) exceeds this bound, then we will obtain a contradiction to E2.8),
unless F(r) is a constant. Letting r approach ioo (or q -*¦ 0), we see that this
constant must be 0. This then proves the proposed identity E2.9).
To calculate this bound, we need formulas for the orders of /1 (mr) and gt (mr)
at cusps. From Part III [3, p. 333, Lemma 0.1 and Table 1; p. 404, eq. A3.11)], we
deduce the following formulas for orders at the cusps r/s, where (r, s) = 1:
48Ord(/,(«T); r/s) = (mr + s, 2J
m
and
24Ord(g!(mr); r/s) - {(mr + s, 2J - l),
m
E2.10)
E2.11)
where (a, b) denotes the greatest common divisor of a and b. We thus use E2.10)
and E2.11) to calculate, for each of the eleven finite cusps in E2.7), the order
of each modular form f\(mr) and g|(mr), m = a,b, c, d, appearing in Entries
43-52. Thus, a total of 88 orders for each entry are calculated. We then calculate
the order of each term QJ, P>, RJ, T', P'Q', T'Q1 appearing in the proposed
modular equations of Entries 43-52. This provides a lower bound for the order of
each finite cusp on the left side of each equation. All of this work was programmed
by Jeff Meyer on Mathematica. For each of Entries 43-52 we provide a table giving
the eleven finite cusps and a lower bound for the order of each cusp on the left
side of each modular equation.
cusp
order
cusp
order
cusp
order
2
15
1
2
2
15
-1
2
21
-2
1
15
-1
T5
-2
5T
-4
10
1
6
1
10
1
3
1
14
2
3
2
5
1
6
2
5
1
3
2
7
2
3
5
1
3
5
2
3
1
7
4
3
2
1
30
1
1
15
1
2
2
21
1
1
15
1
2
15
1
4
21
2
1
30
2
1
15
2
2
21
1
6
1
10
1
6
1
5
1
6
2
7
2
3
1
10
2
3
1
5
2
3
2
7
1
3
1
5
3
2
5
1
3
4
7

384 Ramanujan's Notebooks, Part V
cusp
order
cusp
order
cusp
order
cusp
order
cusp
order
cusp
order
cusp
order
2
21
-2
2
33
-1
2
35
-2
2
55
-3
2
35
_3
2
39
-1
2
65
-3
1
21
4
i
33
-2
i
35
-4
55
-6
35
-3
3»
-2
l
65
6
1
14
2
3
1
22
1
3
п
2
5
1
22
3
5
1
14
3
10
1
26
~Ъ
1
26
3
5
2
7
2
3
2
И
1
3
2
7
2
5
2
11
3
5
2
7
3
10
2
13
~\
2
13
_3
1
7
4
3
1
11
2
3
1
7
4
5
1
11
6
5
1
7
3
5
1
13
2
3
T3
6
5
\
г
21
2
1
33
2
2
35
1
2
3
55
1
2
3
70
1
2
~39
1
2
3
65
1
4
21
1
2
33
1
4
35
1
6
55
1
3
35
1
2
39
1
6
65
2
2
21
2
l
33
2
2
35
2
3
55
2
3
70
2
~39
2
3
65
1
6
2
7
1
6
~TT
sl-
2
7
1
10
3
11
1
10
3
14
1
6
1
13
To
3
13
2
3
2
7
2
3
1
11
2
5
2
7
2
5
3
~TT
2
5
3
14
2
3
1
13
2
5
3
13
1
3
4
7
3
2
II
5
4
5
6
11
1
5
3
7
1
3
2
13
1
5
6
13
In the next table we sum the orders of the eleven cusps and then use E2.8) to
determine the minimal number of coefficients in the ^-expansion of the left side
that we must show are equal to 0 in order to prove the given modular equation.
Entry
43
44
45
46
47
48
49
50
51
52
Order
ii
10
2/
5
'214
21
214
21
ii
11
314
bW
55
471
~W
_Ш
39
H13
65
Number of Terms
3
6
11
11
5
9
13
7
5
13
In each instance we employed Mathematica to compute the ^-expansion of the
left side of each proposed theta-function identity. In each case, we determined
that each of the required coefficients is equal to zero.
36. Modular Equations
7. Modular Equations in the Form of Russell
For Entries 53-57 we define
P = 1 ± (a0I/8 ± {A -
Q = 4((a0I/8 + {A -
385
and
-/3)Г\ E3.1)
E3.2)
E3.3)
We shall give four definitions for the triples P, Q, and R in this section. Russell
[ 1 ], [2] used the same notation, but his definitions of P, Q, and R are different. In
Ramanujan's definitions of Q and R, powers of 2 occur which to do not appear in
Russell's definitions. Moreover, the signs of Russell's P, Q, and R are generally
different from those of Ramanujan.
Entry 53 (p. 302). Let P,Q,andRbe given by E3.1H53.3), with the plus signs
taken. Then, iff) has degree 15 over a,
Q) + R = 0. E3.4)
An equivalent formulation of E3.4) was first proved by Russell [2, p. 388].
Entry 54 (p. 302). LetP, Q, and R be given by E3.1 H53.3),with the plus signs
taken. Suppose that f) has degree 31 over a. Then
P2 - Q =
Although not immediately obvious, Entry 54 is equivalent to Entry 22(ii) of
Chapter 20 (Part III [3, p. 439]) after elementary manipulation.
Entry 55 (p. 302). LetP, Q, and R be givenby E3.1 H53.3), with the plus signs
taken. Suppose that fi has degree 47 over a. Then
P2 - Q- PR1'2 - 2RVi = 0.
Entry 55 is identical to Russell's modular equation of ''»yee 47 quoted in our
book. Part III [3, p. 445, eq. B3.1)]. Note, however, that the notations in Russell's
paper [2], Part HI [3], and here are all different. The equivalence of Ramanujan's
and Russell's formulations is provided in detail in the proof of Entry 56 below.
Entry 56 (p. 302). Let P, Q, and R be given by E3.1И53.3). with the minus
signs taken. Suppose that /} has degree 71 over a. Then
+ Q)

386 Ramanujan's Notebooks, Part V
Proof. In the notation above, Russell's modular equation of degree 71 takes the
form (Part III [3, Chap. 20, p. 445, eq. B3.2)])
/>3 - 4RU\P2 - P + 1) + 2PRV* - R + Л4/3 = О. E6.2)
Comparing E6.1) and E6.2), we find that it suffices to prove that
/3 = -RuiQ,
or
4P-4 + R=-Q.
E6.3)
Using the definitions of P, Q, and R, we see that E6.3) is a triviality, and so the
proof is complete.
Entry 57 (p. 302). Let P, Q, and R be givenby E3.1H53.3), with the plus signs
taken. Then, iff) has degree 95 over a,
4^s^ = 0.
E7.1)
An equivalent version of E7.1) was proved by Russell [2, p. 389]. In fact, the
term —2Л5/3 does not appear in the notebooks, a fact also observed by Ramanathan
[8].
For Entries 58-61 we define
P = 1 - (a/*) - {A - «)A - P))"\ E8.1)
Q = 16((aKI/4 + {A - «)A - p)}]/4 - (eftl - a)(l - fi)}1'*),
and
R =
E8.2)
E8.3)
Entry 58 (p. 302). Let P,Q, and R be given by E8.1 hE8.3), respectively: Then,
ifP has degree 19 over a,
P* -7P2R-QR = 0. E8.4)
An equivalent form of E8.4) was first proved by Russell [1, p. 99], [2, p. 389].
Entry 59 (p. 302). Let P, Q, and R be given by E8.1)~E8J), respectively. Then,
iff} has degree 27 over a,
P9 - RP2B9P*
2Pi
Q2)- \7R2Pi -
=0.
This modular equation was first proved by E. Fiedler [ 1, p. 226] in 1885, although
his formulation is much different. M. Hanna [1, p. 49] translated Fiedler's equation
36. Modular Equations 387
into the form above, but with different signs on P, Q, and R. Hanna s formulation
unfortunately contains three errors; the signs of the terms 29P4 and 11P2Q are
incorrect, and the coefficient 3 of R3 was omitted.
Entry 60 (p. 302). Let P, Q,andR be given by E8.1)-08.3), respectively. Then,
iff) has degree 35 over a,
P4 - Л1/3E/>3 + PQ) + 2P2R2/i -RP- R*'y = 0.
F0.1)
An equivalent version of F0.1) was established by Russell [2, p. 389].
Entry 61 (p. 304). Let P, Q, and R be given by E8.1)-E8.3), respectively. Then,
if f) has degree 59 over a,
P5 - RU\\4P4 + 9P2Q + Q2) + R2/iP(l9P2 + Q)
6RPP2 + Q) + 4PR4/3 - 8Л5/3 = 0.
This result is due to Russell [2, pp. 367, 389]. The sign of 4PR4'* is differ-
different in Russell's two formulations, with that on page 389 being correct. In both
formulations, Russell wrote -42P2R instead of the correct expression 42P2R.
Ramanujan also made a mistake; he wrote -ЗЛ5/3 instead of -8Л5/3.
For Entries 62-66 we define
Q = 64(yZ
F2.1)
F2.2)
and
R = 32-/с*/3A -а)A -p).
F2.3)
Entry62(p.304). IfP, Q, andRaredefinedbyF2.1 H62.3),respeclively,then,
if P has degree 9 over a.
Pb - rtA4P3 + PQ) - 3R2 = 0.
An equivalent form of this modular equation was first established by Russell
[2, p. 390].
Entry 63 (p. 304). IfP, Q, and Rare defined by F2.1 H62.3), respectively, then,
ifP has degree 13 over a,
3 + 8Л) - л/ЛA1P2 + Q) = 0.
F3.1)

and
R = -32RO,
2l6R2P0 = 0,
388 Ramanujan's Notebooks, Part V
Proof. With
P = -Po, Q = -6400,
Russell [1, p. 105] proved that
/>07 - 25*o(lO5Po4 - 27 ¦ U
which in Ramanujan's notation becomes
P7 - Ж105Р4 + 22P2Q + Q2) + 6AR2P = 0.
For simplicity, set P = p2, Q = q, and R = r2. Then, from F3.3),
pH - 105/-V - 22r2qp4 + 64r V - 'V
= (/>7~ Ilrp4 + 8r2p-r<?)(p7 + 11/-/+ 8r2p + r<
Clearly,
F3.2)
F3.3)
Thus,
p1 -
-r<? =0,
which is equivalent to F3.1).
Entry 64 (p. 304). IfP,Q,andR are defined by F2.l)-F2.3), respectively, then,
iff} has degree 17 over a,
Py - Rl/i(\0P2 + Q) + 13Л2/3/> + 12Л = 0.
Proof. With the same notation as in the previous proof, Russell [2, p. 390] proved
that
Pq + DЯоI/3B70о - 20/tf) + 52DЛоJ/3/'о + 384ДО = 0.
It is now an easy task to show that this equation is equivalent to that of Ramanujan.
Entry 65 (p. 304). IfP, Q,andR aredefinedby F2.1)-F2.3), respectively, then,
iff} has degree 29 over a,
>) = 0. F5.1)
This modular equation was essentially proved by Russell B, p. 390]. As in
the foregoing entries, Russell used the altered notation F3.2). Russell has a sign
ambiguity in his formulation of F5.1); corresponding to the last expression in
F5.1), Russell writes (in our notation) ±R*'6(9P2 + Q- 13РЛ|/3 + 15Л2/3).
Entry 66 (p. 310). IfP.Q, and R are defined by F2.1H62.3), respectively, then,
iff] has degree 37 over a,
+ g) _ 3PR2'2 - 25R - M(\9P2 -&) + &. F6.1)
36. Modular Equations 389
As evinced by the two "&'s," Ramanujan evidently failed to determine an exact
modular equation of degree 37. In fact, Ramanujan does not define P,Q, R, and
M, but since 37 s 5 (mod 8), the definitions of P, Q, and R are probably given
byF2.1)-F2.3).
Discussing a probable modular equation of degree 37, Russell [2, p. 363] re-
remarks, "This is of the 19th degree in кХ, к'Х', and is not reducible to a simplified
form. I have consequently made no attempt to form it." Contradicting Russell,
Hanna (I, p. 48] derived a modular equation of degree 37 in the 19/2 degree in кк,
к'Х.'. His modular equation is given in terms of L, M, and N, where L = —P, M =
-Q, and N - -R, where P, Q, and R are defined in F2.1 M62.3). Moreover,
his equation has 18 terms with coefficients ranging up to 923722. In Ramanujan's
notation, Hanna's "leading term" is Pl9/2.
For none of the fourteen preceding modular equations, did Ramanujan record
the definitions of P, Q, and R. We were only able to discern the correct definitions
with the help of Russell's work.
Page 303 is an enigma. For the material on the top two-thirds of the page, we
quote Ramanujan below.
7,23&
3, 11&
1,5&
Q=
R =
Q = V2(~
R = q
Q = <Р\-
R = q
Q= M-^) + V
R = q(n+Wf\-q4)
P2 = P2~Q = qlf\q
15.31&
The conditions at the left margin evidently indicate that the definitions to the
right hold for n = 1 (mod 16), n = 3 (mod 8), n = 1 (mod 4), and n = 15
(mod 16), respectively. These definitions of P, Q, and R do not correspond to
any other definitions of P, Q, and R that might arise in modular equations. In
particular, some of the terms comprising P and Q do not depend upon n, which
evidently denotes the degree of the modular equation, and the dependence of R
on n is too simple.

390 Ramanujan's Notebooks, Part V
At the bottom of the page, Ramanujan cryptically lists four triples of formulas
for P, Q, and R, which we think are E3.1)-(S3.3), E8.1M58.3), and F2.1)-
F2.3), i.e., in more detail,
P = 1 + (a/8)l/fi + {A -
Q = 4 ((a;8I/8 + {A -
{«,8A
Q = 4((a/3I/8 + {A - a)(l -
Д = 4{а/3A -a)(l -/S)}"8
P= l-(a/3I/4-{(l
)'/4 + {A -
P = I - Jafi -
Q = 64
-/3)
Only for the first triple, does Ramanujan give complete definitions. The defini-
definitions of Q and R in the remaining three triples are abbreviated by 4,4; 16,16; and
64,32, respectively. One might surmise that the four triples of P, Q, and R at the
bottom of the page correspond to the four sets of definitions in the top two-thirds
of the page. However, there appears to be very little resemblance between the two
sets of formulas. For example, consider the third triple of formulas at the bottom
of the page. Transcribing P and Q via Entries 10(i), (ii) and 1 l(iii) of Chapter
17 (Part HI [3, pp. 122-124]), and also employing Entries 24{iv) and 25(iv) of
Chapter 16 (Part III [3, pp. 39-40]), we find that
p = i -
<p(q)<p(qn)
and
<p4q)(p4q")
Any connections between these two formulas and those for P and Q in the third
(or any) set at the top of the page appear, at best, artificial.
Observe that in Entries 53-57, the degree n = 7 (mod 8), and that the defini-
definitions of P, Q, and R are given in the first set of definitions at the bottom of page
303. In Entries 58-61, each degree n = 3 (mod 8), and the definitions of P, Q,
and R are given by the third set of formulas at the bottom of page 303, but that in
the third set of formulas at the top of the page, n = 1 (mod 4). Lastly, in Entries
36. Modular Equations 391
62-66, и = 1 (mod 4), and the formulas for P, Q, and R are given by the last
set of formulas at the bottom of the page, but at the top of the page in the last set
of formulas, и = 15 (mod 16).
Finally, we provide a proof of the second equality for Рг given after the last
triple for P, Q, and R at the top of page 303. A brief calculation shows that this
equality is equivalent to the following identity.
Lemma 67 (p. 303).
- /V.
Proof. From the corollary to Entry 31 of Chapter 16 (Part III [3, p. 49]),
Replacing q by -q, we find that
Upon multiplication,
*W(-4) = /V- <710) - <72/V-
But by Entry 25(iii) of Chapter 16 (Part III C, p. 40]),
F7.1)
F7.2)
Combining F7.1) and F7.2), we complete the proof.
8. Modular Equations in the Form of Weber
The next six modular equations are of Weber-type [1]. The definitions of P, Q,
and R given below were not recorded by Ramanujan. We have deduced these
definitions from Weber's modular equations, but we have replaced Ramanujan's
R by /f3. Let n = /i|/i2, where ti\ and n2 are primes, which are not necessarily
distinct. Suppose that д is defined by
+ 1)(«2 + 1) = 8/i = 0 (mod 8).
Define
P = 1 + (-1)"
Q = 4 ((а
(A -
- y)(\ -
(A -
F8.1)
F8.2)
F8.3)
and
R= B56afiyS(.l -cr)(l -
- Й))
|/24
F8.4)

392 Ramanujan's Notebooks, Part V
Suppose that a, /3, у, and S have degrees 1, b, c, and d, respectively. Then using
Entries 10(i), (iii). 1 КО. and 12(i) of Chapter 17 (Part III [3, pp. 122-124]), we
can express F8.2)-F8.4) in the forms
<p(qMqh)<P(qc)<P(qd)
<p(q)<p(qb)<p(qc)<p(.qd)
F8.5)
о = 1
<p(.q)<p(qbMqcMqd)
(- IV
V>4q)<p4qb)<p4ic)<p4qd)
щ.
and
R =
<p(q)((>(qb)<p(qc)<p(qd)
F8.6)
F8.7)
Entry 68 (p. 309). Le/ a, /3, y, am/<5 have degrees 1,3,11, and33. respectively,
and let P, Q, and R be given by F8.2)-F8.4), respectively. Then
P2 - Q- PR-4R2 = 0.
Entry 69 (p. 309). Let a,fi,y, and S have degrees 1, 5,7, and 35, respectively,
and let P, Q, and R be given by F8.2)-F8.4), respectively. Then
P2 -Q-PR-2R1 = 0.
Entry 70 (p. 309). Leta,fi,y, and 8 have degrees 1,5,11, and 55, respectively,
and let P, Q, and R be given by F8.2}-F8.4), respectively. Then
PJ - ЛDР2 + Q) - PR2 + 4tf' = 0.
Each of the three preceding modular equations was first established by Weber
[1]. The first two were not explicitly recorded by Ramanujan, but he must have
found them, for he wrote, "N.B. For 1, 5, 7, 35 same as for I, 3, 11, 33, 4Л2/Э
instead of 2Л2/3."
The next three equations were not discovered by Weber, and we shall use the
theory of modular forms, as described in Section 6, to prove them.
36. Modular Equations 393
Entry 71 (p. 309). Let a, fi, y, and S have degrees 1,3,17, and 51, respectively,
and let P,Q, and R be given by F8.2)-F8.4), respectively. Then
P* - RAP2 + Q)+ HR2P- 12Л3 =0. G1.1)
Proof. We proceed as in the proofs of Entries 43-52. For q - ехр(л7 г), г е Н,
the functions h2(v) := (p(-q2) and Ло(т) := q[/s\Jr(q) are modular forms of
weight 5 on ГB). Thus, F8.5)-F8.7) can be expressed in terms of the modular
forms fi,g\,ho, and Иг, where /( and gt were defined prior to E2.2). From Part
III [3, p. 333], for each positive integer n and cusp r/s, where (r, s) — 1, we
deduce that
and
48Ord(A0(nr); r/s) =
48Ord(/i2(nr); r/s) =
D - (nr, 2J)
D - (,, 2J) .
G1.2)
G1.3)
Using E2.10), E2.11), G1.2), and G1.3), we calculate, for each finite cusp, the
order of each function f\(mx), g\(mr), ho(mr), and/i2(mr),m = 1,3, 17,51,
appearing in the translation of G1.1) into modular forms. We then calculate the
order of each term in G1.1) at each finite cusp, so that we can obtain a lower bound
for the order of the left side of G1.1) at each finite cusp. This was programmed by
J. Meyer on Mathematica, and the following table summarizes the calculations:
cusp
order
2
51
0
1
51
27
2
1
34
0
2
17
0
1
17
9
2
1
2
0
1
9
34
2
0
i
б
0
2
5
0
l
i
27
34
Hence, if F(x) denotes the left side of G1.1), after it is converted into modular
forms,
" ~ 17 •
We thus need to show that the first 20 coefficients in the g-expansion of F(t) are
equal to 0. With the use of F8.5)-F8.7) and Mathematica, this can be shown, and
so the proof of Entry 71 is complete.
For the next two modular equations, the degree n is not the product of two
odd primes. More precisely, n = 9 and 27, respectively. We were unable to find
a complete set of inequivalent cusps for Г B) П Г0(9) and Г B) Л ГоB7) in the
literature, nor were we able to determine complete sets. Thus, we employed the
valence formula E2.3) in a way different from that in Section 6. We take the
two theta-function identities equivalent to the two proposed modular equations
and clear denominators. This then yields identities involving products of modular
forms, with each product having a positive weight and nonnegative order at each

394 Ramanujan's Notebooks, Part V
cusp. Let F(r), g = exp(^ir), denote the left side of one of the two proposed
thcta-function identities (set equal to 0). We show that F is a modular form of
weight r, say, on Г, where Г is one of the two modular subgroups mentioned above.
Ifthecoefficientsof<7°,<j',<72, qv in the expansion of F about q — 0 are equal
to 0, it follows that Ordr(F; oo) > v+ 1. Suppose further that v + 1 > rp wn-
; z) > Ordr(F; oo) > v + 1 > rpr.
G2.1)
This then yields a contradiction to E2.3) unless F is constant, which must be the
case. This constant trivially is equal to 0, and so the modular equation is proved.
Entry 72 (p. 309). Let a, /3, y, and S have degrees I, 3, 3, and 9, respectively,
and let P, Q,andR be defined by F8.2H68.4). Then
P3 - PQ + 3Ry = 0. G2.2)
Proof. Substituting F8.5)-F8.7) into G2.2) and multiplying both sides by w3(q),
where
w(q) - (f>(.q)(p2(qi)<p(q9),
to clear denominators, we find that
p3 - ps + 3r3 = 0, G2.3)
where
P ¦= pig) := w(q)P, s := s(q) := w2(q)Q, and r :- r{q) :- w(q)R.
G2.4)
Each of the three terms in G2.3) has weight 6 and multiplier system identically
equal to 1. By E2.2), if Г = ГB) П Г„(9), then (ГA): Г) = 72. Hence, by
E2.4), rpi = 36. Thus, expanding p3 — ps 4- 3r3 in a ^-series with the help
of Mathematica, we find that the coefficients of q", 0 < n < 36, are equal to 0.
Thus, G2.3), and hence G2.2), are established to complete the proof.
Entry 73 (p. 309). Let a, p, y, and S have degrees 1,3,9, and 27, respectively,
and let P, Q, and R be defined by F8.2И68.4). Then
P9 - R3P\UP2 + Q) + 9Я6Я3 + 6/?9 = 0. G3.1)
Disproof. Substituting F8.5)-F8.7) into G3.1) and multiplying both sides by
w9(q), where
G3.2)
w(q) =
we obtain the proposed equation
p4 - л Vd V + s) + 9r V + 6r9 = 0,
36. Modular Equations 395
where p, s, and r are defined by the same equations as G2.4). Each of the terms
in G3.2) has weight 18 and multiplier system identically equal to 1. By E2.2),
if Г = ГB) П ГоB7), then (ГA): Г) = 216, and so, by E2.4), rpr = 324.
Expanding the left side of G3.2) in a ^-series with the aid of Mathematica, we
find that
p9-rip4(llp2+s)+9r('p:i+6r9 = 46080<j19+414720<?20+H7504<V +••¦ .
Thus, the proposed modular equation is false. That the first 19 coefficients in the
«/-series are equal to 0 is evidence that Ramanujan used a method of comparing
coefficients and did not compute enough of them, or made an error in computing
them.
As in the previous entries, Ramanujan does not provide the definitions of P, Q,
and R in the next entry. In the notation F8.1), и, = n2 — 5, and so F8.1) is
not satisfied by any integer /г. Thus, it might be that P, Q, and R are not defined
by F8.2H68.4). In fact, we employed F8.2)-F8.4) twice, once with the plus
signs and once with the minus signs, in an attempt to determine if these are the
correct definitions for Entry 74, and indeed they are not. We tried several possible
definitions for P, Q, and R, but none were viable. The definitions that came the
"closest" are given by
- (A -a)(l-
C=16 ((apy8)l/A + (A - a)(l - p)O - y)O -
- y){\ - <5))'/4) ,
and
R =
- y){\ -
G4.1)
G4.2)
G4.3)
Translating G4.1H74.3) via Entries 10(i), (ii), ll(iii), and 12(iii) of Chapter 17
(Part III [3, pp. 122-124]), we find that
p= x
Q =
у(-д)<Р2(-д5М-д25)
4 4)
<p(g)<p4g5Mg2i)
<p4q)<P4(q5)<p2(q25)
and
G4.5)
G4.6)

396 Ramanujan's Notebooks, Part V
Entry 74 (p. 309). Let a, /9, y, and S have degrees 1,5,5, and 25, respectively.
Then (possibly) for P, Q, and R defined by G4.I}-G4.3), Ramanujan claims that
P4 - PREP2 + Q)- P2R2 + 3PR3 - R4 - 0. G4.7)
Disproof. Transcribe G4.7) into a proposed theta-function identity by G4.4b
G4.6). Then clear fractions by multiplying both sides by t4(q), where t(q) =
<p(q)<pz{q5)<p(qi0). Setting p := p(q) := t(q)P,s := s(q) := t2(q)Q, and
r := r(q) :— t(q)R, we find that G4.7) is equivalent to the proposed identity
l(q) := p4 - prEp2 +s)- p2r2 + 3/>r3 - r4 = 0. G4.8)
Expanding /(<?) in a ^--series, we find that
%) = 768<78 - 2560?10 + 7680<?12 .
Thus, Ramanujan's claim G4.7), if we have interpreted it properly, is false.
By successively changing the coefficients of p2r2, pr3, and r4 in G4.8), we can
increase the power of the leading term of l(q) by 2. For example, if we replace
Hq) by
l*(q) := p4 - prEp2 + s)- Ap2r2 + Ipr3 - 10r4,
we find that
l\q) = -
Thus, we were never able to alter sufficiently the definition of l(q) to produce a
function that was identically equal to 0.
We reluctantly conclude that Ramanujan either miscalculated some coefficients
in the ^-series for l(q), or did not calculate enough of them.
The next entry does not properly fit in this section, but we place it here simply
because we must again resort to the theory of modular forms to effect a proof.
Entry 75 (p. 298). Let fi have degree 49 over a. Put
P={a
and
Then
(P+R- y/m) - Dfi + APR + 13Я2) = 0,
G5.1)
36. Modular Equations 397
where m denotes the multiplier of degree 49.
Proof. By using Entries 10(i), (ii), 1 l(iii), and 12(iii) of Chapter 17 (Part III [3,
pp. 122-124]), we can transform G5.1) into a theta-function identity. Clearing
denominators, we then deduce the proposed identity
{p + r - tJ - 4s - Apr - 13/-2 = 0,
G5.2)
where
P = q
and
If q = ехр(тпг), where т б И, then >/(t) = qunf(-q2),gi(r) - <p(q),
?г(г) = qlf*if(q2), and Аг(т) — <p(—q2) are modular forms of weight j on
ГB) (Part III [3, pp. 330, 331]). The multiplier system for each term in G5.2)
is the same, and each term has weight 5. By E2.2), if Г = ГB) П Г0D9), then
(ГA): Г) = 336, and, by E2.4), р,- = 140. Thus, by G2.1), to prove G5.2), we
must show that each of the coefficients of q", 0 < n < 140, on the left side of
G5.2) is equal to 0. Using Mathematica, we have indeed shown this, and so the
proof of Entry 75 is complete.
To the best of our knowledge, Entry 75 is the first and only modular equation
of degree 49 that can be found in the literature.
9. Series Transformations Associated with Theta-Functions
Recall from Section 6 of Chapter 17 (Part HI C,p. 101]) the fundamental notation
G6.1)
where
z:=z(x):= 2F,(I,i;l;.
G6.2)
Entries 76 and 77 are beautiful series transformation formulas involving the
variables x, y, and z above and should be compared with Entry 6 of Chapter 18
(Part III [3, p. 153]), which is also given in the first notebook along with Entries 76
and 77. Ramanujan gives still another formula in the same vein, but it appears that

398 Ramanujan's Notebooks, Part V
a faint line has been drawn through the entry to indicate that the entry is incorrect,
or that Ramanujan could not prove his claim. We record in Entry 78 the entry as
Ramanujan gives it. We are unable to discern a general formula for the nth term in
the series on the right side. Then we sketch an attempt at proving (or disproving)
the formula. As we shall see, serious technical difficulties prevent us from utilizing
the ideas that we used in proving Entries 76 and 77 and Entry 6 of Chapter 18 in
Part III.
Entry 76 (p. 281). With the notation G6.1), G6.2),
(-1)" V^ ФпХ"
E
Bn + l)cosh{B«
2 ^ n! B« + 1)
Proof.
for any
Now,
constant c,
1
d
dy
and
G6.3)
2cosh(cy) erv+e-°v Jy
Let с - Bn + l)/2, multiply both sides above by (—l)"/c, and sum on n, 0 <
n < oo. Using also G6.3), we find that
(-1)"
S:=
•I
oo oo
(-1)" Г ±( 1
2n + 1 Jy dt \е<2"+1>'/2 +
gBn+\)t/2 _ g-B"
dt
я=0
dt.
2J
G6.4)
We now evaluate the integral on the right side of G6.4) in another way. Replace
the variable t by yo. For brevity, set и — exp(—yo/2). Now,
oo 2
Я1^"
я=0
oo
n=0
иЗBл+1)
M2Bn+1)J
2Bn-H)m
+
! )
k2Bm+3) J
36. Modular Equations 399
,,2m+\
2m + 1
^ " 2cosh{Bm+l))-0/2[
= Jz2(Ov^(l-O, G6.5)
by Entry 16(i) of Chapter 17 (Part III [3, p. 134]), with x there replaced by t here.
We substitute G6.5) into the right side of G6.4) and make the change of variable
yo = nz{\ - t)/z(t). By Entry 9(i) of Chapter 17 (Part III [3, p. 120]),
dy0 _ 1
dt tO-t)z2(t)'
When yo = У. then t = x, say. We therefore find from G6.4) that
dt
1 Г°° 1
=- z2(.t)y/t(\-t)dy0=-
4 Л 4
_ 1
~ 4
It is easily seen that G6.7) is equivalent to the claim made in Entry 76.
Entry 77 (p. 280). In the notation G6.1), G6.2),
oo / 1V, f- oo /„,ч2 п (i\2
^ B/i + lJsinh{Bn-
Proof. For any complex number c,
dy \e<> -есУ )
2г
^ П)
> 2 /
ec>' - е~с?)
and
Hence,
1
2 sinh(cy)
2 f°° f°° e2" + e~2cl +
~ C Л J. (er' - e-"K
-<// du.
G6.6)
G6.7)
G7.1)
G7.2)
G7.3)

400 Ramanujan's Notebooks, Part V
Now let с = Bn + l)/2, multiply both sides of G7.3) by (-l)"/c2, and sum on
n, 0 < n < oo. Accordingly, we find that
= / / ?
*¦ ->У •>" n=0
eBn+i)t _|_e-Bn+l)i
dt du.
For brevity, sets = exp(-//2). The integrand in G7.4) then becomes
-№.+.) _ ,2*+
(
= y\_n»?—li
n=0
я=0
m=0
6s
M2n+l)
_га(Фи-2)
m=0
w(m-
n=0
-^(W +
1+.
m=0
o2Bm + l)
m=0
?2Bm+l) '
G7.5)
From G7.4) and G7.5), so far, we have shown that
i i\2
+ I)
where, for the sequel, it is more suggestive to set t — yc,.
We now apply Entry 16(x) of Chapter 17 (Part III [3, p. 134]) in G7.6) to deduce
that
/•00 /"OO
/-OO
Ju
We make the change of variable yQ = nz(l- 0/г@- If yo = «. then i = дс0, say.
Thus, by G6.6),
lrr m dtdu. G7.7)
8
- О
36. Modular Equations 401
Replacing u by y0 above, we find, from G6.6), that
dyo 1
When >0 = У> then x0 = x, say. Hence, appealing to G7.7) and then replacing x0
by u, we find that
8 7o \Jo V~t(l-t) /лоA-
tk'w2z(t)dt
tff
i=0 Уо Л
u(\ -i
" 8г ?j; (A + 1/2J 3 4 fe + |,
A + i, i
G7.8)
by Entry 9(iii) of Chapter 17 (Part III [3, p. 120]). Using the definition of 3F2 in
G7.8), setting к + j = n, and inverting the order of summation, we find that
1 ^2, л*+1/2
t+J
Ф1
V*
(>
h Ф1 к
which completes the proof.
By using Entry 29(b) of Chapter 10 (Part II [2, p. 39]), namely,
(л + 1)Г2(л+ 1) L •• ч + 2
we easily find that the right side of Entry 77 may be rewritten in the form

402 Ramanujan's Notebooks, Part V
Entry 78 (p. 280). With x,y,andzas in G6.1) and G6.2),
oo .
1
у
?-Bn + lL
\)y/2)
We are unable to discern a general formula for the nth term of the series on the
right side above.
Sketch of Attempted Proof. Differentiating G6.3), or from the proof of Entry 6
of Chapter 18 (Part III [3, p. 153]), for any complex number c, we find that
d2 { 1 \
dy2 \ecv 4- е~сУ }
e2cy + e-2cy _ ,
which is an analogue of G7.2). After two further differentiations,
,?*v - 76e2<v + 230 - 76efV + e~
Letting с = Bл + l)/2, dividing both sides of G8.1) by Bи +1L, summing both
sides on n, 0 < л < oo, and integrating, we find that
1
i ЛОО pOC лОО ЛС
= 8 Л Л, /Л Л,
те e2Bn+l)v0 _
xE
n=0
Dyt, _|_ 230 —
(eB«+l)vo/2
-dyo dy] dyz dyi-
G8.2)
As before, set и = ехр(-уо/2). The integrand in G8.2) may then be written in
the form
^ «?"+' - 76м3Bп+|) + 230tt5B"+" - 76w7B"+l) + м^2')
m=0
4!
„,2m + 3
_ ,.2Bm + l)
-76-
+230
,.2m+5
,2ra+7
-76-
m=0
_ M2Bm+5)
(-l)"'Bm+lL«2"'4
_ M2Bm+I)
_ M2Bra+9)
- M2Bm+3)
G8.3)
36. Modular Equations 403
Employing G8.3) in G8.2), we find that
8 Л Л, 4 /., 2L
) ° d>« ^2 rf^3. G8.4)
The value of the "second" sum in the integrand on the far right side of G8.4) can
be obtained from Entry 17(viii) of Chapter 17 (Part III [3, p. 138]). Thus,
E
ra=0
=Нг5«)E- 0A-0-5.
G8.5)
The value of the "first" sum in the integrand on the far right side of G8.4) can be
obtained from that of the "second" sum by the process of dimidiation (Part III [3
p. 126]). Thus,
^ (-l)'"Bm + IL /
2 -,am+1)W2 _ i = 3 (г (Od - Od - VF)E + 6Vi + 50 - 5). G8.6)
2-.
G8.7)
Using G8.5) and G8.6) in G8.4), we find that
J ЛОО /.OO /.ОС /-0C
5 = 32 Л / У У
1 Г°° г» /.oo лл0
= зг/ / / / t'
-1* Jy Jy} Jyi JO
by G6.6). We can make similar changes of variables for >>,, y2, and y3, and we
can employ G6.6) (with y0 replaced by y,, y2, and >3)) as we did above. However,
these steps do not enable us to completely evaluate the quadruple integral in G8.7).
Thus, at this point, we terminate our discussion of this apparently deleted entry.
10. Miscellaneous Results
Entry 79 (p. 172). For 0 < x < 1, suppose that
( 1 + x si
V 1 — x si
1 - sin
where 0 < a, /S < 7Г/2.
1 + sin/3 _ 1+sina f 1 +jcsina\2
jesina/
1 - sin a
2x)
Л)
G9.1)
G9.2)

404 Ramanujan's Notebooks, Part V
Proof. By Entry 6(iii) of Chapter 19 (Part III [3, p. 238]), if
tan i (a + 0) = (!+/>) tan a, G9.3)
then G9.2) holds, with x replaced by p. Thus, it only remains to show that G9.1)
and G9.3) are equivalent. In Part III [3, pp. 239-240, eq. F.2)], we solved G9.3)
for p in terms of sin a and sin fi and found that
1 — sin a sin fi — cos a cos fi
p =
sin a (sin fi - sin a)
= {\ + *sina)/(l -Jtsina). Thus, from G9.1),
G9.4)
/A
-sin»)
x sin a —
A -sinj8)(l+sina)
- 1
у/п+1 /A+sin/0H -sina)
+ 1
f A -sin0)A + sina)
After straightforward elementary algebra and trigonometry, we find that
1 — sin a sin fi — cos a cos p
sin a (sin/3 — sin a)
This agrees with G9.4), and so the proof is complete.
Entry 80 (p. 172). For 0 < x < 1, suppose that
1 + sin/3 1 +sina /1 + xsina\2 1 +.x2sina
1—sin0 1 — sin a \l — jrsina/ 1—jc2sina'
where 0 < a, fi < n/2. Then
A
2 Г de f
+ xI / , = /
^o J\-x4sin29 Jo
(80.1)
(80.2)
Proof. Set
Solving for у in terms of x, we find that
1 +6л2+
A-xV _ 8jcA +x2) 4y_
l~\TTx~) ~~ A+JcL ~ A+У
-л2)
2J
_
У ~
+л:2)
(80.3)
We want v to approach 0 as x tends to 0, and so we need to choose the minus sign
above. Hence,
(S0.4)
У =
1+*2
36. Modular Equations 405
We now apply Entry 7(xii) of Chapter 17 (Part HI [3, p. 112]). If
A + у sin2 y) sin /3 = A + y) sin y,
and we use (80.3) and (80.4), then
t" de _ г?
JO f, Г /ч-гч*) ¦ 2„ ^0
(80.5)
.„+„
Jo y/l-ytsi^e
Л ¦
do
7"
Jo
r° v-flw*
where we have applied Entry 17(xii) of Chapter 17 once again and thus need the
condition
A +x2sin2a)sin)/ = (I +jt2)sina. (80.6)
Combining (80.5) and (80.6), we find that
sin I = (l+xJ(l+Jt2sin2«)
sine* A +*2sin2<*J +2*A +jc2)sin2a
It remains to show that (80.7) is equivalent to Ramanujan's hypothesis (80.1). If
we substitute in (80.1) the formula for sin f) obtained from (80.7), we find that
(80.1) and (80.7) are compatible, and so the proof of (80.2) is complete.
Entry 81 is one of many entries where Ramanujan writes "nearly" to indicate
an approximation.
Entry 81 (p. 244). Let, F(x), 0 < x < 1, be defined by G6.1). Put
F(jc2) = t.
Then
l+x
(81.1)
"nearly."
Proof. From Entry 2(vi) of Chapter 17 (Part III [3, p. 95]),
1 . 5 з 369 5 4097 , 1594895
(81.2)

406 Ramanujan's Notebooks, Part V
On the other hand, using Mathematica to expand the right side of (81.1), we find
that
1 + (i _,)(!_, + 2,2)
5 , 369 с 4097 7 398697 ,
* + ^X + ~?~X + ~WX
(81.3)
The first four coefficients in (81.2) and (81.3) agree, while
1594895
and
227
398697
225
= 0.0118829...
= 0.0118821....
Thus, the fifth coefficients in (81.2) and (81.3) are remarkably close. Therefore,
Ramanujan's claim has been justified.
Entry 82 (p. 244). Let F(x) be defined by G6.1). Then
x 1
F(\ -е~х)ъ —
10 + л/36 + jc2
(82.1)
Proof. From Entry 2(vii) of Chapter 17 (Part III [3, p. 96]),
1 1 ,31 5 661
m-e')=2-*X ~3~2пЛ Г~2'»А 15-225" + "' (82'2)
On the other hand, expanding the right side of (82.1) by means of Mathematica,
we arrive at
Ю+у/36 + х1 216°
1 1 з 31
)JC
3187
24* 3-210* ¦*" 15-219A 35-52-223" '
Thus, the first three coefficients in (82.2) and (82.3) agree, while
661
(82.3)
-8
and
315 -225
3187
-6.25376- -x 10
= 6.25383 • • • x 10"8.
3s • S2 • 2"
Thus, the fourth coefficients in (82.2) and (82.3) are amazingly close, and Ra-
Ramanujan's approximation is indeed justified.
36. Modular Equations 407
Entry 83 (p. 271). Let
<P4(u)
where
:=e :=
where z is defined by G6.2). Then, for any real number n,
f^*\u) f z"(x) .
/ du = / -— -dx.
J и J x(l -x)
Proof. From Entries 6 and 9(i) of Chapter 17 (Part III [3, pp. 101, 120]),
du du dy и
Лс ~ d~y"dx ~ x(l -x)z2(x)'
Thus, by a change of variable,
J и "~J и x(i-x)zHx) X ~ J x(l-x) X'

37
Infinite Series
In the last two chapters devoted to the second and third notebooks, we gather
together most of Ramanujan's results on series in the 133 pages of unorganized
material found in these two notebooks. In this chapter, we primarily focus on ex-
exact formulas, while in Chapter 38 our attention is given to approximations and
asymptotic formulas. In Part IV [4], we had disengaged Ramanujan's results on
special functions, partial fraction decompositions, and elementary and miscella-
miscellaneous analysis from the material on infinite series, and devoted individual chap-
chapters to these three topics. Although those three chapters contain a couple of gems,
Chapters 37 and 38 have many more jewels.
We now briefly describe some of the interesting theorems proved in this chapter.
Entries 2-10 are new summation formulas, or applications thereof, akin to the
Abcl-Plana summation formula. Our proofs by contour integration employ the
same idea that is used in the most well-known classical proof of the Abel-Plana
formula.
Entries 15-19 arise from Eisenstein series and provide analogues of more well-
known series identities, some of which Ramanujan had derived elsewhere and
some of which the author [6], [7] had previously proved from different consider-
considerations of Eisenstein series.
Entry 21 is particularly fascinating. Ramanujan offers two transformations for
doubly exponential series that are analogous to the famous theta transformation
formula. That such transformations exist is surprising.
Entry 35 and Entries 36 and 37, which follow from Entry 35, are incorrect. Entry
35 gives a series transformation, reminiscent of the Poisson summation formula.
However, Entry 35 involves the Mobius function, and, indeed, it is extremely
unlikely that such a general transformation would exist. Numerical calculations
demonstrate that Entries 36 and 37 are "close to being true." We conjecture that
Ramanujan employed approximate formulas from prime number theory that he
considered to be more accurate than warranted (see Chapter 24 of Part IV [4]). It
would be interesting to reconstruct Ramanujan's thinking.
On pages 335,340, and 341 Ramanujan makes successively more general claims
about the behavior of partial sums of certain oscillating series. Ramanujan's claims
are very remarkable in their explicit descriptions of the oscillations. We are not

410 Ramanujan's Notebooks, Part V
aware of any theorems in the literature similar to Entry 42, which is a rigorous
formulation of Ramanujan's most general claim.
Beginning with Euler, many authors have written about the convergence of
infinite exponentials
W. J. Thron [1] in 1957 proved a theorem which, it would seem, gives the best
possible general upper bound for \an | that suffices for the convergence of iterated
exponentials. However, in his third notebook, Ramanujan offers a slightly better
upper bound for |а„ | and also claims, at least in one sense, that his bound is the best
possible. These claims were recently proved by G. Bachman [1], and we present
his elegant, difficult proofs in Section 50.
We conclude our introduction by stating some definitions, notation, and well-
known results that are needed in the sequel.
Recall that the Bernoulli numbers Bn, n > 0, can be defined by the generating
function
\z\ < 2n.
@.1)
л=0
As usual, let ?(z) denote the Riemann zeta-function. For each positive integer л,
(-l)"-'B7rJ"S2n
С Bл) =
@.2)
which is a famous formula of Euler. Also, for each positive integer n (Titchmarsh
[3, p. 19]),
f(l-2n) = -^. @.3)
2n
For any complex number г, the functional equation of f (г) is given by (Titchmarsh
[3, p- 22])
-z)/2)f(l -z),
which can be written in the equivalent form (Titchmarsh [3, p. 25])
f(z) = 28т(^г)B7г)г-'ГA -z)?(l -z). @.4)
We shall need the Euler-Maclaurin summation formula (Olver [1, p. 285]). If /
has In + 1 continuous derivatives on [a, b], where a and b are integers, then
b pb
?/(*) = / /@Л + |(/(в) + /(*I
k=a J°
Г<2к-»(а)) + Rn, @.5)
37. Infinite Series 411
where, for n > 0,
@.6)
where Bn(x), 0 < n < oo, denotes the nth Bernoulli polynomial.
For each nonnegative integer n, the rising factorial (а)„ is defined by
(а)я := a(a + l)(e + 2) - • • (a + и - 1) =
(д + n)
Г(а) '
If С is a simple closed contour, / (C) denotes its interior. The residue of a function
f(z) at a pole a is denoted by Ra. (Usually, the function /(z) is understood, and
so there is no ambiguity by not specifying / in the notation for a residue.)
In Sections 17 and 18 of Chapter 13, Ramanujan states a version of the Abel-
Plana summation formula and some variations (Part II [2, pp. 220-222]). At the
top of page 335, Ramanujan offers another version of the Abel-Plana formula, A
rigorous proof of this formulation can be found in Henrici's book [1, p. 274].
Entry 1 (p. 335). Let <p(z) be analytic for Re z > 0 and suppose that either
or
2
n=0
Jo
(p(x)dx
is convergent. Assume furthermore that
lim \<p(x ± iy)\e~2"y = 0,
У-Ю0
uniformly in x on every finite interval, and that
/ \q>(x±iy)\e-2n4y
Jo
exists for every x > 0 and tends to 0 as x tends to oo. Then
f00 4>(iy) - <P(~i
<p(x)dx+i 2,v i
Jo e > - \
„=0
Entry 2 (p. 335). Let n be any complex number such that Re и > 1. Let z" have
its principal value. Then, i/Re x > 0,
Proof. Let fit) = /" le '*. Although /(/) is not necessarily analytic at / = 0,
the proof of the Abel-Plana summation formula may be easily extended to include

412 Ramanujan's Notebooks. Pan V
this function. We therefore find from Entry 1 that
Aу)п~хе~'ух — (—iy)n~{e'yx
= / t"-'e-'xdt + i
Jo Jo
dy
e2ny _ 1
2-п-д; _ g-.Ti(n-l)/2+;vjr\
2»> - 1
(\nn - yx)
"'
which completes the proof.
Entry 3 (p. 334). Assume that <p(z) is analytic for Re z > 0. Let a be real with
0 < \a\ < 1. Suppose that
lim \<p(x±iy)\e~"}t =0,
V-»OO
uniformly on any finite interval in x > 0. Suppose also that
f°° ^C* + iy)
I dy
J-<x cos(tt(x + iy)) + cos(;ra)
exists for each nonnegative number x and tends toOasx tends to oo. Assume also
that the integral below converges. Then
sin(^a) ^j
Proof. Consider, for positive integers m and N,
/<m,A0:=
w(ix)
coshGr;c) + cosGra)
-dx.
'Cltm cos(^z)
where C/v.ra is a positively oriented rectangle with horizontal sides passing through
±i/V and vertical sides passing through 0 and m. We apply the residue theorem.
Since
cos(ttz) + cosGra) = 2 cos {j^(z + «)} cos {j^Cz — <*)),
the integrand has simple poles at z = 2n + 1 ± a for each nonnegative integer n.
Straightforward calculations give
<pBn + 1 ± a)
7Г sinGia)
Hence, by the residue theorem,
2/ ,
sin(jra) 0<2nf?±a<
- <f>Bn + 1 - a)).
37. Infinite Series 413
Letting N tend to oo, then letting m tend to oo, and using our hypotheses, we find
that
(f+П
\Jioo Jo /
<p(z) dz
cos(ttz) +
Upon parametrizing the two integrals above, we complete the proof.
The next result is a limiting case of a lemma in Section 18 of Chapter 13 (Part
II [2, pp. 221-222]), and so it is unnecessary to give a proof here.
Entry 4 (p. 335). Let <p(z) be analytic for Re z > 0. Assume that
lim \<p(x ± iy)\e-nyl2 = 0,
)'-*OO
uniformly for x in any compact interval in [0, oo). Suppose also that
\<p(x±iy)\e-">/2dy
Jo
exists for all x > 0 and tends toOasx tends to oo. Assume that the integral below
exists. Then
Entry 5 (p. 335). Let <p(z) be analytic for Rez > 0. Assume that
lim |<p(x± (у)|е-л-у = 0,
>--юс
uniformly for x in any compact interval on [0, oo). Assume that
f \<p(x±iy)\e-n4y
Jo
exists for every x > 0 and tends toOasx tends to oo. Then, provided that the
integral below exists,
Ш0)
Jo S"
о sinhGrjt)
-dx.
Proof. For positive integers N and w, let См,т denote a positively oriented,
indented rectangle with horizontal sides passing through ±i N and vertical sides
passing through 0 and m + \. The indentation is a semicircle Ce of radius e > 0

414 Ramanujan's Notebooks, Part V
centered at the origin and lying in the left half-plane. Applying the residue theorem
to
/(z) := -
sinGrz)'
we find that
^- f
Letting N tend to oo, then letting m tend to oo, and invoking our hypotheses, we
deduce that
T- I / + / + / ) /<*> dz = V(-l)V(n). E.1)
2ni \Jiao Jc. J-,( ) ^o
Setting z = (e'°, n/2 < 9 < Зтг/2, we find that
1 Г i [W <p(ieie)eewi d6
hm -— / /(z) dz = hm — /
Hence, using E.2) in E.1), we deduce that
1 г
= т- /
2л- Л/
E.2)
= _1 Г
2i Jo
2
sinhOry) 2/ Jo sinh(^y) 2
This completes the proof.
Entry 6 (p. 335). Let <p(z) be analytic for Re z > 0. Suppose that
lim |р(л ± /»k"TV = 0,
y—>oo
uniformly for x in any compact interval on [0, oo). Assume that
Г \<р(х ± iy)\e~'4y
Jo
exists for every x > 0 and tends to 0 as x tends to oo. Then, provided that the
integrals below exist,
- <p(,-ix) ,
Proof. For positive integers N and m, let CjJ 2m denote the positively oriented
indented rectangles with horizontal sides passing through 0 and ±i N, respectively,
37. Infinite Series 415
and with vertical sides passing through 0 and 2m. The indentations are semicircles
Cj( and C~(, of radius e, 0 < e < 1, centered at 2j - 1, 1 < j < m, and lying
in the upper and lower half planes, respectively. Thus, observe that
Снлп, u Cno* = UN, -iN) U [-IN, -iN + 2m]
U [-/AT + 2m, iN + 2m] U [iN + 2m, iN] Uj=] Cj,(,
where Cj,f = C*( U C~f is a negatively oriented circle of radius e and center
2j - 1,1 < j < m.
Applying Cauchy's theorem, we find that
<p(z)dz
F.1)
Now,
dz
ze'"i ,
and
=4°
.. f viz) dz .. Г
lim f -— = lim I
V{2j - \) dd =-<pBj - \)
- 1 + ee'°)ceiei
enHBj-l)+eap(W))
1 Г
Thus, from F.1),
Now let N tend to oo, then let m tend to oo, and use the given hypotheses. Hence,
r-'°° viz) dz
2f>B,--l)=
n
enii
F-2)
Since
2 2
-1 H : = i tan (lяx) = 1
e~nn + \ V2 ' e"n

416 Ramanujan's Notebooks, Part V
we find that
/ I + )<p(z)dz
1 Г°°
= - / q>{x)\i\ata(lnx) + \ +
2 Jo
<p(x)dx.
= f
Jo
F.3)
Also,
Г0 <p(z) dz
Г0 <p(z) dz Г10° <p(z) dz =_. Г <Р('У) dy . f* y(-iy) dy
Putting F.3) and F.4) in F.2), we complete the proof.
The first two formulas on page 269 arc remindful of the Abel-Plana summation
formula, but, in fact, a stronger version of this formula is needed, because the
functions to which we would like to apply the formula have poles on the imaginary
axis or in the right half-plane.
Entry 7 (Formula A), p. 269). Let u, 0 > 0 with ар = 4л2, and let Re/i > 2.
Then
- sin fan) PV
?l
cot (i
where PV denotes the principal value of the integral.
Proof. In proving the Abel-Plana summation formula of Entry 1, one proceeds
as in the proof of Entry 6, except that now
<p(z)
is integrated over C^ m+1/2> where now the right vertical sides pass through m +
5, where m is a positive integer. Furthermore, the semicircular indentations of
radius €,0 < t < ^, are centered at each positive integer j,l < j < m. On
the left vertical side, we need a semicircular indentation at the origin, with the
upper quartercircle being part of Cj|in+l/2 and the lower quartercircle belonging
t0 ^V
37. Infinite Series 417
We now set
- 1
G.1)
where the principal branch of z" is taken. Observe that <p(z) has a simple pole
at z = 2nik/p, for each nonzero integer it, and also has a singularity at z = 0.
Except for the singularities on the imaginary axis, <p(z) satisfies all the remaining
hypotheses of the Abel-Plana summation theorem, Entry 1. However, because
of the singularities, principal values need to be taken for the integrals appearing
in the Abel-Plana formula. We thus will proceed with the necessary calculations
in applying Entry 1 and make the necessary modifications to accommodate our
function <p(z), defined by G.1).
First,
' sin G*<" - D) + 2' sin
= x"~]i
- sin(^Ttn)cot(jfix)).
Thus,
(PV /
JO
fii-n X
-dx= -
Jo e2** - 1
G.2)
Recall that (Gradshteyn and Ryzhik [1, p. 370, formula 3.411, no. 1])
"-' . 1
Hence, from G.2) and G.3),
L
Re a > 1, Reb > 0. G.3)
P"
dx.
We now calculate the contributions from the poles. First considera pole 2nik/p,
where к / 0. On C^M+l/2, we make a semicircular indentation of radius e > О

418 Ramanujan's Notebooks, Part V
in the right half-plane. For к > 0, we thus need to calculate
z"~ldz
lim
ez _ 1)(е-2,т,г _
= lim
- 1)'
G.5)
since a/> = 4л2. Second, we examine the contribution about —2nik/f} for A > 0.
We thus need to calculate
,lino/
= lim I
znXdz
~ We2™ - 1)
_ \\
- 1)'
G.6)
Recall that, in the proof of Entry 6, we needed to take the difference of the integrals
over С% 2ra and C^ 2m. In modifying the proof of Entry 1, we need to calculate
the difference of G.6) and G.5). To that end,
an/2k"-]
G.7)
Lastly, we examine the singularity at z = 0. Since Re n > 2, it is easy to see
that the contributions of the two quartercircular indentations tend to 0 as their radii
tend to 0.
Hence, on the right side of A.1), we must add, by G.7), the additional expression
G.8)
-/ ectk _ ] '
In conclusion, by the modified form of A.1), G.4), and G.8),
^ k"-{ Г(и)?(и) „.,,__чГ(и){(и)
ОС _И-1
37. Infinite Series 419
COS(itf/l),
2
G.9)
Р" ы\ е"к ~ ]'
Multiplying both sides of G.9) by y/fi" and rearranging, we complete the proof.
Entry 8 (Formula B), p. 269). Let x > 0 and -1 < n < 2. Then
' кй + 4л4
- 2cos(\nn)
' .т(л ./2)"-
2
f
Jo
(e2"' - l)(f4 + 4л4)
„-2-T*
- cosBttj-)
(8.1)
Proof. We apply the Abcl-Plana summation formula of Entry 1, but modifications
are necessary because of poles. Let
z"
Observe that <p(z) has simple poles at z = л%/2ехр((тп + 2я|Л)/4), 0 < к < 3.
When к = 0, 3, the poles lie in the right half-plane.
First, a straightforward calculation shows that
(8.2)
Second, setting t = xu*/2, we find that
(p{t) dt =
f
Jo
¦L
/ ^
o и
"-2\„ sec(^n),
(8.3)
by, for example, the calculus of residues or tables (Gradshteyn and Ryzhik [ 1, p.
340, formula 3.241, no. 2]). Therefore, by (8.2) and (8.3),
f
Jo
t"
"'- Dfr4 + 4jc*)(g_4)
Returning to the proof of the Abel-Plana summation formula, or to the proof of
Entry 6, we see that we must modify the proof by accounting for the contributions

420 Ramanujan's Notebooks, Part V
of the poles of
<p(z)
alz- xy/2c\pGii/4) and of
Viz)
at z = Jtv^2exp(—эт|'/4). Denoting these residues by R' and R", respectively, we
find that
R' =
and, by a similar calculation,
„ _
Thus, by the residue theorem, we obtain on the right side of A.1) an additional
contribution of
2ni(R' - R") = -(
я r „_2 /2e2"xcosBnx + \тгп)-2со&(\тгп)\
= ^(*v2)n у i + e*"* _ 2eb" cosB7r;t) /
_ n j- „_2 lcosBnx + \nn) - e'2"* cos(^«)\
= -(jrV2) I coshB^x) - cosBttx) J\
5.5)
Hence, using (8.4) and (8.5) in a modified form of A.1), we conclude (8.1) to
complete the proof.
Entry 9 (Formula E), p. 283). Let x be a positive, nonintegral number, and let
0 < n < 1. Then
oo in
in __я
w/ierc principal values are taken.
7nd7
1-- 1)(г2+л2)
2)
(9.1)
Proof. We shall apply a modified form of the Abel-Plana summation formula
from Entry 1. This modification is necessary because the function in our application
has a singularity at the origin and a simple pole on the real axis. We thus will
indicate the modifications in the proof of the Abel-Plana formula that need to be
made.
37. Infinite Series 421
Let
г"
г"
z + x z - x
By straightforward, elementary calculations,
and
<p(k) =
2za{in-(-i)n)
2k"
V?--x2'
(9.2)
(9.3)
for each positive integer k.
We now indicate the alterations that we mentioned above. For positive integers N
and m, let C^ m denote the positively oriented indented rectangles with horizontal
sides passing through 0 and ±iN, respectively, and with vertical sides passing
through 0 and m + |. The indentations are quartercircles of radius ( about the
origin in the upper and lower half-planes, respectively, and semicircles C+ and
C~, of radius e about x in the upper and lower half-planes, respectively. Because
и > 0, the limits, as e tends to 0, of the integrals around these quartercircles equal
0. The union of C+ and С ~ is a negatively oriented circle of radius € centered at x.
We therefore obtain contributions on the "right side" of the Abel-Plana summation
formula equal to
(е-^'г- l)(z-*)
0 (x +(eie)"-leieiede
W» - l)ee'''
inx"
inx
.n-1
'
<JcotGtx) -!) + («
+ I))
= - nx"
i. (9.4)
Hence, using (9.2)-(9.4) in our modified Abel-Plana summation formula, we
find that
г"
+ 4sin(ijr/i)
Jo
where PV designates the principal value of the integral. However, by (8.3) and a
result in Sanson« and Gerretsen's book [1, p. 133],
/
Уо
OO ,1-1
dz + PV
+ x Jo
/"OO ,1-1
I -rfz = я
Jo z - x
cscGr/i)A"-' - n со1(тгя)л"

422 Ramanujan's Notebooks, Part V
= лх" ' tan(^nn).
(9.6)
Putting (9.6) in (9.5), we readily deduce (9.1) to complete the proof.
Entry 10 (Formula F), p. 283). Let x be positive, and let 0 < n < 1. Then
e2trx _ J
zndz
Г i
Jo (e2"' -
(Ю.1)
Proof. As in the previous proof, we apply the Abel-Plana summation formula
under appropriate modifications.
Let
z + ix z - ix
Then, by elementary calculations,
г2-*2
f &m(\nn)z"
7* -r2
and
2k"
k2 + x2'
A0.2)
A0.3)
for each positive integer*. Also (Gradshteyn and Ryzhik [1, p. 340, formula 3.241,
no. 2]), for 0 < n < 1,
/ (p(z) dz = 2
Ja Jo
sec(ijrn).
A0.4)
In modifying the proof of the Abel-Plana summation formula for the present
application, we take quartercircular indentations around the origin in the two rect-
rectangular contours. By the same argument as in the proof of Entry 9, we get contri-
contributions of 0 when we let the radii tend to 0. We also take semicircular indentations
of radius e, Cf+ and C~, in the right half-plane about the poles г = ±ix, respec-
respectively. On the "right side" of the Abel-Plana summation formula, we then obtain
contributions of
—— + lin
"" (ix +eeie)n-]eiewde
/„-7ni(ix+e exp(iS)) _ |V»iD
' ''m / 2^J(-iJr+6exp(.9)) _ 1\ W
37. Infinite Series 423
ТТЛ"
е2лх
2л-д:
л-
Г
1
7ГДГ""'(-|)"
r i 00.5)
Using A0.2H 10.5) in our modified Abel-Plana summation formula and divid-
dividing both sides by 2, we arrive at A0.1) to complete the proof.
Entry 11 (Formula B), p. 268). Let <p(z) be an entire function. Let CN denote a
positively oriented rectangle with its horizontal sides passing through ±iN and
its vertical sides passing through 2N + 1 and -2N, where N is a positive integer.
Assume that, for p > 0,
lim /
-dz = 0,
A1.1)
j + l)sin(^z)
where f (z) denotes the Riemann zeta-function. Then, if a, fi > 0 andafi = тс2,
„-/¦ L0)
where Bj, j > 0, denotes the jth Bernoulli number.
Proof. We integrate
A1.2)
around the contour C/v. Observe that f(z) has simple poles at г = 1, because
C(z) has a simple pole at z = 1, at z = 2n, for each nonnegative integer и, and at
z = ~2j - 1, for each nonnegative integer j. Note that /(z) is analytic at z = 2л,
when л is a negative integer, because f (z) has a simple zero atz — 2n (Titchmarsh
[3, p. 19]). We next calculate the residues.
First,
Ri =
A13)
since ГC/2) = </тт/2. Second,
(П.4)
since f @) = -i (Titchmarsh [3, p. 19]). Third, for each positive integer л,
л! Dj6)"
2л! /?"
A1.5)

424 Ramanujan's Notebooks, Part V
by @.2). Fourth, using @.3), we find that
(-\y+ln(i-2j-lM-2j-
R-2j-\ =
2Bj)\ П-j
2By)! B/ + 2)
20 + 1)! • A16)
Thus, applying the residue theorem, using our calculations A1.3)-( 11.6), letting
N tend to oo, and employing A1.1), we find that
=0.
Letting и = j + 1 in the latter sum, multiplying both sides by —2a~'/4, using the
hypothesis afi = jt2, and rearranging, we complete the proof of A1.2).
Entry 12 (Formula E), p. 269). lfn is any positive integer, then
oo r.4n
?f sinh2Grfc)
<„ / в oo i.4n-l \
~ я \ 8« ?- e2"k -\J
Entry 12 was communicated in Ramanujan's 110, p. xxvij first letter to Hardy
and was later proved by С. Т. Ргеесе [1] in 1928. The first proof in print, however,
appears to be by M. B. Rao and M. V. Aiyar [1] about four or five years earlier. Б.
Grosswald {2] and the author [6J have also found proofs.
Entry 13 (p. 273). lfn is an odd positive integer, then
^coth(TTfc) 1 /3 .. „ я .. Л 22"-2
where
270
3
2'
1
«20 =
191
2310'
2730'
907
1J4 = ,
2 147
1
340'
Proof. This entry is a sequel to Entries 25(i), (ii) of Chapter 14 (Part II [2, p.
293]), where only the first two cases are presented. All six evaluations above can
be deduced from the general formula (Part II [2, p. 293, eq. B5.3)])
1 cothGrfc) = 22пл-2п+| YVl
* " «r=0
I B2k B2n+2-2k
BJt)! B/1 + 2
A3.1)
37. Infinite Series 425
where n is a positive odd integer, and B,, j > 0, denotes the j th Bernoulli number
It is now routine to check that for n = 1, 3, 5,7,9, and 11 Ramanujan's claims
agree with A3.1).
For references to proofs of A3.1) and special cases thereof, see Part IIГ2 d
293]. 'P'
Ramanujan writes the next entry in terms of a function <p "defined" by
A4.1)
«=o
This series does not converge for any finite value of г, and so <p is not well defined.
We will therefore find a function <p which has an asymptotic expansion, as z tends
to oo, given by A4.1). Several functions may have the same asymptotic expansion,
but the function <p defined below will be shown to satisfy Ramanujan's claim.
Define, for г > 0,
<piz) := viz,x) =z I
Jo
where x is any complex number. Now,
/, , .w ^ Mni-t)"
oo o-n
e~lz dt
A4.2)
n=0
n\
Id < i.
Applying Watson's Lemma (Olver [1, p. 71]), we find that, as z tends to oo,
Viz)
n=0
which agrees with A4.1).
Entry 14 (Formula A), p. 276). Let <p(z, x) = <piz) be defined by A4.2). Then
for 0 < Re a < 1, and any complex number x,
I , ^ (-l)n-|Q2"CBn)
2n — x
Tn
A4.3)
Proof. Using A4.2) and inverting the order of summation and integration by
absolute convergence, we find that
„—2лno,
2n
о f°
I
g — 2nnat
(I +0
-dt
t)
r00 dt
= ita I
Л A +/)-Г(е2ли('+|)- 1)'

426 Ramanujan's Notebooks, Part V
Thus, A4.3) is equivalent to the formula
J
2l4Bи)
— x
па
f°° dt_
Jo а+ОЧе2™
A4.4)
f
We now temporarily add the restriction, Re x < 0. Then
dt Г°° dt
Гх dt Г1 dt
~ Jo t*{e2«al - 1) ~ Jo tx(e27"" -
1)
A4.5)
by a well-known representation for f (s) (Titchmarsh [3, p. 18]). Employing the
functional equation @.4) of <(z) in A4.5), and then using A4.5) in A4.4), we see
that we are required to prove that
ла _ J_ уЧ (,-ir a-(,
x - 1) ~ 2x ?-! 2n-x
o t*(e*"»-l)
=0, (.4.6)
where Re x < 0.
Using the generating function @.1) for the Bernoulli numbers and inverting the
order of integration and summation by absolute convergence, we find that, for
Re x < 0 and 0 < Re a < 1,
f' dt _ 1 Г1 _{_x f^ IWnat)"
7o t4e2»°>-\) 2 Jo ^ n\
BnBna)n 1
n\ n — x
ita I i
ла
-dt
i
n=0
2(x - 1) + 2 f^ Bи)!Bи - x)
by Euler's formula @.2). Hence, we have proved A4.6) for 0 < Re a < 1 and
Re л < 0. Using analytic continuation, we find that A4.3) holds for all complex
x.
On pages 277-278, Ramanujan records five enigmatic formulas, all of the same
type and each apparently arising from Eisenstein series. These formulas are num-
numbered (9), A0), A1), A2), and A4) in a long list of results, but the remaining ]
37. Infinite Series 427
formulas in this series have no relation to these five formulas. All five formulas
involve the Bernoulli numbers В„. For each of the next five entries, we precisely
quote Ramanujan, even though his convention for Bernoulli numbers is different
from ours in @.1). We then determine those values of n for which the proposed
claim might have validity. For each formula, there is a sequence of values of n for
which the result is classical; in most cases, the theorem can be found elsewhere in
Ramanujan's notebooks. After discussing the classical case, we reformulate Ra-
Ramanujan's claim for those values of л for which Ramanujan's claim is completely
new. We lastly, in each case, prove the new theorem.
Proofs of Entries 15-19 were first proved in a paper with P. Bialek [1].
Entry 15 (Formula (9), p. 277).
ft"
\Bn\ \В„\ (лп\ I 1
= cos I — I \
2n n V 4 / I 2"/2
5«/2
2cos(rctan '5) 2cos(ntan~' \)
10"/2
13"/2
A5.1)
"where 2, 5, 10, 13, ... are sum of squares of numbers that are prime to each
other."
In Ramanujan's convention all the even indexed Bernoulli numbers are positive,
in contrast to the usual definition in @.1), and so we have inserted absolute value
signs around his Bn 's.
It appears to be difficult to discern the pattern in the numerators on the right side
of A5.1), but a natural pattern will emerge in the proof below. If n is a positive
odd integer exceeding 1, В„ =0, and so A5.1) cannot be valid, as the left side of
A5.1) is positive. Numerical calculations indicate that A5.1) is apparently false
if n — 1, even if we assume that B\ = 5, instead of — | from @.1). We thus
conclude that Ramanujan apparently intends n to be an even positive integer.
Let л = 4m + 2, where m is apositive integer. Then A5.1) reduces to the claim
E
k=\
k4"'
B.
Am+2
еЪк - 1 8m+ 4'
A5.2)
Indeed, A5.2) is correct. To the best of our knowledge, A5.2) was first proved by
J. W. L. Glaisher [1] in 1889, although an equivalent formulation was established
in 1881 by A. Hurwitz [1], [2] in his thesis. Moreover, A5.2) is found in Section
13 of Chapter 14 in Ramanujan's second notebook [9, p. 171]. For proofs of other
generalizations of A5.2) and for references to the many proofs of A5.2) that can
be found in the literature, see our paper [6] and book [2, pp. 261-262J.
In concluding our discussion of A5.1), we remark that the instances when
n = 0(mod4) appear to be the only ones remaining for which A5.1) may be
correct and new. Indeed, A5.1) is then of great interest, for in these cases a curious

428 Ramanujan's Notebooks, Pan V
infinite series appears on the right side of A5.1), and there are no comparable
results in the literature.
Before we state Theorem 15.1 it is necessary to provide a discussion about
solutions to
gcd(c,d)=l.
A5.3)
where gcd(c, d) denotes the greatest common divisor of с and d. We partition the
solutions of A5.3) into three classes. First, suppose that с Ф d and cd Ф 0. Then
each solution (c, d) of A5.3) generates eight solutions, namely,
±(c, d).
±(c,-d), ±(d,c), ±(d,-c).
(i)
The case с — d — 1 generates four solutions, namely,
±A,1), ±A.-1). (ii)
There is one further case, namely, с -\,d = 0, which generates the four solutions
±A,0), ±@,1). (Ш)
We shall say that the solutions (c\, ^i) and (c2, d2) of A5.3) are distinct if they
do not simultaneously belong to the same set of eight solutions in (i), or the same
set of four solutions in (ii), or the same set of four solutions in (iii).
Recall that solutions (c, d) of A5.3) exist if and only if the prime factors of I
are all of the form 4k + 1, except for the prime 2, which may occur to at most
the first power (I. Niven, H. S. Zuckerman, and H. L. Montgomery [1, p. 164]).
Although not needed in the sequel, we also recall (G. H. Hardy and E. M. Wright
[1, pp. 241-242]; Niven, Zuckerman, and Montgomery [1, p. 167]) that if r(l)
denotes the number of representations of I as a sum of two squares, then
Of course, in A5.3), we have imposed the restriction gcd(c, d) = \.
Theorem 15.1. Let m be a positive integer. Then
0? i,4m-l n. /_11>"R.
8m
4m
I2
c+d! \
)
A5.4)
where the sum on the right side of A5.4) is over all integers I > 2 representable
by A5.3), and where for each I, the sum is also over all distinct solutions (c, d)
ofA5.3).
Observe that the four terms in curly brackets on the right side of A5.1) arise
from A5.4) when с, Л = 1, 1; 2, 1; 3, 1; 3,2, respectively.
Proof. First,
oo 1,4/n-l
k=\
37. Infinite Series 429
A5.5)
where ak(n) = Yld\n dk- Now if n is an integer with n > 2, and if Im(r) > 0
(Rankin [1, p. 194, eq. F.1.4)]),
¦ dJn'
A5.6)
(c,d)=l
where the prime / on the summation sign indicates that the term with с = d = 0
is omitted from the summation. Thus, by A5.5) and A5.6),
к
B
um
A5.7)
Let tan ' z denote the principal branch of the inverse tangent relation, i.e.,
|argw| < я/2. Then
(ci + dy*" = Г7" exp(-4m/ tan (c/d)).
A5.8)
where I is defined by A5.3). Since the sum on the right side of A5.7) converges
absolutely, we can rearrange the terms in any order. So, we group terms according
to increasing values of I.
We now sum the terms in each of the cases (i)-(iii), described prior to the
statement of Theorem 15.1. For fixedc,rf, withc,</ > 0, the eight terms in A5.7)
for case (i) equal, by A5.8),
(ci + d)*" ' (-ci + d)* ' (di + сL """ (-di + cL
= 4ReI -
since
{(ci + d)*" ' (di + cL
= 4Г2га {cos Dm tan (c/d)) + cos Dm tan (d/c)))
= U~2m cos Dm tan'1 (c/d)),
tan"
+ ran'1 (d/c) = я/2.
We now show that
cos Dm tan (^ )) = (-1)'" cos ( 4m tan
A5.9)
A5.10)
(]5.il)

430 Ramanujan's Notebooks, Part V
Observe that
tan
Hence,
c-d
= 1.
tan
-i
-l + tan ) = -+kn,
A5.12)
for some integer k. Thus, A5.11) follows easily. Hence, by A5.9), the sum of the
eight terms in case (i) equals
A5.13)
\
In case (ii), the sum of the four terms equals
2 2
+
= 4 Re
A + iL™ A -
In case (iii), the sum of the four terms equals
4.
Using A5.13M15.15) in A5.7), we deduce A5.4).
22m
A5.14)
A5.15)
Entry 16 (Formula A1), p. 278).
ABn\
—
2 cos (n tan"' A) 2 cos In tan"' 4)
10"/2
7A6.1)
As before, it appears that Ramanujan intends n to be an even positive integer.
If n = Am + 2, where w is a positive integer, A6.1) reduces to the evaluation
A6.2)
sinh(&7r)
This result is due to A. Cauchy [1, p. 362] and is also found in Glaisher's paper
[1]. Many proofs of A6.2) can be found in the literature; see the author's paper [7,
p. 337] for a list of several authors. If n = 2, A6.1) and A6.2) are false. In fact,
"' к 1
sinh(fctf) 4tt '
a result also due to Cauchy [1, p. 361]. See the author's paper [7, p. 337] for
another proof and references to further proofs.
37. Infinite Series 431
For other generalizations of A6.2), see our paper [7] and book [2, pp. 294-295].
If n = 0 (mod 4), A6.1) is new, and we state a precise version of this in the
next theorem.
Theorem 16.1. Let m be a positive integer. Then
EV , =(-'
Z_/ ekn _ е-кл
k=\
Am
2 cos Dm tan (^
«even
t
I
A6.3)
where the sum is over all even positive integers ? > 2 that can be represented by
A5.3), and, for each fixed ?, the sum is also over all distinct odd solutions (c, d)
ofA53).
Note that the second and third summands on the right side of A6.1) arise from
the terms with с = 3, d - 1 and с - 5, d = 1, respectively, in A6.3).
Proof. First,
oo /•_i\*-li,4m-l oc oo
Y, J! *-,. -ED
*=l e e k=\ r=0
where
A6.4)
A6.5)
E
n/dodd
Second, we repeat the argument made in the proof of Theorem 15.1 with the
added stipulation that the indices in the Eisenstein series be odd. We thus find that
(c.d)=\
c.d odd
2m
, A6.6)
where the summation on the right side of A6.6) is over all even I > 2 that are
representable by A5.3), and where, for each fixed ?, the sum is also over all distinct
solutions (c,d) satisfying A5.3).
Third, we need an analogue of A5.6), where in the Eisenstein series on the right
side of A5.6) we sum only over odd с and d. To that end,
¦dr)
,2n
c.d odd
U:d)=\
L-.d odd

432 Ramanujan's Notebooks, Part V
37. Infinite Series 433
= t;Bn)(l-2-2n)
'(г.<Л=1
c.d odd
Put
Then
oo .
= у !
oo oo
E'
„
c.dodd
= E - E' - E
с even Jeven c.d even,
- G2n(r) - вгА
Writing A5.6) in the form
= 2f Bи)
where
we deduce from A6.8) that
1
2">
- 2n f 2?Bи) + Дь, ?СГ2»
Дл, ?а2я_,(*) {A +2-2л)е2
A6.8)
t=i
= D2n
A6.10)
where
and
Since, from @.2) and A6.9),
if it is even,
if к is odd,
if4|Jt,
otherwise.
we conclude from A6.7) and A6.10) that
e"*T = B2n - 1)^
A6.11)
c.dodd
where
*2»-i<*) = -2""
а2„-\(к),
-B2n + l)cr2n_1(Jt/2)
-d2" + \)a2n-[(k/2)
if*= 1,3 (mod 4),
, if it = 2 (mod 4),
A6.12)
+ 22яа2н-Лк/А) + a2n.i(k), if ft a 0 (mod 4).
We now return to the definition of /2,,_i {k) given in A6.5) and relate it to the
definition of hin-\(k) above.
First suppose that к is odd. Then k/d is odd foT all divisors d of it. Hence,
rf|t d\k
by A6.12).
Second, suppose that к = 2 (mod 4). If Jt/rf is odd, then d is even. Thus,
Write d = 2di. Then
d\k
d even
E
A6.13)

434 Ramanujan's Notebooks, Part V
Since к = 2 (mod 4), it follows from the standard product formula for а„(к)
(Hardy and Wright [1, p. 239]) that
Thus, by A6.12),
hm-Лк) = (-22" - 1 +22'-1 +
by A6.13).
Third, suppose that к = 0 (mod 4). Then,
= - E
rf|*
i/rfodd
= -22"-V2,,_,(fc/2) = />,,_,(*),
E <*2"
rfl*
t/deven
Define the integer a > 2 by 2" \\k. Then by the aforementioned product formula
iovaln-\(k).
A6.14)
On the other hand, from A6.12), when к = 0 (mod 4),
Ab,-i(*) = - B2" + l)aa,-i<*/2) + 22" f2,.,^,)
\ L I
by A6.14). .
In conclusion, for all fc we have shown that h^-xik) - /ъ<-\(к)- Using this
fact in A6.11), setting г = i and n = 2m, and combining A6.11) with A6.4), we
find that
* = 1
R,
16m
A6.15)
<-,J cxld
Combining A6.6) and A6.15), we complete the proof.
37. Infinite Series 435
Entry 17 (Formula A4), p. 278).
~ (_!)*-!*—1 \En\
cos
- + cos ( ^r
2 V 3
2 cos (W tan-'^
__
3"/2
A7.1)
As before, Ramanujan evidently intends n to be even. Thus, set n — 2m. If
^ 0 (mod 3) and m > 1, then
2m
COS
3 / 4m'
A7.2)
This result was apparently first established in print by ML B. Rao and M. V. Aiyar
in papers published in 1923-1924 [1], [2]. Thus, even the special case A7.2) was
first proved by Ramanujan, although he never published a proof. See also Berndt's
paper [6, p. 157, Prop. 2.8], which also contains some generalizations of A7.2).
If m = 1, A7.1) and A7.2) reduce to the claim
^ (-!)*-'& B2 _ 1
24'
B2
4
A7.3)
Now A7.3) is false. In fact (Berndt [6, p. 159, Prop. 2.13]),
(-I)**
1 1
24 4л-л/з'
which was also first established by Rao and Aiyar [1].
Thus, it remains to prove A7.1) for n = 0 (mod 6).
Before stating Theorem 17.1, we need to offer some remarks about the solutions
of
I = c2 - cd + d\
A7.4)
gcd(c,d) = \.
We consider three cases.
First, suppose that с Ф d, cd Ф 0, and that B,1) docs not appear in the list
immediately below. Then each solution (c, d) generates 12 solutions, namely,
(i)
±(c,d), ±(d,c), ±(c-d,c), ±(c,c-d), ±(d,d-c), ±{d-c,d).
Second, for 1 = 3, there are only six solutions, because, for example, if с =
2, d = 1, then с -d-d, and so (c, d) and (с, с - d) are not distinct. The six
solutions are
(h) ±B.1), ±A.2), ±A,-1).
Third, for I = 1, there are again only six solutions, namely,
(Hi) ±A,0), ±@,1), ±A,1).

436 Ramanujan's Notebooks, Part V
We shall say that two solutions (ci%d\), (сг, d2) of A7.4) are distinct if they
are not both simultaneously in any of the three solution sets given in cases (i)—(iii>
above.
Those integers I that can be represented by A7.4) have the representation
where a — 0 or 1, the primes pj are distinct and have the form 3^ + 1, and a, is
a positive integer, 1 < j < r (Niven, Zuckerman, and Montgomery [1, p. 176]).
Although not needed in the sequel, we note in passing that if r(t) denotes the
number of integral solutions to I = c2 - cd + d2, then by a theorem of Dirichlet
[1],
where </y,j(Oij = 1,2, denotes the number of divisors of ? of the form 3k + j.
Theorem 17.1. Let m be a positive integer. Then
6m 4m
[J i c' J A7.5)
where the sum is over all integers I > 1 that are representable by A7.4), and
where, for each fixed I, the sum is also over all distinct solutions (c, d) of A7.4).
Proof. First, if со = expB^i/3),
_ _ ^
k=\
\ _ e2niwk
r.6m-1 2niu>kr
*=1 r-\
л=1
1
12m ' 24m f ^^ (ceo + dNm '
C(c.d)=l
by A5.6).
Next,
(ceo + dN™ = tim exp I 6mi tan"'
A7.6)
A7.7)
where I is given by A7.4). We shall group terms in A7.6) according to increasing
values of I. This rearrangement is justified by the absolute convergence of the
double series in A7.6).
37. Infinite Series 437
We consider first the 12 terms on the right side of A7.6) that arise from a typical
value in case (i). This sum of 12 terms equals
2 2 2
(ceo + dN™ (dco + сN ((с - d)eo + сN"
2 2 2
(ceo + (с - <*))<™ + (dco + (d- c)Nm + ((d-c)eo + d)("'{ll g)
Observe that
tan tan
,/cV3\
+ tan"
cVb
Id - с + 2c - d
1 -
3cd
Bd - c)Bc - d)
Thus,
tan
-1
ш
л
for some integer j. Hence,
arg(c<» + dfm + xg(dco + cNm =
for some integer к. Therefore, (cw+dy6™ and (dco+c)'6™ are conjugates. Using
A7.4) and A7.7), we find that the sum A7.8) equals
-6mi tan"
4
/Зш
+ COS
A7.9)
By a calculation similar to one above, we find that
tan
-1
for some integer j. Hence, the sum in A7.9) equals
с
,2d
¦4-c))-
A7.10)

438 Ramanujan's Notebooks, Part V
Another elementary calculation shows that
_,
tan ' I — I -tan 4 —= I = -- + jn,
\2d-cJ \V3(rf-c)/ 6
for some integer j. Thus, from A7.10), the sum of our 12 terms equals
For case (ii), a similar argument shows that the six terms total
6(-l)m3-3m.
For case (iii), an easy argument shows that the six terms total
6.
Using A7.11H17.13) in A7.6), we find that
A7.11)
A7.12)
A7.13)
2- ~k«S _(_
i)*
,
24w
3*"
+ ,2(,1ГЕ;|
< * JA7.14)
where the sum is over all ? > 7 that can be represented by A7.4), and where,
for fixed I, the sum is also over all distinct solutions of A7.4). The desired result
A7.5) now follows upon simplifying A7.14).
Formulas A0) and A2) arise from a different Eisenstein series. We thus begin
by deriving an analogue of A5.6). An alternate proof may be obtained by using
Fourier series of Eisenstein series of level 4 (Schoeneberg [1, pp. 154-157]).
Lemma 18.1. Let lm t > 0, and let n be a positive integer exceeding I. Then
= i22"-2B2"-l)^
Proof. Recall the partial fraction decomposition
„,мтB*+1)/2.
*=O
37. Infinite Series 439
Differentiate both sides with respect to г a total of 2n - 1 times to obtain, after
some rearrangement,
00
*=0
In
Replacerby Bj + \)z, multiply both sides by (— 1)-*and sum on j,0 < j < oo.
We then deduce that
oo oo
y^yv \)'+kBk + 1J"-1ел'BУ+
;=0 *=0
G
(
^Z "zJ + i)T+ ZK l} A8.1)
Note that the summands (-l)/+(r"'/(Bj + l)r +2ifc- IJ" in A8.1) are invariant
upon the replacement of j by -j - 1 and fe by -Jfc + 1. We also observe that the
double sum on the left side of A8.1) can be rewritten in terms of а-щ-1 • Thus, we
may rewrite A8.1) in the form
r=Q
/ 0
= i,-(-l)-'Bn-l)!(-
*~« «2J +
/2Ч2"
= i|(-l)"-'Bn - 1)! ( - j A - 2~2")CBn)
j.k— — oo
IJ
Upon using Euler's formula @.2), we complete the proof.
Entry 18 (Formula A0), p. 278).
)*B*+ I)"
^cosh{B*+lOr/2}
= 242- -1I*1 sin (^
n V 4
1 2cos(ntan"' \) 2cos(ntarr'2)
IJ"
A8.2)
As with the previous formulas, most likely, Ramanujan intended n to be an even
positive integer. If we set n = 4m, where m is a positive integer, in A8.2), we find

440 Ramanujan's Notebooks, Part V
that
= 0.
A8.3)
This result was first proved by Cauchy [1, pp. 313, 362]. Ramanujan [2), [10,
p. 326] offered A8.3) as a problem to the Journal of the Indian Mathematical
Society. In Section 14 of Chapter 14 in his second notebook [9], Ramanujan
recorded A8.3) as a corollary of a beautiful, more general theorem (Part II [2, p.
262]). For references to the many proofs of A8.3) and statements and proofs of
more general theorems, see the author's paper [6, pp. 176-178] and book [2, pp.
261-262].
The next theorem gives a precise version of A8.2) when n = 2 (mod 4).
Theorem 18.1. If m is a positive integer, then
= (_iy»
B
- 1)
4m+2
l ) A8.4)
where the summation on the right side of A8.4) is over all even positive integers
? > 2 that are representable by A5.3), and where, for fixed I, the sum is also over
all distinct pairs (c, d) satisfying A5.3).
It is easily checked that the second and third displayed terms in Ramanujan's
formulation A8.2) arise from the terms when с = 2>,d = 1 and с = 5,d — 1,
respectively, in A8.4).
Proof. We first transform the left side of A8.4). To that end,
*=0
*=0
it=0 n=0
r=0
-Br+l)R/2
(_!)>+*
4Dm + 2)
В4т +2
)i + 2*
\4m+2
(-D(C
4Dm + 2) с^ж (ci + d)*"*2'
U.d)=l
c.d odd
A8.5)
37. Infinite Series 441
by Lemma 18.1 with r = 1 and n = 2m + 1. The sum on the right side of A8.5) is
similar to the Eisenstein series in A6.6). The only differences are that the power
of (ci + d) is 4m + 2 instead of 4m, and that the series above contains the extra
factor of (-l)^)/2 in its summands.
We now consider cases ОМ"), as we did in the proof of Theorem 15.1.
The sum of the eight terms in case (i) equals
(di
(-di
((_l)(<-+ (J)
(ci + d)*"+2 + (di +cLm+2)
= 4i(-l)((+rf)/2r2/"-'lm (exp (-/Dm + 2)tm-\c/d))
+ exp (-1 Dm
+ sin(D/n + 2)tan-'W/c))}
= -8/(-1)(г+<"/2Г2т-' sin(Dw + 2)tan-'(c/d)),
by A5.10). Using A5.12), we find that
sin(Dm + 2)tan"l(c/rf)) = (-l)'"cosH4»i + 2)tan-1 (C~-\\ .
Thus, the eight terms in case (i) have the sum
-8i(-l)"'+(c+</)/2r2m-1 cos (Dm + 2) tan (^^
to
(I8-6)
Next, by the same type of reasoning, we find that the four terms in case (ii) sum
A8.7)
2)л-/4]=4((-1)"'2-2т-1.
Putting A8.6) and A8.7) in A8.5), we conclude that
_ _
4Dw + 2)
where the summation on the right side is over all even positive integers t > 2 that
can be represented by A5.3), and where, for each fixed ?, the sum is also over all
distinct solutions (c, d) of A5.3). The theorem now follows after a small amount
of simplification.

442 Ramanujan's Notebooks, Part V
Entry 19 (Formula A2), p. 278).
2cosGrn/6)
A9.1)
As before, we assume that Ramanujan intended n to be an even positive integer.
If n = 6m, where m is a positive integer, then we deduce from A9.1) that
\6m— 1
= 0.
A9.2)
The evaluation A9.2) was first achieved by Cauchy [1, p. 317]. Ramanujan re-
recorded A9.2) as part of Entry 18(iii) of Chapter 17 of his second notebook. For
two proofs of A9.2), see the author's paper [7, Corollary 7.6] and book [3, pp.
140-141]. For references to other proofs, generalizations of A9.2), and further
results of this sort, see the last two cited references.
Before stating Theorem 19.1, we need to say a few words about solutions to
? = 3c2 + d2,
gcd(c,</) = 1.
Each solution (c, d) of A9.3) generates four solutions, namely,
±(c.d), ±(-c,d).
A9.3)
A9.4)
We shall say that (cudi) and (c2,d2) are distinct solutions to A9.3) if they
belong to different sets of four solutions given by A9.4). We remark that the
number of positive odd solutions (c, d) to 4л = 3c2 + d2, where n is odd, equals
d\.i(n) - </2.з(п) (L. K. Hua [1, p. 309]).
Theorem 19.1. Let mbea positive integer such thatm ф 0 (mod 3). Then
^
о cosh I Bk + 1OГл/3/2
2m
A9.5)
where the sum on the right side of A9.5) is over all even positive integers I which
can be represented by A9.3), and where, for each fixed I, the sum is also over all
distinct solutions (c,d)ofA9.3).
37. Infinite Series 443
Proof. By a calculation like that in A8.5) and by Lemma 18.1 with x = (V3, we
deduce that
\2m-l
*=o cosh I Bk
= i2MBlm-l) —
Sm
«27
2k
A9.6)
< -, J odd
Now,
(ciy/Ъ + dy2 = Гт exp {-2mi
A9.7)
where ? is given by A9.3). We group terms according to increasing values of I.
The sum of the four terms arising from A9.4) equals, by A9.7),
— 4( Im
= -4i(-\)(c+J)/2rnsin
Thus, from A9.6) and A9.8), we complete the proof of Theorem 19.1.
A9.8)
Comparing A9.5) with A9.1), we see that the trigonometric sums on the right
sides have quite different shapes. The first three terms in A9.1) evidently arise
from the values c, d = 1, 1; 1,3; 1,5, respectively. However, we note that 28 has
two representations, 3 • I2 + 52 and 3 • 32 + I2. Easy calculations show that the
first two terms on the right side of A9.1) agree with the first two terms on the
right side of A9.5). However, there is a discrepancy between the third terms. This
discrepancy exists if we take either term arising from the two representations of
28, or if we take the sum of the two terms, from our sum A9.5).
We shall now establish an alternative representation for the sum on the right
side of A9.5). This will give a result which is "close" to that of Ramanujan and
perhaps indicate where Ramanujan erred. An elementary calculation shows that
tan
+ tan"
3c + d

444 Ramanujan's Notebooks, Part V
and so
tan
-i
-I-tan
л
, - - +кл,
гс + d / з '
for some integer k. Thus,
sin I 2m tan
I —- I I = si
sin | 2m \ — - tan
_,
- c)
Bmn\ ,Л
— sin I —— I cos ( 2m tan
; I 2m tan I
3c
- cos
Bтл
= (-I)
m-\
Thus, we obtain "half" of what Ramanujan probably found, because for c, d =
1,1; 1,3; 1,5, the first term on the far right side of A9.9) yields precisely the
trigonometric functions given by Ramanujan in A9.1), except for an additional
factor of 2 in the second and third terms in A9.1). For с = d = 1, the second
term on the far right side of A9.9) vanishes. For c, d = 1, 3, a simple calculation
shows that the first and second terms on the far right side of A9.9) are equal, while
for c, d - 1,5, the second term does not equal the first term.
Entry 20 (Formula A5), p. 278). Ifn is a positive integer, then
/ ft — I \
~~t Гп ~~ 7Г I \1tl *—* А**Л> — (—Wk ] '
where B;, j > 0, denotes the jth Bernoulli number.
Proof. We shall easily show that Entry 20 is equivalent to a result of the author
[6, p. 163, Cor. 2.22], namely,
+
_1 = 2t. B0.1)
12n ??sin2(ibrp) 12n'
where n is a positive integer and p = ехрBя«/3). By a straightforward calcula-
calculation,
8т2(*,гр) = ?(i _ cosB7r*,o)) = i(l - (-D* cosh(**>/3)).
Using this and elementary manipulation in B0.1), we complete the proof.
37. Infinite Series 445
Ramanujan's statement of Entry 20 does not contain an equality sign.
Ramanujan next offers two puzzling transformations for doubly exponential
series. We shall state them exactly as Ramanujan wrote them and then reformulate
them.
If afi = lit, then
4 *-«-=¦
*=o
where
2 + L k\
, B1.1)
Second, if afi = л/2, then
аУ^(-1)*е~"е' "" = a \-
?—' I 2
where
B1.3)
B1.4)
In B1.1), у denotes Euler's constant, and, in B1.2) and B1.4), Bh j > 0,
denotes the jth Bernoulli number. We emphasize that in Ramanujan's notation,
all even indexed Bernoulli numbers are positive.
The definitions of(p(P) and ir(fi) given in B1.2) and B1.4), respectively, are
certainly enigmatic. Appearing in the arguments of the trigonometric functions
are apparently asymptotic series as fi tends to oo. Thus, the definitions of (p(f))
and $(&) are imprecise, and so Ramanujan's claims are unclear. Nonetheless,
we shall show that B1.1) and B1.3) are correct, if B1.2) and B1.4) are properly
interpreted.
The work on Entry 21 which follows was done jointly with J. L. Hafner [2].
We begin by defining functions G(f)) and B{fi) by
Hip" + 1) =
= (ipyfi+l/2e-ifij2*e-iBll'). B1.5)
Then (Whittaker and Watson [1, pp. 252-253], as 0 tends to oo,
and
G{fi)
139
571
288/32 5184O(«/9K 2488320^"
B1.6)
Ramanujan less explicitly gives the asymptotic expansion for B(fi) in the argu-
arguments of the trigonometric functions in B1.2) and B1.4).

446 Ramanujan's Notebooks, Part V
We now state rigorous formulations of B1.1) and B1.3), and then immediately
show that Ramanujan's aforementioned claims are consequences.
Entry 21 (Formulas D), E), p. 279). Let n, a, and fi be positive with afi = 2jr.
Then B1.!) holds, where
= Щ'~ят {sin (p log i - 0 + ^ Re {G(P))
B1.7)
as p tends loco.
Let n, a, and p be positive with afS = л/2. Then B1.3) holds, where
= " Me~"m (cos f/»log - " /? + 7)
у Р IV" 4/
- sin (fi\ogP--p + *L\ im {G(P)}
B1.8)
as P tends to 00.
We first show that Ramanujan's definitions B1.2) and B1.4) are compatible
with the far right sides of B1.7) and B1.8), respectively. As p tends to oo,
./- * - cos (pbg?-fi-l- Bip))
V p sinh(jr0) \ n 4 /
log ^ - p + \ -
37. Infinite Series 447
Thus, B1.2) and B1.7) are in agreement. The argument showing that B1.4) and
B1.8) agree is similar. We now proceed to prove Entry 21.
Proofs of B1.1) and B1.7). First,
f-fk\(e*«-l) ~
,-kja
k=i
k=[
1=0
] = \ * = 1 j = \
Thus, the proposed identity may be written in the equivalent form
oo
k=\
B1.9)
Second, we apply the Poisson summation formula (Titchmarsh [2, p. 60]) to the
function
f(x):=e-"e' + e-"e" - 1. B1.10)
Observing that /@) = 2e~" - 1, we find that, for a, p > 0 with ap = In,
a U
Be-" - 1) +
f(x)
+
f(x
- l
dx.
B1.11)
JO ~ JO
Comparing B1.9) and B1.11), we see that it remains to prove that
OO /-OO ОС yOO
-y -Iogn + 2J2<P(W) = / fix)dx+2y^f f(x)cos(kpx)dx.
k=\ JO t_| ^0
B1.12)
Observe from B1.10) that fix) is even. Setting и = e*, we find that
/ f(x)dx = - I fix) dx
Jo ? J-oa
= -/ (e + e ' — 1) —
2/o «
1 / f in 1 — ?~nu r°° ^—"»
2 V Л) « Л/л "

448 Ramanujan's Notebooks, Part V
+ / du - / du
Jo u J\/n u /
1 / /"' 1 - p~x Г°° e~x
= U-\ l—L-dx+f e—dx
2 V Jo x Ji x
o X
Since (Part I [1, p. 103])
у = I dx - I dx,
Jo x Ji x
we find that
/•oo
/ f(x) dx = -[-
/ dx - / dx - —]
Ji X Jo X Jo X J
= \y-y-Y--
= - у - log n.
Using B1.13) in B1.12), we find that it suffices to prove that
f(x)cos(kf}x)dx.
k=\
B1.13)
B1.14)
Set
/(x)cos(/»x)</jc.
By B1.14), it now suffices to prove that Itf) = <p@), where <p(fi) is defined by
B1.7). Letting и - ex, we find that
7 = 1 Г f(.x)coS@x)dx=l- Г (e-nu + e~"/u - l)c^
2 J-oo l Jo
Integrating by parts, we find that
= — Г
2^ Уо
= — ( / e~"" sin(^ log u) du - I —5- sin(/6 log м) rf« )
37. Infinite Series 449
say. Setting / = \/u in /2, we deduce that /2 = -/,. Hence,
P P
1))
Hence, / = /(K) = <p@), by B1.7). This completes the proof of B1.1).
Lastly, from B1.5),
9@)
= jlm Ш)]/2
Hence, the second equality in B1.7) follows, and the asymptotic formula follows
by using B1.6).
Proofs of B1.3) and B1.8). First,
v^ (-1)*"
1\*„< ос / \\к„к оо
-\) n ^—> (.— 1; n ^—«
or i e-ka\ ~ 2^i Ь\ Ла L~i
j=o
Thus, the proposed identity B1.3) can be recast in the form
00 00
(-1
B1.15)
Next, we apply the Poisson summation formula for Fourier sine transforms
(Titchmarsh [2, p. 66]) to the function

450 Ramanujan's Notebooks, Part V
Thus, for a, P > 0 and afi = тг/2,
*=o
B1.16)
We recall that
()*
B1.17)
Using B1.16) and B1.17) in B1.15), we find that it suffices to prove that
00
1
B* + l)P
)¦
Set
-.
By B1.18), we now see that it suffices to prove that 1@) = ^(P), where ф(Р) is
denned by B1.8).
Setting x - eu and integrating by parts, we find that
/ = Г (e-ne' - e-"e ' + l) sin(^x) dx--
f°° du 1
= / {e~nu - e-"'u + 1) sin@ log к) -
Jo u P
= -- Г (e~"U + \e""") cos(^logM) du
P J\ V u )
= ~ [ (e-'" + \e-"") cos(^logr) dt,
p Jo \ ' /
where we set и = 1/r. Hence,
t = ~ / (e-'" + ^e-'I/'Jcos(^logO^ = - — (/1+/2),
say. Letting / - l/м in /2, we easily find that /2 = Л. Consequently,
/ = -?/,= -J He-"'costflogt) dt
P P Jo
37. Infinite Series 451
1
Yp
= 11е{яГ(#+1)}.
Thus, we have shown that I(P) = \lr(p), by B1.8). This completes the proof of
B1.3).
The remaining two claims in B1.8) follow as in the proof of B1.7).
Entry 22 (Formula B), p. 280). Let, as usual,
0<x,
= Г'(х)/Г(х). Then, if
Пч log*
3 б
— x
x2) 2 fa sinh2Grn)
sinBnkx) з ^~
Ic2+n2 " fa k(k2 +
kx)\
n1))'
B2.1)
Using Entry 8 of Chapter 30 (Part IV [4, p. 374]), we obtain a formula for
Euler's constant y, which is equivalent to B2.1), namely,
1
7Г log|2sin(;rx)|
-E
л ^ log \n4 - x4\
2fa sinh2(*n)
00 cosB7rfo)\
ti
+n2
,3 \P
*=1
In collaboration with J. M. Borwein and W. Galway, the author originally proved
B2.2) by showing that the derivatives of both sides of B2.2) equal 0 and then letting
x tend to 00 to show that both sides of B2.2) are equal to -y. Shortly thereafter,
D. Bradley [ 1 ] found a more natural proof by working directly with the double
series on the right side of B2.1), and therefore we shall give Bradley's elegant
proof. Bradley has shown that Entry 22 has several interesting consequences. In
particular, Ramanujan's famous formula for ?Bn + 1) (Part II [2, pp. 275-276,
Entry 21(i)]) follows as a corollary.

452 Ramanujan's Notebooks, Part V
Proof of Entry 22. We begin with the partial fraction expansion from Entry 3 of
Chapter 30 (Part IV [4, p. 359]),
7T2CSC2GTJ-) 7Г 11 ,
2ЯХ3
By the product rule for differentiation,
d лсоЦлх) tt2
T? sinh (лк)(х
;г2соЦ;гл:)
4 -
B2.3)
dxe2*x-\ e2nx-\ 2sinh2(*;t)
and
tf2cotGrjc) n .... .. d
,—- = 5 + -log 2sinGrjc) —-
dx 2sinh2Grx) 2sinh2(jr;c) 2 dx
Hence, we can rewrite B2.3) in the form
, л 1 1 d n соЦлх) d 7rlog|2sinGrA;)|
*(*+ )=Ь~2*+2^ + Ь e2»" - 1 + ~d~x 2sinh2(;rjc)
But,
л d <\
log |2sinGr;t)|— csch (nx).
л d 2 d 2пег"х ( d \
csch {vex) — —- r = — I
2dx dx(einx_x\1 \dx)
B2.4)
к=\
Using this in B2.4) and integrating both sides with respect to x, we deduce that
я 1 1 я cot(ttjc) я log |2 sinGrx)|
nx+l) = -l0gx + -- — + -I--r+ ^^^
00 ->t w ^?_ 1л« It4 _ И|
- S(x) + С
¦*2) 2 ^rf sinh2Grfc)
where С is a constant of integration to be determined and
/OO OO
log I
-f
Jo
k=\
B2.6)
37. Infinite Scries 453
Using the Laplace transforms
00
L
e *' sinfaO dt = ¦ " and
«2 +n2
/•OO
/ e~kl cos(nt
Jo
and the well-known Fourier series (Gradshteyn and Ryzhik [1, p. 46, formula
1.441, no. 2])
cosB7r?y)
in B2.6), we find that
OO
Six) = BтгJ J2k2e-
= - log I2sm(*y)|, у real,
u log |2sin(^(x + u))\du,
Jo
fa
2 -2
fa
nkx f0* _*, v^ si
Jo fa
sinB;rnjc)sin(nO -
= 2я
fa
fa
e~kl sin(nt)dt
A cosB7r«x) /•«• _4, \
- > / e cos(n/) rf/
\ fa
fa
B2.7)
where the inversions in the order of summation and integration are justified by
absolute convergence. Substituting B2.7) in B2.5), we find that
я 1 1 л-соКл-д;) 7rlog|2sinGrx)|
-logx + -- — + 1S7rr+ 2.h2
2—, 1„2лк
2k
л ^ log \k4 - x*\
- д:2) 2 ?r
By letting x -> +oo in B2.8), we can evaluate the integration constant, С In fact,
С must equal zero, as a comparison of B2.8) and B2.1) reveals. Since several

454 Ramanujan's Notebooks, Part V
terms in B2.8) vanish as x -> +00, it suffices to show that
lim
- "- log, + \
3 2
Set x = N + 5, where N is a positive integer, and write
+
41°g;c
logd - k*/x4)
sinh2(;rfc)
= D1og*) V
+ O(x~2) + О
= D log x) 2
Now, by Stirling's formula, as x —> oo,
+ O(x~\
Employing B2.10) and B2.11) in B2.8), we see that
я
я л ^ log I*4 - A4|
+ 1) - - logx + - ? *' 2
3 2 j^f sinh (жк)
= A --+2пУ 1 |
\ 3 fa sinh2 (nk))
as x tends to +00. However (Bemdt [2, Prop. 2.26]),
3 ?? sinh2(jrifc) 3
Using B2.13) in B2.12), we find that
log |jc4 - Jt4|
B2.10)
B2.11)
B2.12)
\ - 1-) =0. B2.13)
6 2J
as x tends to +00. Thus, the limit B2.9) holds, and the proof of Entry 22 is
complete.
37. Infinite Series 455
Entry 23 (Formula D), p. 281). For each positive integer n, let
q :=e
,2,tit
where Im г > 0. Then, iftn is a positive integer,
•5Н4И1-2) „ ,.4 v
24Dm + 2)Dm + 1)'
Proof. In [7], [10, p. 140, eq. B2)], Ramanujan proved that, for Im г > 0 and
any integer n greater than 1,
5)Bи-2)
24Bn + 2)Bn + 1)
B3.1)
(We remark that the notation Sn has a meaning in Ramanujan's paper [7] different
from that in the notebooks.) By a theorem of A. O. L. Atkin [1], ?4n+2@ = 0,
where Е2„(т) is the Eisenstein series defined by (Rankin [1, p. 194])
Bin
Thus, when r = i, the desired result follows.
It is interesting that Ramanujan undoubtedly discovered the theorem, S^+2 (i) =
0, more than 50 years before a proof was published.
The next three entries are found in Ramanujan's paper [6], [10, pp. 47-49].
Since the details of the proofs were not completely given by Ramanujan, we do so
below. The constants c\, cj, and cj below were not explicitly given by Ramanujan
in the notebooks.
Entry 24 (Formula E), p. 281). For each positive integer n,
k=)
oo
iEb^
*=0
Proof. Let
- c, - \n»2 - i«'/2 - I f] j
where C] is a constant such that <p\{\) =0. Thus,
1) = -(и + 01/2 - \nV2 - \nx'
= -A(n
- |n3/2 - А
\(n + \)V2 + I

456 Ramanujan's Notebooks, Part V
= 0.
Since ^i(l) = 0, by induction, ip\(n) — 0 for every positive integer n. From Part
I [l,p. 156], ci = -f C/2)/Dя), and so the proof is complete.
Entry 25 (Formula F), p. 281). For each positive integer n,
«r=l
Proof. Let
_ ^ _ 2,5/2 _. „,,*_. „./2
l:=0
where c2 is a constant such that ^(l) — 0. By the same sort of calculation as in
the previous proof, (рг(п) - <рг{п + 1) = 0. Thus, by induction, <рг(п) = О for
every positive integer n. From Part I [1, p. 156], c2 = -3? E/2)/A6л2), and so
the proof is complete.
Entry 26 (Formula G), p. 281). For each positive integer n,
15
X>5/2 =
Proof. Set
Jk=O
37. Infinite Series 457
where c,i is a constant chosen so that ^3 A) — 0. A somewhat laborious calculation
shows that 4Рз(л) — <рт,(п + 1) = 0 for each positive integer n. Thus, by induction,
<Ps(h) = 0 for every positive integer n. Hence,
*=0
t=0
which is the formulation given by Ramanujan in [6]. Using Entry 24 above, we
deduce that
*=]
64л-
By Entry 1 of Chapter 7 (Part I [1, p. 150]), c3 = f(-5/2). By the functional
equation @.4) of f(z), c3 = f (-5/2) = 15f G/2)/F4jt3). Using this in the
equality above, we complete the proof.
Entry 27 (Formula C), p. 282). Let
<p{x)
¦У. —*-5
B7.1)
where, in the latter representation, \x\ < 1. Then, for every real number x,
4
л
B7.2)
An equivalent formulation of Entry 27 is given by Ramanujan in his paper
[5, eq. A2)], [10, p. 41], where he writes that B7.2) is "very easily proved by
differentiating both sides with respect to x." It seems unlikely that Ramanujan
discovered B7.2) by this means, but we do not have a better proof and proceed
accordingly.
Proof. Let f(x) and g{x) denote, respectively, the left and right sides of B7.2).
Then, by straightforward differentiation,
fix) =
TTX
\+e-"x 2cosh(±jrx)

458 Ramanujan's Notebooks, Part V
and
The differentiation under the summation sign is justified because the differentiated
series converges uniformly for all real x. But (Whittaker and Watson [1, p. 136])
7Г
B7.3)
4cosh(±jr;c) ;^Bл-ЫJ+л:2'
Hence, f'(x) = g'{x). Since /@) = 0 = ?@), the proof of Entry 27 is complete.
Entry 28 (Formula D), p. 282). We have
n=0
where
which is Catalan's constant.
= ic> B8Л)
(-1)"
42'
Proof. Set x = B/лО logB + V5) in Entry 27. Then if S denotes the left side of
B8.1), we find that
S=-
. B8.2)
Setting и = 7Г/12 in the double angle formula cosBm) = 2cos2m - 1, we find
that cos(?r/12) = \y/l + Jb. Thus, sin(;r/12) = j^-v^and tan(^/12) =
2 - л/3. Hence, B8.2) becomes
5 = i
- ^ B - >/з)) - \ logB + V3). B8.3)
Now in [5, eq. GI, [10, p. 40], Ramanujan briefly sketched a proof of the equality
2?(l) = 3<pB - V3) + i* logB + V3). B8.4)
Using B8.4) in B8.3), we deduce that
S=
Since ^A) = C, the proof of B8.1) is complete.
Lastly, we provide details for the proof of B8.4). Ramanujan claims that B8.4)
follows from the equality
sinDn + 2)x
0 < x < n/l, B8.5)
n=0
37. Infinite Series 459
by setting x — 7Г/12. Accordingly, we find that
pB - V3) + iL iogB + -Л)
J
2 I2 32 2-52 2-72 92 2 • II2
i/j__j_ 1 _j_ J L
2 у 12 32 52 72 92 II2
I
Thus, B8.4) has been proved, and it remains to prove B8.5).
To prove B8.5), let f{x) and g(x) denote the left and right sides, respectively,
of B8.5). Then
/'(*) =
n=0
. cosDn + 2)x
2n+ 1 '
since the differentiated series converges uniformly on any compact subinterval of
@, к/2). Also,
g'(x) = -log(tanx) =:
cosDn
n=o
by a familiar Fourier series development (Gradshteyn and Ryzhik [ 1, p. 46, formula
1.442, no. 2]). Thus, f'(x) - g'{x), when 0 < x < ir/2, and so f(x) - g(x)+c,
where с is some constant. Letting x = л/4, we find that
v^ ("I)"
By B7.1), с = 0. Hence, B8.5) has been proved.
Entry 29 (Formula A0), p. 286). If-I < x < 1, then
n=0
(-1)"
Bи + IJ
B9.1)
B9.2)

460 Ramanujan's Notebooks, Part V
Proof. By analytic continuation, Entry 27 is valid for all complex x with \x\ < 1.
Replacing x by ix in Entry 27, we find that
- 2ix
Taking the real part of each side, we deduce that
))
B9.3)
Now
-2Re (ix tan

x Re log (
д: Re log
x log
( -icos(\nx)\
I -. I
\l +s\n{{nx)J
COS(j^X) \
—. I
l+sin(±7TA:)/
B9.4)
Using B9.4) in B9.3), we complete the proof of B9.1).
To prove B9.2), which is eq. A2) in Ramanujan's paper [5], [10, p. 41], we first
note that the right side of B9.2) equals
With f(x) denoting the left side of B9.2), we see that /@) = 0 = g@). Thus, it
suffices to prove that f'(x) = g'(x). Now, by a straightforward calculation with
the use of B7.1) and B7.3),
nx sec
«'(*)= —
— nx
7TX
4 tan
GГ
— 71X
-x)}
nx
and the proof of B92) is complete.
37. Infinite Series 461
Entry 30 (Formula A1), p. 286). Let a > 0 andO < 0 < I, with
Then
я=0
n=o
Proof. Observe that both sides of B9.1) are even functions of x, so that we can
replace x by —x. Thus, by Entries 27 and 29,
= % (
tan (\
~ \ {e-*a>2) - 2a tan (е'™'2)
= 1 {<p(e"a/2) - <p(e-"a/2)} - 2a tan (cot ЛтгA +
Now from Ramanujan's paper [5, eq. D)], [10, p. 40],
- log(e™/2).
C0.2)
Hence,
This completes the proof of C0.1).
Entry 30 is equivalent to eq. A7) of [5], [10, p. 41], which Ramanujan stated
without proof. He also [4], [10, p. 329] submitted this formula as a problem to
the Journal of the Indian Mathematical Society. The proof of C0.2) was only
briefly sketched by Ramanujan in [5]. The editors of his Collected Papers [10, pp.
336-337] supplied a proof in more detail.
Entry 31 (Formula A), p. 288). For x > 0, let
2
F(x) — X
cosh {Bл + l)x} + cos (Bn +

462 Ramanujan's Notebooks, Part V
Then, if a, fi > 0 withaf) = n2/2.
F(a) =
C1.1)
This is a particularly beautiful result. Ramanujan evidently did not possess
a complete proof, for he recorded the result (in abbreviated notation) as "The
difference between the two series (a/J = n2/2) F(a) and F(@) is 0?" As we shall
see, upon examining our proof below, Bn+ IK can be replaced by Bл + IL,
for any nonnegative integer m.
Proof. Let
fit) : =
(cosh(az) + cos
Observe that f(z) has simple poles at z = 2n+1, Bи +1)/, Bn + 1)л A ±i)/Ba),
for each integer n.
Let {Си), N > 1. denote a sequence of rectangles having vertical and hori-
horizontal sides and centers at the origin. We shall choose the rectangles so that, as N
tends to oo, the sides tend to oo but remain at a bounded distance away from the
poles of /(z). It is then easy to see that
lim f f(z)dz=0. C1.2)
"-« Jc,
We apply the residue theorem. By straightforward calculations, for each integer
n,
1)}
and
(Bn
тт (cosh {Bи + l)a) + cos {Bn + l)a|)
2(-l)"B« + lKsech{^Bn+l)}
l)a}+cos{Bn + l)a)
sech {Bn
= «
Brt+lb
= R-
C1.3)
C1.4)
+ i)Ksec {Bи
i)Bn + IK sec \{Bn
- sin \[Gn + \)n(\
i)| sech j^Bw +
(« - l)cosh{ijrBn
{^Bn + l))(cosh{Bn
Furthermore, we see that
+ cos(Bn
C1.5)
C1.6)
37. Infinite Series 463
Hence, applying the residue theorem to the integral of /(z) over CN, employing
the calculations C1.3)-C1.6), letting N tend to со, and using C1.2), we deduce
that
cosh{Bn + l)a]+cos{Bn + l)a]
32 ~ (-l)"Bn + lKsech{i7
—— = 0
ла n^o cosh И2и + 1))8} + cos {Bn + 1)>3}
Multiplying both sides by яа2/8, we arrive at C1.1) to complete the proof.
If we divide both sides of C1.1) by 01 and let a tend to 00, and therefore let f)
tend to 0, and replace Bn + IK by Bn + IL, as we may, for any nonnegative
integer m, we deduce that
= 0.
This result is due to Cauchy [ 1, pp. 313, 362] and is a special case of another
theorem of Ramanujan (Part II [2, p. 262]).
Entry 32 (Formula C), p. 288). //a, /3 > 0 with afi = 7г2/4, then
f* (-1)" = n_
h a)) 8
fzZ Bи + 1) (cosh {Bn + l)a} + cos {Bn
„^ <-1)»со*(Bл-
^ Bn + 1)cosh {\лBп + 1)} (cosh {Dn + 2)fi) + cos{Dn + 2)/?})'
C2.1)
Proof. Let
f(z) : =
z(cosh(az) + cos(az))cos(|7rz)
Note that /(z) has simple poles at z = 0, 2n + 1, and Bn + 1)лчA ± i)/Ba),
for each integer n. Let (C/v} denote the same sequence of rectangles as described
in the proof of Entry 31.
We now calculate the residues of f(z). First,
C2.2)
Second, by an easy calculation,
тгBл +
C2.3)

464 Ramanujan's Notebooks, Part V
Third, by straightforward calculations,
(-1)"
nBn+ 1) cos {Bл
2n + 1)}
C2.4)
By C2.4) and an elementary calculation,
4(-l)" cosh {Bn + l)/3) cos {Bn + \)P)
2п + Dcosh {ijrBn + l)}(cosh {Dn+2)p)+cos{Dn
C2.5)
We now apply the residue theorem to the integral of /(z) over CN. It is easy to
show that
C2.6)
lim f
/(г) dz = 0.
Hence, from C2.2)-C2.6),
1
2
o-i-if;-
(-1)"
1) (cosh {Bn
(-l)"cosh{Bn
cos {Bи + l)a})
л ^L, Bn + 1) cosh \\nBn + l)} (cosh {Dn+2)P) + cos{Dn
Equality C2.1) now readily follows.
If we let a tend to oo, and hence p tend to 0, in C2.1), we find that
OO / 1 \rt rr
•s—* __ (-1) _ *
which is a special case of another theorem of Ramanujan, Entry 15 of Chapter 14
(Part II [2, p. 262]).
If we let P tend to oo, and therefore a tend to 0, we find that C2.1) yields the
well-known evaluation
^ (-1)" *
C2.7)
n=0
Entry 33 (Formula D), p. 288). Ifufi = л2/2, where or, p > 0, then
(-l)"cos{Bn+ \)a)
n я-
^ Bи + 1)(cosh {Bn+l)a}-cos {Bn + l)a}) 8 32a2
^л sin(w/3) sinh(n/3) coth(n7r) ^3>1^
= ?-i n (coshB«j8) + cosBnP))'
37. Infinite Series 465
Proof. Let
fit) : =
cos(az)
z(cosh(az) -
We observe that /(z) has a triple pole at z = 0 and simple poles at г = 2л + 1,
for each integer n, and sa z — пл(±\ + i)/a, for each nonzero integer n. Let
{C/v}, N > 1, denote a sequence of rectangles having centers at the origin and
horizontal and vertical sides approaching oo as N tends to oo. The rectangles are
also chosen so that the sides remain at a bounded distance from the poles of /(г)
as N approaches oo.
We now calculate the residues of /(z). First, after a modest calculation,
8a2
C3.2)
Second, for each integer и,
2(-l)"cos{Bn
7TBn + 1) (cosh {Bn + l)a} - cos (Bn -
Third, straightforward calculations yield, for each nonzero integer n,
±(-1)"СО8{И7Г(±1+|)}
2^insinh(n^)cos(/i^(±l + i))
Using C3.4), we find that
2coth(n7r) sin(n^) sinh(nK)
C3.3)
C3.4)
for every positive integer n.
Lastly, we apply the residue theorem. It is easy to show that
lim
Thus, by C3.2), C3.3), C3.5), and C3.6),
1 4^2, (-l)"cos{Bn
C3.6)
8a2 2 я ^ Bn + 1) (cosh {Bn + l)a} - cos ((In + l)a})
4
71
~
n=] .. ^w.y2nP) + cosBnp))'
After a slight amount of rearrangement, we deduce C3.1).
If we let a tend to 0, and therefore p tend to oo, in C3.1), we deduce the
well-known result
(-1)"
32-

466 Ramanujan s Notebooks, Part V
At the top of page 289, Ramanujan writes, "The difference between the series
в r^ sin(n2<9) 1 ^ (-If
—— and - > -
The sentence is not completed. Observe that, by Stirling's formula, the series on
the right diverges for all О Ф 0. Most likely, Ramanujan realized that the series
diverges and stopped here, or that he proceeded formally and could not evaluate
the requisite integrals. These integrals do not appear to have evaluations in closed
form. Note below that Ramanujan has a (possible) misprint in the first expression
of the quote above. This entry should be compared with formula C) on page 274
(Part IV [4, p. 298]).
Entry 34 (Formula A), p. 289). Formally,
4
(-1)"
An ^ n{e2n* - 1) 4 f^0 Bn + 1)! Bn + 1)
sin(jr20)cosB*«x)
+
x{e2n* -
dx,
C4.1)
where Bj, j > 0, denotes the jth Bernoulli number.
"Formal Proof. Apply the Poisson summation formula (Titchmarsh [2, p. 60])
to
/(*) :=
sin(jc2<9)
x{e
,2лл _
Note that /@) = в/Bп). Hence,
в ^ sin(w20)
2fl) _ Г00 %т{хЧ
- 1) ~ Jo x{e2"* -
) *
C4.2)
Now, by Entry 16(iv) of Chapter 13 (Part II [2, p. 220]),
sinU26>)
f0
^(-lyfl2"-1-1 Г00 jc^1
- 1- Bи + 1)! Л e^^l
C4.3)
я=0
Note that the inversion in order of summation and integration has not been justified.
Indeed, the series on the far right side of C4.3) diverges. Putting C4.3) in C4.2),
we complete the "proof."
37. Infinite Series 467
Ramanujan begins page 312 by stating two series identities involving the Mobius
function д(п). Не then offers a general claim, which contains the previous two
results as special cases, and which is an analogue of the Poisson summation for-
formula, with fi(n)/n as coefficients. We first formally state Ramanujan's general
claim. We then show that the two examples follow from the general claim. Next,
numerical calculations show that Ramanujan's two examples are false but that the
errors are numerically very small. Lastly, we indicate how G. H. Hardy and J. E.
Littlewood corrected Ramanujan's claims.
In our calculations below we employ the prime number theorem in the form
X^| fj,(n)/n — 0. Perhaps Ramanujan used a stronger, but incorrect, version
of the prime number theorem, which would account for the excellent numeri-
numerical agreement in the two applications. As we saw in Chapter 24 (Part IV [4]),
Ramanujan made several errors in deriving approximations to n(x) that involve
д(и).
The author [8] has established a general arithmetical Poisson summation for-
formula which can be applied to series with fx(n) as the coefficients. However, it
is considerably more complicated than Ramanujan's claim. Similar arithmetical
summation formulas were derived via contour integration by the author [5] and
by P. V. Krishnaiah and R. Sita Rama Chandra Rao [1].
Entry 35 (p. 312). Let afi - 2л, where a, /? > 0. Let
-f
Jo
(p(x)cos(nx) dx.
Then
a ^ ti{n)<p(ot/n)
2
Entry 36 (p. 312). Ifp>0,
^ д(и)и
1Р2 + п2
Proof based on Entry 35. Apply Entry 35 with <p(x) - \/{x2 + 1). Now, by
contour integration or tables (Gradshteyn and Ryzhik [1, p. 445, formula 3.723,
no. 2]),
f
Jo
x2+
Thus, by Entry 35,
H(.n)
+ 1)
'Г

468 Ramanujan's Notebooks, Pan V
or
Л" ^ М(П) _2пНпа)
— — / €
Entry 37 (p. 312). lfp>0,
a *-f n
W = 1
_ (?Д
p 1 л
Proof based on Entry 35. Replace a and fi in Entry 35 by y/a
tively, so that
5Л0=?
2 $^\ п
where afi = 4л2. Let <p(x) — exp(-jc2). Then
Thus, by C7.1),
. respec-
respecC7.1)
or
We now numerically examine Entries 36 and 37. If 0 < p < 1,
OO /„\ OO
00 ,./~\
r-
where we have used the fact that ??1, д(я)/я = О, which is equivalent to the
prime number theorem.
37. Infinite Series 469
Next, using this same fact, we find that
я
fx{n)e 2ni(ni'> л y, fx(n) yv 1 / 2n\
n P ~$ n ?*k\\ np /
2n\K
1
p/ Jfc! ?(*+!)
Thus, Entry 36 may be rewritten in the form
00 / ,\k „Ik _ 00 / o_
E\— I) p Я ^—\ / Ln
ttik-i- П ~~ ~n 4-> \ "IT
C6.1)
Setting P = 5 and summing the first 50 terms of each series via Mathematica,
we find that
^ (-0*
2M f B* + 1)
= -0.1600806325...
and
t=1
Since the series are alternating, and since
1
2102CA03)
<i98xi°1 and Шй<4-65xio0'
these calculations show that C6.1) is false for p = \.
If p > 0, since Xl^ti (*¦(")/n — 0,
Also, by a similar calculation,
Thus, Entry 37 may be rewritten in the form
/F^> (-1)**»
C7.2)
Setting p = 1 and summing the first 50 terms of each series above by Mathe-
Mathematica, we find that
(-0*
= -0.4805338008 ...

470 Ramanujan's Notebooks, Part V
and
= -0.4805622889....
.102
Since the series are alternating, and since
- < 6.45 x 107 and V7rJr < 5.85 x 106,
E1)! CA03) E1)! rA03)
we conclude that C7.2) is false for p — 1.
Now, in fact, during his stay in Cambridge, Ramanujan told G. H. Hardy and J.
E. Littlewood about Entry 37, and they discuss the formula and the more general
claim, Entry 35, in their paper [1] (Hardy [3, pp. 20-97, especially pp. 57-63]).
Assuming that all of the zeros of ? (s) are simple and that the series on the far right
side below converges, Hardy and Littlewood proved that
A - \p)
C7.3)
where the latter sum is over all complex zeros p of t, (s), arranged according to
increasing moduli. Thus, Ramanujan's claim in Entry 37 must be altered by the
latter expression on the right side of C7.3). This is then another instance where
Ramanujan's ignorance of the complex zeros of ?(s) led him astray.
Hardy and Littlewood also briefly address the more general formula in Entry
35. Although they do not give a complete proof, they clearly demonstrate that
Ramanujan's claim must be modified by a similar sum over p.
Returning to C7.3), Hardy and Littlewood [1, p. 161] showed that the estimate
= O(p
-1/4+4.,
C7.4)
as p tends to oo, where e is any positive number, is equivalent to the Riemann
hypothesis.
Titchmarsh [3, pp. 186-187] also provides a proof of the corrected version
C7.3) of Ramanujan's Entry 37 and briefly mentions C7.4) (with the exponent e
unfortunately missing) as well [3, p. 328]. We are very grateful to Richard Brent
for correcting some inaccuracies in our discussions of the past two entries in an
earlier version of this chapter.
Although Entries 35-37 are false, it would be exceedingly interesting to dis-
discover Ramanujan's heuristic arguments, since Hardy and Littlewood's approach
is through contour integration.
The next result generalizes Entry 35 and is stated in the notebooks without the
latter series on the right side of C8.2).
If <p(z) = 1 and we omit the latter series on the right side of C8.2), we obtain
C6.1).
37. Infinite Series 471
If we set tp(z) = 1 / Г(± [z + 1|), replace p by ,//>, and again ignore the second
series on the right side of C8.2), we deduce C7.2). To see this, we first note that
if n is odd, <p(—n) = 0. If n = 2m is even, we observe that
22 22тГ(\[2т + \\)
Bm)! *(-!
Bm)! VJr
(-1)"
m\
Hence, C7.2) readily follows.
Entry 38 (p. 312). Let <p(z) denote an entire function, and put
where p is any fixed nonzero complex number, and where {(?) denotes theRiemann
zeta—function. For simplicity, assume that each nonreal zero of?(z) is simple. Let
Сц denote the positively oriented circle of radius N + | centered at the origin,
where N is a positive integer. Suppose that
Then
lim f f(z)dz = 0.
"-We
¦^ (— D"<pBn + Dp2n+l v^v (—1)"*3(—л) Bп\п
у = к у [ — i
*—' ?Bл + 1) " n\ f (л +1) V P /
C8.1)
<p{p)p"
C8.2)
where the sum on p is over all nonreal zeros of l, (г) arranged according to in-
increasing moduli.
Proof. We apply the residue theorem to the integral in C8.1). Observe that /(г)
has simple poles at г = 2л + 1, for each integer л, at г = — 2л, for each positive
integer n, since С, (-2л) = 0 (Titchmarsh [3, p. 19]), and at each nonreal zero p
of ?(г). By a straightforward calculation, for each integer л,
2(-D»<pBn+Dp2»+1
—— . (Зъ.3)
C8.4)
C8.5)
From the functional equation of ?(z), @.4), for each positive integer л,
z + 2n (-1)лBл-Jп+|
lim
Hence,
R-2» =
BnJ"+]<p(-2n)p-2n

472 Ramanujan's Notebooks, Part V
Lastly, for each nonreal zero p of ?(z), a simple calculation gives
<P(P)PP
C8.6)
Thus, applying the residue theorem, using C8.3), C8.5), and C8.6), letting N tend
to oo, and employing C8.1), we find that
2 y> (-\)n<pBn+l)p
2n+l
n=0
?(-2я -
P
where the sum on p is over all nonreal zeros p of f (z) arranged according to
increasing moduli. Now, by the functional equation @.4)
?(-2и - 1) = 2(-1)л+1Bтг)-2л-2ГB/1 + 2KBn + 2).
Using this in the second sum in C8.7), we deduce that
(-l)>Bn+l)p2
In + 2)Bn + 1)!
<p(-2n)
(т)
n
+ 2
C8.8)
The first two series on the right side of C8.8) can be combined into one series,
and since 1/f A) = 0, the proof of C8.2) is complete.
In Ramanujan's formulation of Entry 39, the latter series on the right side of
C9.2) does not appear.
Entry 39 (p. 312). Let <p(z) be an entire function, and put
/W := „,i.
where p is any fixed nonzero complex number, and where f (г) denotes the Riemann
zeta-junction. Assume, for simplicity, that all nonreal zeros p of$(z) are simple.
Let CN be the same circle as in Entry 38, and assume that
lim f
/(z) dz = 0.
C9.1)
37. Infinite Series 473
Then
¦A/
E-
+ 2
where the sum on p is over all nonreal zeros p of f (г) arranged according to
increasing moduli.
Proof. The proof follows along the same lines as the proof of Entry 38. The
function /(г) has simple poles at z = 2n + 1, for each nonnegative integer n, at
z - -2л, for each positive integer и, and at z = P, for each nonreal zero p of
?(z). Note that the zero of cos(^jtz) at z = 2n + 1 is cancelled by the zero of
1/ T(j(z + 1)) at г = 2и + 1 when и is a negative integer. A simple calculation
yields
n2n+l
C9.3)
By C8.4) and the duplication formula (or the functional equation) of the gamma
function, for each positive integer n,
BnJn+1<p(-2n)p
-2n
For each nonreal zero p of f (г), we easily see that
C9.4)
C9.5)
Invoking the residue theorem, using C9.3Ы39.5), letting N tend to oo, and em-
employing C9.1), we conclude that
г.) /яЛ2"
! VP/
= 0,
where the sum on p is over all nonreal zeros p of f (г) arranged according to
increasing moduli. The last equality is equivalent to C9.2).
Entry 40 (p. 324). For each positive integer n,
1
1
y= y
f^ Dk - 2K - D* - 2) 2 fr{ In + 2k - Г
D0.1)

474 Ramanujал's Notebooks, Part V
Proof. The following proof is due to R. Sitaramachandrarao [1].
For any nonzero number a and positive integer n, define
Then it is easy to see that
f^ Dk- 2K - D* - 2)
By Entry 1 of Chapter 2 and Example 4 in Section 5 of Chapter 2 (Part I [1, pp.
25, 31J),
2я
<pB,2n)-(pD,n)=
1 4n 1 ^
2n + к 4л + 1 2
.7 V 1-1 V i
к 2 ,,4- A;
Аг=2л + 1
4n
,2л. 2n-l
1 t—> 1 ^
? 2 ^—' *• ^^ Ik 4- 1 '
D0.3)
Putting D0.3) in D0.2), we obtain an equality that is easily seen to be equivalent
to D0.1).
In the entry immediately following Entry 40, Ramanujan proposes an equality
between two finite sums of inverse tangent functions,
gtan' {ъгткп) - ?"""'
However, a line has been drawn through the right side. Indeed, it is easily checked
that this claim is false, in general.
Entry 41 (p. 324). For each positive integer n,
\ 3
) •
Proof. We induct on n. For и = 1, D1.1) is trivial.
Assume that D1.1) is valid. We thus will prove that D1.1) holds with n replaced
by n + 1. Recall that, for 0 < xy < 1,
tan
D1.2)
37. Infinite Series 475
Thus, by induction,
?--¦( ' У
f? VBJt+l}V3/
Bn+2k+\)j3
+tan-
+ tan
-i
)'
Thus, it remains to show that
tan
= tan
-i
By D1.2),
tan"'
-i
= tan
= tan
— tan
V3(8n + 4)
УЪBп+\) \
Bи + 1)*-1/
= tan
VBw + l)V^/
9Bn + 1)" -
4-tan^
IWf)'
D1.3)
by another application of D1.2). Thus, D1.3) has been proved, and the proof is
complete.
On pages 334, 335, 340, and 341, Ramanujan offers four related claims about
products of certain alternating series. In particular, on page 335, he asserts that
"a\ - 2а\аг + BЯ|Дз + a\) - Ba\a4 + 2а2вз) + • ¦ • oscillates between (a, -
а2 ¦+• аз — • ¦ • J ± (я/2) limn-юо najr For example,
2 1\ / 2_ _2_\ /_2_ J2_ 1\ _
% sfl) \s/5+\/8 3/

476 Ramanujan's Notebooks, Part V
oscillates between
and
/J 1_ J_ _
+2
л;
2'
We state this claim more precisely. Suppose that, for 0 < x < 1, /(дг) has the
power series representation YlT=i(~^n~la"x"' where an > 0,1 < n < oo. Let
g(x) denote the power series obtained by forming the Cauchy product of f(x)
with itself. If L := limM00 na\ exists and is finite, then the even and odd indexed
partial sums of g(\) tend to /2A) + (n/2)L and /2A) - (n/2)L, respectively. In
particular, if an - 1/Vn, then the even and odd indexed partial sums of g(l) tend
to {A - \/2Хф j + Л-/2 and {A - \/2)<(±)} - тг/2, respectively, where ?
denotes the Riemann zeta-function. The case when L = 0 implies that the series
#A) converges. In the case when L — -foo, the result states that the even and odd
indexed partial sums diverge to +oo and —oo, respectively.
On page 340, Ramanujan states a generalization of the foregoing result for the
Jith power of f(x), where к is any positive integer exceeding 1. Finally, on page
341, Ramanujan offers a similar theorem for the product of it (possibly distinct)
alternating series Cl.f-ir'^i-Er^f-l)""'^, ¦ ••, IXit-D'1" V un-
under the hypothesis that limn_>oo nk~'an|Дп2 • • • ank exists or is +oo. The statement
on page 334 is the special case к = 2 of the last claim.
We shall prove Ramanujan's assertions under appropriate assumptions. Our
results are possibly not as general as Ramanujan intended. Slightly stronger theo-
theorems can undoubtedly be established at the cost of additional technical details in
the proofs. (See the remarks following our proof of Entry 42.)
Ramanujan's results are quite remarkable for their explicit description of the
behavior of the partial sums of certain alternating divergent series. We know of no
other comparable results in the literature.
The results in this section first appeared in a paper by the author and J. L. Hafner
[1].
We begin with a simple lemma concerning the asymptotic behavior of a certain
finite sum.
Lemma 42.1. Let a and p denote constants with 0 < a, fi < 1,
1
and
n>2.
D2.1)
37. Infinite Series 477
Then, as n tends to oo,
wAere f denotes the Riemann zeta-function.
Proof. Define, for x > 0 and я > 1,
V X ~ xa(\ -x)*> I" A -;
1 1 1
+ 1
and
*.<*) =
1
xa(n -
na(n -x)? ni+P'
D2.2)
D2.3)
Note that
Since, for у > О,
<p(J)(x) =
uniformly for x in the given ranges, it follows that
<i,
nL
- j, n/2<x<n.
D2.4)
We now apply the Euler-Maclaurin summation formula @.5) to gn (x). Recalling
that Вг(х) denotes the second Bernoulli polynomial, we find that
-1 /41-t
^8n(k) = / gn(x) dx + Ifo(
Re, D2.5)
where
Rn = j2 Ш" ~ О - g'nO)} ~ [ j B2(X - lx))g
n-l
D2.6)
as n tends to oo, by D2.4). (Note that the notation Rn has a meaning here different
from that in @.6).) Using D2.4) and D2.6) in D2.5), we deduce that
О
О
D2.7)
as n tends to oo.

478 Ramanujan's Notebooks, Part V
Recalling the definitions D2.2) and D2.3) of cp and gn, respectively, and setting
x = nt in the integral above, we find that
Г l /"'
/ gn(x)dx= / <p(t)dt
Jo n p Jo
1
1
1
+ 1 .
„а+Л-i [ rB-a-/3) I-a 1-/3
where we have used the classical integral representation for the beta function.
Hence, by D2.7), as n tends to oo,
4=1
ГB-а-/3) I-a 1-/3
+ 1
D2-8)
Now, from the definitions D2.1) and D2.3), we deduce immediately that
4=1
n-l
n~I
k=\
П — \
D2.9)
Recall (Part I {1, p. 150]) that for any complex number г Ф -1, as n tends to oo,
D2.10)
+
where Bhj>2, denotes the ; th Bernoulli number. Employing D2.8) and D2.10)
in D2.9), we conclude that
С(П) =
„a+fi
1 [ и1"" п-а / 1
-= С(а) + + — + О ( —
я" I 1 -а 2 \""+
which completes the proof of Lemma 42.1.
Our next lemma extends the result to multiple sums.
37. Infinite Series 479
Lemma 42.2. Let к > 2, let a\, a2,..., a* be constants such that 0 < otk <
• • • + аи — к + 3. 77ien у* > 1 о/м? there exist constants Ьц and fikj such that for
each j, 1 < j < 3*~' - 1, «i + a2 H ha* - it + 1 < fikj < 1 and
П-] n—fti—l
E
пТпТ-1 ¦ ¦ ¦ n22(" - "* )"'
Г(к - a, - a2
4и иА' V">''/
D2.11)
to oo.
Proof. Naturally, we induct on k. The case к = 2 is just a restatement of Lemma
42.1 with a = ct, and fi = a2 (assuming fi < or). In this case, we have Ьг\ =
f (a2), fr22 — ?(<*]), )?2i ^ аь А22 — a2> "nd /2 = 011 + 1.
Suppose that Lemma 42.2 is valid with к replaced by it — 1, where к — 1 > 2.
Then for some constants bk-t,j, Pk-ij, with 1 < у < 3*~2 — 1 and 0 < a\ +
or2 H l-tt«:-i — к +2 < /?*-i,; < 1 and yit-i = a2 +аз +¦ •
it follows that
n-l
n-l f
n* = l l
E* *';',;,_,,+ »(',,.,)U.
>=i ^ 'J*
Applying Lemma 42.1 a total of 3* 2 times, we deduce that
, , ГA -а,)ГA -а2)---ГA -at_i)
Г(к - I - «i - a2 at-i)
- a, - a2
- 2)
- en - a2
2)
+ °
о

480 Ramanujan's Notebooks, Part V
ГA -аг,)ГA -ог2) • ГA -щ)
Г(к — 0Г| — а2 — ... — ак)п+а2'
k-k + \
nYk
Here the set (bkj : 1 < j < 3* ' - 1 ( comprises the numbers
V{k -
-а2).
a, -
Г(* - 1 -a, -a2
and, for 1 <j <3*-2- 1,
Furthermore, the set [fikj : 1 < j < 3*"' — I) is composed of the numbers
«i + «2 + • • ¦ + otk — к + 2, a* (with multiplicity 3*~2), ft-i.; 4- a* — 1, and
Pk-\j, where 1 < ./ < 3*~2 - 1. Lastly, we observe that
Yk = «2 + «з + h or* - к + 3
= inf {a, +a2+ •
l,y*-i + a* - 1},
which justifies the exponent in the final 0-term.
We are now in a position to state and prove one form of Ramanujan's assertion
on page 341 in his second notebook [9]. Our result is the special case of the general
claim to which we referred in the introductory paragraphs of this section, when
ani = и-* and \imn.KX>nk-lanian2 ¦¦¦а„к = 1-
Entry 42. Let к > 2. Suppose that a,,a2,... ,ak are constants such that 0 <
a* < «*-i < • • • < ai < 1 andct\ +a2-\ 1-a* — к - 1. Letck(n),n > k, be
defined by D2.11). Then the even and odd indexed partial sums of
-!)% (и)
D2.12)
n=k
tend to Sic + 5 Г* and Sk - \ Г*, respectively, where
Sk = (-1
-О-
Г* =
- a2) •
- ак).
37. Infinite Series 481
Proof. Let 0 < г < 1 and suppose that N is a positive integer. Then by Lemma
42.2,
(-1)дгг''г f H)V
щ=1
00
n=k
3* " — I
tk 1+г D2.13)
as N tends to oo, uniformly for 0 < z < 1. The term o(l) arises from the fact
that the series ?~*(-г)"/'лА' converges uniformly on 0 < z < 1. Letting г
tend to 1- in D2.13), we see that the left side approaches 5b while the right side
alternates like
n=k
as N tends to oo. The proof is now complete.
We next state a simple corollary of Entry 42.
Corollary. Letck(n) be given by D2.11), with aj = 1 - 1/*, 1 < j < k. Then
the even and odd indexed partial sums of D2.12) tend to
S4 + ir*(l/ft) and Sk-{rk(l/k),
respectively, where
5fc = (-1)* {A - 2^*K(I - I/A)}* -
In particular, if к = 2, the even and odd indexed partial sums of D2.12) tend to
S2 + jt/2 and S2 - n/2,
respectively.
This corollary is an immediate consequence of Entry 42, when, for к = 2, we
recall that Г( i) = Jn. Note that this last case is the example that we mentioned
in the opening paragraph of this section.

482 Ramanujan's Notebooks, Pan V
We now offer several remarks.
From the proofs of Lemma 42.2 and Entry 42, we can furthermore conclude
that if qti + a2 H \-ctk > к - I, then the series in D2.12) converges to Sk, while
if «i + a2 H +ak < к - 1, then the even and odd indexed partial sums of this
series tend to +oo and —oo, respectively. This observation partially explains the
meaning of Ramanujan's assertion on page 334, "The product of the two series
(a\ — at + из — • • • )(b] — ?>2 + by — ¦ ¦ ¦) is convergent, divergent, or oscillating
as lim(I^№ nanbn is zero, infinite, or finite, when a,, and bn do not contain any
logarithmic functions."
It seems that Ramanujan is tacitly assuming that an = (a + *ni )")""'• an<'
bn = (b + ent)n~a2, where a, b, a\, and a2 are constants with a\, ai > 0, and
where fn\ and („2 tend to 0 as n tends to oo. Ramanujan evidently assumes similar
hypotheses for the aforementioned claims on pages 335,340, and 341. Indeed, our
theorem can undoubtedly be generalized by replacing \/п"' by (a, + 6nj)n~"',
where, for each j, 1 < j < k, uj is a constant and й„ j is a suitable function
approaching 0 as tij tends to oo. We have been able to show that this claim is
correct under the assumption that enj, 1 < j < k, is monotonic, though this is
probably not the weakest assumption under which Ramanujan's assertion would
hold. We forego giving the details of the proof of this more general result.
Adolf Hildebrand has kindly pointed out to us that Ramanujan's weak assump-
assumption is not sufficient for the conclusion of Entry 42 to hold, even in the case к = 2.
He observes that the series in D2.12) cannot oscillate ifoB«) -ckBn- 1) > &„
for sufficiently large n, with Sn > 0 and J2T= i &¦ — °°- ^e reconstruct his
example here. In the notation of the previous two paragraphs, set
а„ -Ьп-
—^=A + („), if n is even,
—=., if и is odd,
where е„ tends to zero, Y1T=\ (l = °°< ап(^ €" satisfies the relation
= €„
D2.14)
uniformly for -Jn < к < n, as n tends to oo. For example, we can take („ =
(log log n) /2.
Define, in analogy with D2.11), с2(л) = c(n) — ?"~| "kOn-k- If n is even,
then
= c'(n) + 2d(n) + e(n).
37. Infinite Series 483
say, where
Л-1
c\n) =
= ж + О(п-1'2),
frf
according to Lemma 42.1 and the fact that F(i) = y/n. To estimate e(n) we
proceed as follows:
e{h) =
E + E
к even к even
* even
The first and last sums can be estimated trivially as O(n l/4) = o(e2). Now we
use relation D2.14) to express the second sum as
= \c'(n/2)(Bn+o(e2n)).
к even
Consequently, e{n) — ne2/2 + o(e%).
A similar and simpler argument shows that
We then deduce that for n even
c(n) = nil + С + €2/2) + o(Bn).
On the other hand, if и is odd, then
c(n) = 2
D2.15)
D2.16)
*i
к even
by D2.15).
Thus, by D2.16) and D2.17),
and hence
D2.17)
сBи) - cBn - 1) = \
и) - сBи - 1))
= oo.
In light of the remarks above, it seems to be very difficult to determine the most
general conditions under which Ramanujan's claim is valid.

484 Ramanujan's Notebooks, Pan V
We quote Ramanujan in the next entry.
Entry 43 (p. 348). eax can be expanded in ascending powers ofebx — e" and
consequently eax can be expanded in ascending powers of ebs sin jc and hence
many transcendental equations can be solved.
The content of Entry 43 has been thoroughly discussed in Part 1 [ 1, pp. 308-312].
Entry 44 (p. 350). Let n be complex, с be real, and b > 0. Then, ifO<x <
пе~Ьх sin(c;t) ^ dk [ e~hx sin(ct)\*
where
enx = 1 +
n(n + kb) [(n + kbJ + BcJ} {(n + kbJ + DcJ} • • •
x[(n+ kbJ + (k- 2Jc2}, if к is even,
n {(« + kbJ + c2\ {(n + kbJ + CcJ} •¦¦{(«+ Щ2 + {k- 2Jc2},
if к is odd.
See Part I [1, pp. 309-310) for a proof.
At the top of page 352 Ramanujan writes, "If an nth degree series can be
expressed in terms of M and N only, then
du nLu
can be expressed in terms of M and N only." The degree of a series is vaguely
defined in Chapter 15 (Part II [2, pp. 320-321]). The identity of the function и
is not divulged. However, L, M, and N are undoubtedly the Eisenstein series
L(q), M(q), and N(q) defined at the beginning of Section 4 of Chapter 33. Al-
Although the meaning of Ramanujan's claim is unclear, he gives a two line "proof of
his assertion. But, Ramanujan's "proof" appears to have only a shadowy connec-
connection with his claim, and so we shall let the next entry encompass what Ramanujan
sketchily proves.
Entry 45 (p. 352). Let f be any differentiable function and set u(q) =
Mn/4f(Mi/N2), where M and N are the Eisenstein series mentioned above.
Then
du^ _ nL(q)u(q)
4 dQ 12
is a function of only M and N.
37. Infinite Series 485
Proof. We shall employ Ramanujan's differentiation formulas ([7, eq. C0)), A0,
eq. p. 142), Part II B, p. 330])
dL L2 - M
dM LM - N
dN LN - M2
Thus, using D5.1), we first observe that
d(M3/N2) M2dM M^dN
Я ~ qWlq ~ q~№~dq
dq
_ tf_ (LM-N\ _ \P_ (LN -M2
~ N2\ 3 )~ ~N>
D5.1)
Hence, by D5.1) and D5.2),
12
N2
D5.2)
which indeed is only a function of M and N (and not of L), as claimed by Ra-
Ramanujan.
Corollary (p. 352). We have
d(L*/M) 2
dN
and
<HL*/N)
dM
2N2
There is a misprint in the notebooks; Ramanujan wrote 3M instead of ЪМ2 in
the first equality. It is not clear why Ramanujan used the appellation, "Corollary,"
here.
Proof. By D5.1),
4L3 dL Li dM
d(L*/M) _ d(L4/M)/dq _ ~M~dq ~ W~dq
dN ~ dN/dq ~ dN/dq

486 Ramanujan's Notebooks, Part V
~M~\ 12 ) M2\ 3 /
LN -M1
!(-*+?)
2Ll
LN-M2 ЪМ2'
The proof of the second equality in the corollary follows along the same lines.
Entry 46 (p. 353). For Re a > 0,
и!
„J
D6.1)
Proof. Observe that each side of D6.1) has simple poles at jc = ±i(a + n) for
each nonnegative integer n. It then suffices to show that the residues of each pole
are equal. For if F(x) denotes the difference of the left and right sides, F(x) is
then an entire function which tends to 0 as x tends to oo. By Liouville's theorem,
F(x) is a constant, which obviously equals 0. Thus, D6.1) is established.
If Ra denotes the residue of a pole a on the left side of D6.1), then it is easy to
see that
(-1)"
Ri(a+n) = Т. ; " = —R-i(a+n)- D6.2)
i{a 4-и)
The calculation of the residues on the right side of D6.1) is more difficult. Now
Ra denotes the residue of a pole a on the right side of D6.1). We have
•«>n;=
ул (*+ и)! (a)t-H, (Jt2 + (a + nf)
(a - xi)n+M2i(a + n)
x~it,a+n)
(a)nn\ (x2 + (a + nJ)
(a + xi)n+i(a - xi)n+l2i(a + n)
(n + l)k(a+n)k
\+xi+n + l)k(a-.
(a),,n\
\)k
(a + xi)tt(a - xi)n2i(a + n)
x
Г \,n + I,a + n "II
la + xi+n + l,a - xi+ n +I JL=,(a
(д)„я!
37. Infinite Series 487
l,n + l,a + n
1,2й + 2и
'"]
2а + 2п + 1
ГBй+2п+
Bа
я)(-1)ТBа+п)ГBа+2я+
Г(а)ГBа + 2иJ/(а + п)ГBа + п)Г(а + п
(-l)"Ba + 2n) (-1)"
D6.3)
where we applied Gauss's theorem (Bailey [I, p. 2], Pan II [2, p. 25]). Thus,
D6.3) agrees with D6.2). The calculation of /?_,«,+„) is similar. Thus, the proof
is complete.
Observe that the right side of D6.1) equals
1
3F2
LI. я
Entry 47 (p. 355). For\x\ < 1,
<x> _2n + l oo ri _
i -jt2n+i = 2-i i-x" '
n=0
D7.1)
Proof. The left side of D7.1) may be written in the form
Bn+l)m
л=0
Arrange the terms in the array
x x3 x5 x1 x9
X2 ХЬ ХЮ X14 XX%
X*
X4
X*
X9
xn
JC15
X
X
X
15
20
25
X21
X28
_3S
X27
xi6
JC45
Summing these terms by the column-row method (Part III [3, p. 114]), we arrive
at the right side of D7.1).

488 Ramanujan's Notebooks, Part V
Entry 48 (p. 364). For \x\ < 1 and positive integers n > 2, the polylogarithm
Lin(*) is defined by
oo к
k=\
Then, for x >0,
-Jtzlog(l -
D8.1)
Proof. Trivially, D8.1) is valid for x = 0. It therefore suffices to show that the
derivatives of both sides of D8.1) are equal.
Differentiating both sides of D8.1), simplifying, and dividing both sides by 2x,
we find that we must prove that
4=1 '
1
1 7гдс
2 ~ ~2~
nxe
2"x
D8.2)
For \x\ < 1, the left side of D8.2) equals
1 °°
J
k=\ [n=0
On the other hand, if В„, 0 < n < oo, denotes the nth Bernoulli number, and
|л| < 1, the right side of D8.2) equals, by @.1),
1 71X
2 ~~2~
n=0
n=2
oo
by Euler's formula @.2). Thus, D8.2) has been verified for 0 < x < 1. Thus,
D8.1) has been established forO < x < 1. But both sides of D8.1) are analytic
for all complex x with Re x > 0. Hence, by analytic continuation, D8.1) is valid
for all complex x with Re x > 0.
In notation slightly different from that of Ramanujan, he claims on page 365
that if
D9.1)
37. Infinite Series 489
then
D9.2)
This result is also found on page 370 of Ramanujan's lost notebook [11]. As we
shall see, Ramanujan's assertion is incorrect. In our corrected version below, the
proposed equality is independent of the value of x.
Entry 49 (p. 365). //|*| < 1, then
D9.3)
Proof. Recall that, for \x\ < n (Gradshteyn andRyzhik [1, p. 42]),
1
cothjc = - +
2k~x
x t{ Bft)!
where В„, 0 < n < oo, denotes the nth Bernoulli number. It follows from Euler's
formula @.2) that, for |jc| < 1,
nx2 coth(jrjt) = i +
Upon replacing x by / and integrating over @, *) for \x\ < 1, we find that
Jo П' C° П' ' ~ ~2 + frf k + l
which is easily seen to be equivalent to D9.3).
Now suppose that D9.2) were true for |jc| < 1. Then
..
П-fOO
lim
n—»oo
OO /
n
k=]
-iy
1,2 v^ С)
OO / «w_| ,
2 \ ¦> 1 l> л
СBу)л>+1

490 Ramanujan's Notebooks, Part V
where the inversion in order of summation is justified by the absolute convergence
of the latter double sum. Replacing jc by jc2, we deduce that
,2 oc
к=\
)krBk)x7k+2
D9.4)
Comparing D9.4) with D9.3), we see that Ramanujan's claim is equivalent to
asserting that, by D9.1),
Jo
t2 coth t dt = 0,
which is obviously false.
In connection with the Lagrange inversion formula, in Chapter 3 Ramanujan
studied infinite exponentials
a" E0.1)
(Part I [1, p. 77]). L. Euler [1], [2] was evidently the first person to seriously
examine E0.1), and he showed that E0.1) is convergent if and only if e~' <a<
Upside down, on page 390, Ramanujan offers a theorem about the convergence
of the more general infinite exponential
a . E0.2)
Before stating his result, we mention some further relevant papers. As we indicated
in [1, p. 77], many authors have written about the convergence of E0.2), and an
extensive bibliography on such results is contained in A. Knoebel's comprehensive
survey paper [1]. Most authors assume that {an} is a real, positive sequence and
establish convergence when e'e < a,, < e'/e, for n sufficiently large. D. F.
Barrow [11 appears to have been the only one to venture outside this interval.
Writing а„ = ei/e + e,,, where en > 0, he showed that E0.2) converges if
and diverges if
e
lim €nn2 < ——
я-к» 2e
lim е„и2 > ——.
<-»« 2e
E0.3)
E0.4)
Furthermore, writing an = e~e - е„, where en > 0, he proved that necessarily
limn^oo<n = 0 and that limn_oo «*e« = 0, for some q > 1, is a sufficient
condition for convergence. For complex an, the most general result is due to W. J.
Thron [1] who proved that E0.2) converges if \an\ < e>/e for n sufficiently large.
We now state Ramanujan's claim on page 390. He asserts that E0.2) is conver-
convergent when
1 +loglog«* < I ji + TTli^ + („lognloglog*J + •" I ' EO3)
37. Infinite Series 491
and is divergent when the left-hand side is greater than the right side when any 1 is
replaced by 1 + e. This statement needs some clarification. First, the series on the
right side of E0.5) is finite for each n and persists as long as the iterated logarithms
remain positive. Ramanujan evidently had in mind a test for the convergence of
E0.2) when а„ > 1. If for n sufficiently large, E0.5) holds, then Ramanujan
claims that E0.2) converges. On the other hand, if one of the numerators 1 in the
series on the right side of E0.5) is replaced by 1 + <., for some fixed number e > 0,
and if there exists a subsequence а„, tending to oo for which the left side of E0.5)
is greater than or equal to the right side with one of the numerators 1 replaced by
1 + e, then Ramanujan claims that E0.2) diverges.
An easy calculation shows that Barrow's theorems E0.3) and E0.4), even in
the stronger form when the inequality < in E0.3) is replaced by <, are contained
in Ramanujan's assertion E0.5) with the right side of E0.5) truncated after the
first term.
In the remainder of this section we follow the analysis in Bachman's paper [ 1 ].
We shall establish a version of Ramanujan's claim for complex an. So that the
exponentiation is unambiguous, we assume that the sequence of complex numbers
\bn), 1 < n < oo, is given and set
«„:=«*",
n > 1.
E0.6)
With this definition,
n > 1,
E0.7)
is well defined. We first give the following test for the convergence of [En] for
complex exponents.
Theorem 50.1. Let {an} and {Е„) be given by E0.6) and E0.7), respectively. Set
an=ew"\ п>1, E0.8)
and define {?„}, и > 1, by E0.7) with а„ in place of a,,. Then, if{En) converges,
{En} must converge as well.
The test above is of independent interest. In particular, Thron's result [ 1 ] follows
from Barrow's theorem for real exponents a,,, 1 < an < ex/e, and Theorem 50.1.
To state Bachman's results concerning Ramanujan's test for convergence, we
introduce the following notation for iterated logarithms. Setting x\ = e and
L\(x) := L(x) := log*, x > e,
we recursively define jc* and Lk, for к > 2, by xk := e4-', and
Lk := Z,*_,(L(jc)), x > xk.
Entry SOa (p. 390). Let [a,,) and [ Е„} be defined by E0.6) and E0.7), respectively.
Then {Е„) converges if there exist positive integers ко and no, such that for all

492 Ramanujan's Notebooks, Part V
n >n0,
1 4- log | log an | = 1 4
Hi 1
+ h
+
1
(nLx{n)L2(n)
)
E0.9)
Entry 50b (p. 390). Let (?„) be defined by E0.7), where the sequence {а„) is
real, an > 1, for every integer n, and
J 1
+ (nL,(n)L2(«)J +"
1 +logloga,, >-j-
1
(nL{(n)L2(n) ¦ ¦ ¦ Lk<l-\(n)J (nLdn)L2(n) ¦
E0.10)
for n > no, for some positive integers ко and no, and for some ( > 0. Then the
infinite exponential {En) diverges.
We first set some convenient notation and establish three useful lemmas. The
first lemma reduces the principal case of our problem to an equivalent problem
that is easier to attack. Set
[xux2,...,x,,):=xxii and
.
Also set
to(x) := - and ?„(*) = ,..,,. ——,
x xL,(x)L2(x)-Ln(x)
n>\. E0.11)
Lemma 50.2. Let{xn),n > 1, be a sequence of real numbers such that xn > 1.
Define another sequence {Xn),n > 1, by
E0.12)
Then [x\,x2,...] converges if and only if there exists a sequence [Yn),n > 1,
such that Yn > -1 and such that the inequality
1 + Yn > A + Xn)er"*1 E0.13)
holds.
Proof. Since xn > 1, the sequence [jci,*2, ... 1 is monotonically increasing.
Hence, to prove that it converges, it suffices to show that it is bounded. By E0.12)
37. Infinite Series 493
and E0.13),
[xux2, ...,*„]< [jci, *2,..., xn, e14"^1 ] < el+r'.
In the opposite direction, suppose that [*i, jc2, ...) converges. Since, х„ >
1, then [xn,xn+l,...) also converges for each и > 1. Denoting the limit of
[xn, x,l+l,... ] by el+r", we observe that Yn > -1 and that
Thus, we deduce E0.13) with equality, and this completes the proof of the lemma.
The next two lemmas are the primary ingredients in the proofs of Entries 50a
and 50b.
Lemma 50.3. Let Tk, Ck, and Xk be defined by
к
Tk := ^2,tj(n - 1
>=0
and
E0.14)
E0.15)
E0.16)
where tj(.n) is defined in E0.11), and where к > 0 and n > 2 are any integers
for which the right sides of E0.14) and E0.15) are defined. Then there exists a
sequence of integers [n^] such that, for n > nk,
„ < Xn < Cn . E0.17)
Proof. Let к > 0 be fixed. For brevity, set Т„ = Tk and Xn = Xk. By E0.14) and
E0.11), Tn = ОкA/п) < 1, for и > пь, say, and where the notation Ok indicates
that the implied constant is dependent on к. For such n, we can expand the right
side of E0.16) in a Taylor series about 0 and so find that
\+Xn = (\+Tn)e-r"'
= 0+ Tn) A - Tn+X + \(Tn+lJ - i(
= \+Tn- Гп4_, + i(Tn+1J - Г„7„+1
Now, by E0.14) and E0.11), expanding Т„ about n + 1, we find that
Tn = Гп+1 - Гд'+1
Ok(n~4).
E0.18)
E0.19)

494 Ramanujan's Notebooks, Part V
for some number f, such that и < % < n + 1. Note that
y=0 ;=0 i"=0
E0.20)
y=0 \ J = i <=0 /
n3 \ л'log и/
and
E0.21)
E0.22)
y=o
Substituting E0.19M50.22) into E0.18) and simplifying the resulting expression,
we find that
Hence, by E0.20), E0.14), and E0.15), we deduce that
4- ± + Ok
= I v й(И) + JL + o, f —j^—
2^ ; Зи3 \n3logn
_L + Ok
33
E0.23)
Thus, for и > пк, E0.23) implies E0.17), and this completes the proof of Lemma
50.2.
Lemma 50.4. Let Г„* and Xkn be defined by E0.14) and E0.16), respectively.
Moreover, let jc * be defined by
Then
E0.24)
E0.25)
Proof. We begin with the observation that it suffices to show that there exists
a sequence of integers {n'k) such that E0.25) is valid for each n > n'k. Indeed,
37. Infinite Series 495
assuming this, we have, for each i > n'k,
by E0.24) and E0.16). To exhibit the existence of such a sequence [n'k], we first
observe that, by E0.24), Lemma 50.2, and Lemma 50.3, any infinite exponential
[¦**• xn+\>•••]' witri n 2: "*. is convergent, where [nk} is a sequence defined in
the statement of Lemma 50.3. Denote the limit of such an infinite exponential by
e1+5*. Then E0.25) will follow if we can show that
E0.26)
for all n > n'k > nk.
To this end, we define, for integers к > 0 and n > л*, the numbers f* by
We will deduce E0.26) from the three inequalities,
S* > Sen > 0, *>?;«> sup(nb n,),
(* > 0,
and
E0.27)
E0.28)
E0.29)
E0.30)
where n'k is sufficiently large, where in the case к = 0, L0(x) = x. Indeed, assume
that E0.26) fails for к = 0 and some n > n'o. Then, by E0.29), /° > 0, and so,
by E0.30) and E0.14), we find that
for some m > n, where m is sufficiently large in terms of <°. But, by E0.27),
this implies that S° < 0, which contradicts E0.28). Thus, E0.26) is valid with
к — 0 for all n > n'o. Proceeding by induction on it, assume that E0.26) holds up
to it - 1. Assume, to the contrary, that E0.26) fails to hold for some к > 0 and
и > п'к. By the same argument as used above, we find that
/* > tk(m - 1),
for some m > n that is sufficiently large in terms of t*. This, together with E0.27)
and E0.14) shows that
E0.31)

496 Ramanujan's Notebooks, Part V
by the inductive hypothesis, provided thatm > n'k_y, which we may assume. But
since E0.31) contradicts E0.28), we conclude that E0.26) and therefore E0.25)
hold. Thus, it remains to prove E0.28)-E0.30).
To that end, assuming that [nk) is increasing, as we may, we find that, for к > I
and n > rik > ri(,
Xk > Xi > 0,
and so
xk > xln > exl\ E0.32)
by E0.17), E0.15), and E0.24). Recall that Euler [1], 12] showed that the infinite
exponential with constant exponents e1/e converges to e. This fact, together with
E0.32), yields E0.28).
To prove E0.29), we first observe that, for m > n > nk,
\xk xk xk\<\xk xk xk el+r"*'l -f?1+r»
by E0.24) and E0.16). Hence, 5* < Tk, and so E0.29) holds by the definition
E0.26) off*.
For the proof of E0.30), we first observe that, by the definition of Sk and E0.24),
Hence, 5* satisfies E0.16) with Tk replaced by Sk. We now fix к and write Sn, Т„,
and tn for Sk, Tk, and tk, respectively. From our last observation it follows that
Substituting Sm = Tm — tm and m — n, n + 1 into the last identity, we find that
-?V = 1-«-'"'. E0.33)
By E0.29), E0.28), E0.14), and E0.11),
0<tn < Tn
E0.34)
Hence,
1 _<,-'»" =/n4
provided that n > n'k, where n'k is sufficiently large. Using this bound on the right
side of E0.33), we deduce that
1+W./2
tn+\ > 'n-
+Ta
Hence, for any integers m > n > n'k,
'Ft' 1+/.Ч-1/2
tm > /- 1 1 ,+7} ¦
E0.35)
37. Infinite Series 497
We use E0.35) twice. First, by E0.35) and E0.34),
m-\
tm > 'л П ttt = '"
i=n ^ '
i
m-\
for m > n > n'k, where n'k is sufficiently large. Now, by E0.14) and E0.11),
m—I m—\ к к /•m—2
?/
i=n i=n ;=o
к к
J2 {Lj+dm - 2) - Ly+iOt - 2)) < ^Z.y+,(m - 1).
i=o j=o E0.36)
Hence, by E0.11),
tm > tn exp I -
= /„«*(«- 1),
E0.37)
for any integers m > n > n'k. We now reiterate the argument above but this
time using E0.37) instead of E0.29) on the right side of E0.35). Employing also
E0.36), E0.34), and the inequalities tnik(i)/2 < Tt/2 <t 1/r, we find that
='"exp
(m-l
n(Lk+i(m - 1) -
;=o
where the second inequality above holds for m > n > n'k, where n'k is sufficiently
large. Thus, E0.30) is established, and the proof of Lemma 50.4 is complete.
Proof of Theorem 50.1. We assume at the outset that an Ф 1, for each n > 1,
for otherwise both [01,02,...] an<^ [<>i > <b,... ] converge trivially. Fix a positive
integer n and, for any complex number г, set
and
/U) := U
dz
g(z) := -^-l
dz
2 an,z].
E0.38)
E0.39)

498 Ramanujan's Notebooks, Part V
For each m > n,
[a\,a2, ... ,am] - [a\,a2, ... ,а„] =
_ }M
|Д„ + |.Пл+ 2 am\
Setting
and
и := [ап+\,а„+2, ¦¦¦,ат]
w : = [an+i,an+2. • • •. om],
we find, upon estimating the right side of E0.40), that
f(z)dz. E0.40)
E0.41)
E0.42)
\[а\,а2 ат]-[щ,а2, ...,ап\\ = / f(z)dz
= I [ /A+(и- 1H<*A + (м- 1H
1Л
< и-\\ Г
Thus, by E0.38), E0.6), E0.39), and E0.8),
f(z) = b\[a\,a2,. ..,an,z\^-[a\,a2, ...,an,z\ = I »*№.
and
g(z) =
k=\
E0.43)
Hence, by E0.6), E0.8), E0.41), and E0.42), we obtain the inequalities
П
Y\k,ak+\ а„,0 -t+ut)]\
n
П \bt\[&k, ak+ , an, A - I + \u\t)]
-t +Wt),
E0.44)
37. Infinite Series 499
which is valid for 0 < t < 1. Applying E0.44) to the right side of E0.43), we
find that
|[аьа2,...,ат]- [a,,a2,... ,an]\ < \u - 1| / ^A + (w - 1H <*'
Jo
= l-^—^- ( 5d + (и» - 1H^A + (w - 1H
_ I" - H Г'
ID — 1 J|
w~] ' E0.45)
by E0.39) and E0.42). Observe that w > 1, since an ф I, and so an > 1.
Moreover,
2,an+i,..., am])k
k=l '
E0.46)
Theorem 50.1 now follows from E0.45), E0.46), and the Cauchy criterion for
convergence.
Proof of Entry 50a. By Theorem 50.1, it suffices to consider real exponents
a,, > 1. In such a case, [а\,а2,...,а„] is monotonically increasing, and so it
suffices to show that it is bounded. Define the sequence [cn) — {c*u+l) by
/ 1 + Cn*«+1 \
cn=expl I, и>
where C*0+l and и^+| are defined in the statement of Lemma 50.3. Setting Cn —
СУ, we find that, by E0.15), E0.11), and E0.9),
i + log log с = logo + С) = С - \cl + O{Cyn)
ко
;=0
for n > no, sufficiently large in terms of #o, as we may assume. Therefore, for
n > «o, we have an < с„, and so
[ann,a,l(l+ ,an] < [с„0,с„1)+| cn].
Thus, it suffices to show that the infinite exponential [c,,0, cn<1+i,... 1 converges.
By Lemma 50.2, this, in turn, is equivalent to the existence of a sequence Sn, n =

E0.47)
г*0+|
500 Ramanujan's Notebooks, Part V
«о. "о + 1. • ¦ •. such that Sn > -1 and
But, by Lemma 50.3,
1 + С = 1 + C*o+1 < 1 4- X* =A4- rnto+
Hence, E0.47) is satisfied with Sn = Г*о+1, и > щ. This completes the proof of
Entry 50a.
Proof of Entry 50b. We proceed by contradiction. Suppose then that the infinite
exponential [a\, 02,... ] is convergent. Then, since an > 1, so is [а„, an+i • • • • 1
convergent for any n > 1. Denote the limit of the latter infinite exponential by
el+s". Define also a sequence {An} by
+А
Then
e1T"" =
E0.48)
E0.49)
In the remainder of the proof, n always denotes an integer such that n > и0.
For such n, it follows immediately from E0.10) that А„ > 0, since а„ > e^1'.
Moreover, by E0.48), E0.10), E0.11), E0.15), and E0.17),
А„ > log(l + An) = 1 +loglog<3n > С„*° + ^?l(n) > Сл*«+| > Xn\ E0.50)
for n > no, where n^, which depends on it and e, is sufficiently large. Thus,
an > xkn\
where jc^ is defined by E0.24). Therefore, by the definition of Sn and Lemma
50.4, we find that
Sn > T*.
For brevity, set Tn = Г„*°, Хп = X%>,
Bn = An-Xn, and Rn = Sn
by E0.51). By E0.50), E0.23), E0.15), and E0.11),
вп > О + !<(„) - x1? = \al(n) -3^ + 0
E0.51)
E0.52)
E0.53)
where n > n0, and where и0 is sufficiently large.
The remainder of the argument in Bachman's paper [1] is incorrect. We are
very grateful to A. Hildebrand for supplying the following elegant argument to
complete the proof. We begin with a lemma.
37. Infinite Scries 501
Lemma 50.5. For M,N > n0,
Proof. From E0.11), we observe that
Thus,
М к
к М
y=0n=,
Since ?j(x) = L'j{x)ILj(x) is decreasing forx > no,
L,(n-\) "/,_, Lj(x)
Thus,
- 1)
(и-1)
-f^JN L,{x)
and so the proof is complete.
Recall from E0.18) and E0.49) that
1 + Х„ = A + Tn)e-r^ and 1 + A,, = A + Sn)e~s'-'.
Thus, from E0.52) and E0.54),
В„ \ + А„ \+Sn _,
E0.54)
1 +
1 4- Х„ 1 + Xn \+Tn
Hence,
Since Л„н > 0 by E0.52), equality E0.55) first implies that
Rn > Bn
1 4- Т„ - l+Xn'
and then, with the use of the inequality,
h
log(l
1 4-х

502 Ramanujan's Notebooks, Part V
secondly implies that
Bn
1 (l±X»R -B\
Лл — On — С l\n Dn,
<
where, in the last inequality, we used the fact that Я„+, > О from E0.52), and
where, in the last equality, we used E0.54). Thus,
Iterating this inequality, we find that
7=1
By Lemma 50.5, for n > n0,
E0.56)
E0.57)
Hence, using E0.53) and E0.57) in E0.56), we conclude that
Since m > 0 is arbitrary and since ??, €»(я + у - 1) diverges, we have reached
the desired contradiction, and so the proof is complete.
38
Approximations and Asymptotic
Expansions
One of the primary areas to which Ramanujan made fundamental contributions,
but for which he received no recognition until recent times, is asymptotic analysis.
Asymptotic formulas, both general and specific, can be found in several places
in his second notebook, but perhaps the largest concentration lies in Chapter 13.
Several contributions pertain to hypergeometric functions, and an excellent survey
of several of these results has been made by R. J. Evans [1]. The unorganized
pages in the second and third notebooks also contain many beautiful theorems in
asymptotic analysis. This chapter is devoted to proving these theorems and a few
approximations as well.
On pages 270-273 of the second notebook, Ramanujan examines some related
functions that can be considered as hybrids of the Riemann zeta-function and
hypergeometric functions. Some of these results were established in a paper with
Evans [2]. Entries 2-8 contain accounts of Ramanujan's findings described on
these pages.
In Entries 12 and 13, Ramanujan determines the asymptotic behavior of some
multivariate exponential series. These results are in the spirit of several theorems
that can be found in Chapter 15 (Part II [2, pp. 303-314]).
In Entry 16, Ramanujan derives the asymptotic expansion of
as / tends to 0 +. A complete description of the asymptotic expansion of this false
theta-function involves Euler numbers. All of the coefficients in Ramanujan's
asymptotic expansion appear to be integers, and this was recently proved by W.
Galway [1] using a formula from Ramanujan's lost notebook [11].
Entry 17, which can probably be generalized, gives the asymptotic expansion
of a function which is a hybrid of a theta-function and a hypergeometric function.
This is also related to the aforementioned material in Chapter 15.
Entries 18 and 19 present families of approximations to certain finite sums and
certain infinite series, respectively. These approximations arise from the orthogo-

504 Ramanujan's Notebooks, Part V
nality, respectively, of the discrete Hahn and discrete Charlier polynomials. The
results are quite remarkable, for no one had previously realized that Ramanujan
must have, in essence, discovered these orthogonal polynomials.
Entry 23, which can also undoubtedly be generalized, provides an asymptotic
formula for a certain Lambert series as x tends to 0.
We quote Ramanujan in the first entry of this chapter.
Entry 1 (p. 265).
X
= 0.1015314 + loglogU2 + x + 9);
x = oo, в = j;
x = l, 0=0.46811;
130489 quadrillion terms to get the value of 5.
A.1)
A.2)
A.3)
A.4)
Proof. Applying the Euler-Maclaurin summation formula, @.5) of Chapter 37,
to /(jc) = 1/(jc log*), we find that, for some constant c,
= с + log log* +
1
2* log*
as jc tends to oo. From Entry 14 of Chapter 7 (Part I [1, p. 166]),
log log*) =0.7946786,
lim (Y —Ц- - log log x ) = 0.
— I. A-5)
A.6)
i.e., the constant с in A.5) equals 0.7946786. On the other hand, observe that
/ 1
logM +- +
= B1ogx)
2* log* 2*2log
gjc \*3log*/y
as x tends to со. Thus,
+ O
Vj^log2*,
A.7)
Now Iog2 = 0.6931472, and 0.1015314+ 0.6931472 = 0.7946786. Hence, by
A.6) and A.7), the constant terms in A.1) and A.5) are in agreement. We also see
that if we set в = | in A.7), then the first three nonconstant terms in A.5) and
A.7) agree. This then proves Ramanujan's assertion A.1).
38. Approximations and Asymptotic Expansions 505
When * — 1 and 9 — 0.46811, the right side of A.1) should approximately
equal 0, if Ramanujan's claim is correct. Since
loglogB.46811) = -0.1015315,
the right side of A.1) to seven decimal places is -0.0000001, which justifies
Ramanujan's claim in A.3).
The assertion A.4) is recorded in a different color or different shade of ink
in some "empty space" further down the page. With an American interpretation,
130489 quadrillion = 1.30489 x 1020. However, in the United Kingdom and areas
under its former dominion, one quadrillion equals 1024 instead of 1015. Therefore,
Ramanujan claimed that 1.30489 x 1O29 terms will give a sura exceeding 5. This
is in agreement with work of Hardy [2, p. 61] and R. P. Boas Jr. [1, p. 244], [2,
p. 156] who showed that 1.3 x 1O29 terms are required to exceed a sum of 5. See
also Part I [1, p. 328].
We are grateful to R. P. Brent for showing us the advantage of log log(*2 +* +в)
over loglog(*2) through his analysis in A.5) and A.7).
Some of Entries 2-8 below, recorded on pages 270-273, are not approximations
or asymptotic estimates, but since all the results are connected and asymptotic
expansions are the focus, we prove and discuss all of them here.
Entry 2 (formula A), p. 270). For p > 0,
^—' In 4
k=\
j=\
-. B.1)
Proof. Let S denote the double sum on the right side of B.1). Then
s = Z<b
k=\ »i=0 ;=1
oo * , i
U-m)l
B.2)

506 Ramanujan's Notebooks, Part V
We examine the contribution of the double finite sum on j and m. Inverting the
order of summation, we find that
^ 1 ^
(-1)"
к-т
(-1)"
-m-ny. n\
Hence, substituting B.3) into B.2), we have shown that
B.3)
-. B.4)
Let Ak denote the inner double sum in B.4). Inverting the order of summation
and employing Vandermonde's theorem (Bailey [1, p. 3]), we deduce that
(-1)"
(-iy(P + *y * {-к),
fc! U (" + JV-
{-1)т(Р + кГ /A (-fc); _ \
k\ m! \]~ (m + 1), /
(-l)"'(p + fe)m / (m)k _ \
Thus, in view of B.1) and B.4), it remains to show that
/t=| m=0
k\m\(m
B.5)
where we have employed the binomial theorem and inverted the order of summa-
summation. Comparing the coefficients of p" on both sides of B.5), we sec that it suffices
38. Approximations and Asymptotic Expansions 507
to show that
(-1)"
(» + !]
n > 0.
B.6)
Multiplying both sides of B.6) by (-l)"n!, we see that B.6) is equivalent to
B.7)
It is an easy and well-known consequence of the binomial theorem that, for
each j > 2, the inner sum on the far right side of B.7) is equal to 0 (Gradshleyn
and Ryzhik [1, p. 5, formula 0.154, no. 3]). Hence, the right side of B.7) reduces
to l/(n + 1), as desired. Thus, the proof of B.6) and, consequently, of B.1) is
complete.
Ramanujan's formulation of Entry 2 is slightly imprecise, because he only offers
the first three terms of the sum on к on the right side of B.1). Note that the first
three values of A* are 1, 1. and 3.
Ramanujan next expands the inner sum on the right side of B.1) in powers of
к with coefficients that are powers of I/p. Perhaps Ramanujan was striving to
obtain an asymptotic expansion for the left side as p tends to oo. This procedure
does not lead to the desired end, because the contribution of fem, m > 1, to the
sum on m yields a divergent series. At any rate, we next establish Ramanujan's
expansion of the inner sum.
Entry 3 (p. 270). Let 1 < к < \p\, where к is an integer and p is any complex
number. Ifm and n denote integers with m > 0 and n > 1, define
t+m-n /- _ I \
«„(»«. ^) := E (- 0m"r ( _ \s(r + и - m, r + 1),
C.1)

508 Raman ujan's Notebooks, Part V
where s(a,b),a,b > 1, denote the Stirling numbers of the first kind. Then
P
Moreover, ifn < к andm + 1 < n < 2m, thenan(m, k) = 0.
Proof. The Stirling numbers s(.j, r), j, r > 1, of the first kind may be denned by
(L.Comtet[l,p.213,eq.
(* -
, r)kr~{.
C.3)
r=l
Thus, using C.3) and inverting the order of summation, we find that
By the generalized binomial theorem, for it < \p\,
Thus, letting I = m + 1 — r and i = j — r, we find that
* *
EED-^r)^ +Je
r=\ j=r 1=0 У \ I
's(j.r)/m+ j -r
m=0 r=\ i=0
oo *— I i
oo It—I
m=0 1=0 P r=0
oo m+k < *+m-n
- T{-l)m-rs(r+n-m,r+l)(n M.
3.4)
C.5)
where we have set n — m + i + 1 in the last step. Hence, the proof of C.2) is
complete.
Lastly, from C.5) and C.1), we see that it remains to show that the inner sum
on r vanishes if n < к and m + 1 < n < 2m. Alternatively, from C.4), we will
38. Approximations and Asymptotic Expansions 509
show that
Y(-l)m-rs(r +i + \,r
C.6)
for each j such that 0 < j < m — 1 and »i < ifc — i — 1.
From Comtet's book [ 1, pp. 227-228], s(r + i + 1, r + 1) is the coefficient of
t' in the power series expansion of
\e'-\J r\
Therefore,
is the coefficient of t' in the expansion of
Thus, the sum in C.6) is the coefficient oft' in the expansion of
i+i
m!
for certain numbers a,, /' > m. Thus, for 0 < i < m — 1, the coefficient of t' is
indeed equal to 0. Thus, C.6) has been established, and the proof of Entry 3 is
complete.
Entry 4 (Formula B), p. 271). For each p > -1, define в - 9P by
1 1
(p+lHp + 2)
4
(p + l)(p + 2)(p + 4)Cp
Then O.i = -2.5856, в0 = 0.0069,6», = 0.4137, апйвоо = \.
D.1)

510 Ramanujan's Notebooks, Part V
Proof. Multiplying both sides of D.1) by (p + 1) and then setting p — — 1, we
find that
Thus,
12e
4e-9
- 20 = -2.585603,
which is in agreement with Ramanujan's claim.
Second, letting p tend to 0 in D.1), we find that
Solving for 90, we find that
10 -л2
3-2-4 2-4B3+6»0)
-23 = 0.006912,
)
which again is in agreement with Ramanujan.
The case p — 1 is more challenging. Letting p — 1 in D.1), we see that
S:=
_ i 1Л 1 2
~ ~ё + ё\2~6 + 45~
15B6
Solving for $i, we find that
6», = —
-26.
D.2)
45e(S- 1) -+-28
To determine 6\ to the accuracy demanded by Ramanujan, we need a very precise
determination of S.
Write
1.
Then
where
We are grateful to W. Root who calculated A and found that
A =0.16410279790586.
D.3)
(=i
106 oo
V^ Mk and В = 7 Mb.
38. Approximations and Asymptotic Expansions 511
Hence, from D.3),
= 0.7692402231818 + B.
Since (A:/(Jt +1))* — 1 /e monotonically decreases to 0 and has a value at к = 106
that is less than 2 10, we deduce that
OO г. 1Г.-7 Г0
< /
k2 Jw
Hence,
5 = 0.769240223182.
Returning to D.2) and using the value of S calculated above, we find that
6, = 0.413696,
which agrees with Ramanujan's calculation.
We have calculated S to more accuracy than needed to establish Ramanujan's
claim. However, it seems to us that a calculator or computer is necessary to cal-
calculate S to the precision needed to determine в\ to the accuracy indicated by
Ramanujan. So, we wonder how Ramanujan computed 5.
The proof that 9Ж = | is considerably deeper than the previous calculations.
We show later that this result follows from Entry 7.
Entry 5 (Formula C), p. 271). LetO< p <a. Then
^ (а + я)"-' ^^(-р)пи„(а)
where, for n > 0,
Furthermore, for n > 1,
un_i(q) - un(a+ 1) _ a
un(a) - un{a + 1) n'
E.1)
E.2)
E.3)
Proof. By the generalized binomial theorem, for \p\ < a,
(a + n)"
+ и)»
n=0
(a + nJ
*=0
it!
Ш'

512 Ramanujan's Notebooks, Part V
~ 2-i i-i
*=0
from which E.1) follows.
Next,
ия_,(а)-и„(в
+!)„-,
(*)„
n+2
(и - 1)! . ^ (* + !)„_
¦{(a+k)-k)
[(к + п)-к)
it!
Thus, E.3) has been established.
Entry 6 (Formula D), p. 271). Let un(a) be defined by E.2). Then as a tends to
oo,
1
1
и(п- 1)\ I
1 n n(n-l) w(/i-l)(w-2)
+ l~30+ 12 + 4 + 16
I n
\ 2
/1 n n
\42 ~ 12
5n(n-\) 5n(n-l)(n-2) n(n - \)(n - 2)(n - 3)\ 1
+ + ) a*
24 32
(n-1) 5и(и-1)(и-2) 5и(и - 1)(и - 2)(я - 3)
1
32
18 16
и(и - 1)(п - 2)(/i - 3)(и - 4)
64
ln(n - 1) 7и(л - 1)(я - 2) 35л(л - 1)(п - 2)(и - 3)
+ ( т^ —. Ь ~ +
24
72
- 3)(w - 4)
96
64
38. Approximations and Asymptotic Expansions 313
n(n-\)(n- 2)(л - 3)(я - 4)(я - 5У
128
F.1)
First Proof. Recalling the definition C.3) of the Stirling numbers s (л, г), we find
that
Л + 1
(A 4- 1)„ = (-1)" VJs(«+ l,r)l
Г = 1
r-l
r=0
= (-1)"
!> r
1, г
/=o w F.2)
Recall that the Hurwitz zeta-function f (s, a) is defined for Re s > 1 and a > 0
by
Thus, from E.2) and F.2),
*=o
и„(а) = (-1)" J>
*=0
r=0
= (-1)
T]j(« + 1, r + 1) У" fr)(-l)r-ya>?(n
r=0 7=0 V/
+ 2 + у - r, a).
F.3)
By the Euler-Maclaurin summation formula, @.5) of Chapter 37, as a tends to
oo, for Re s > 1,
m=0
where Bm,m > 0, denotes the /nth Bernoulli number. Employing this asymptotic
expansion in F.3), we deduce that, as a tends to oo,
r=0 j=0
Inverting the order of summation on j and m in F.4), we are led to the inner sum
l)i(r\(-m+n+J ~rV
-гУ.уу (-r)j(m + n - r + 1);

514 Ramanujan's Notebooks, Part V
by Vandermonde's theorem (Bailey [1, p. 3]). Observe that if 0 < m < r, where
r > 1, then the sum above equals 0. Extracting the term for m = 0, we find that,
from F.4), as a tends to oo,
1, г
r=0
(л
F5)
We now calculate the coefficients of a~'v, 1 < A/ < 8.
First, the coefficient of 1 /a equals
s(n+l,n+ 1)
(я +1)! n + Г
in agreement with F.1).
For the remainder of the calculations, it will be convenient to use the following
formula (C. Jordan [1, p. 150, formula C)]):
2m
F.6)
F.7)
where Cio = -1, С,.* = 0 if к / 0, and
Cm+i.k = -Bm - к + l)(Cmjt + Cm *_i).
The coefficients Cm,k for 1 < m < 6, 0 < fe < 5, are found in a table on page
152 of Jordan's book [ 1 ].
For n > 1, the coefficient of a~2 equals
(я- 1)! n(n+\) 1 1
-s(n + \,n)
(w + 1)!
2 (n+l)n 2
If л = 0, the series in F.5) also yields \ for the coefficient of a 2.
For n > 2, the coefficient of а~г equals
)и(и-1)
n-2 2_n 1
"T" + 3 " 4 6"
For n = 0, 1, we obtain the same value.
For the remainder of the calculations, we assume that n > N — 1. In each case,
the same formula for the coefficient of a~N also holds for 0 < n < N — 2, from
an examination of the double sum in F.5).
38. Approximations and Asymptotic Expansions 515
The coefficient of a, if n > 3, is equal to
-s(n + 1,и - 2)-7
4
(я - 3)(п - 4)
8
п(п — 1)
We employed MACSYMA in F.5) to calculate the remaining coefficients as
polynomials in n. We then collected terms to write Ramanujan's coefficients as
polynomials in n to verify that the polynomials agree for each coefficient. The
coefficients of a, a, and a'1 are, respectively,
s(n + l.n-3)
4! (п-4)!
(п+1)!
15/13 + 15п2 - Юн-8
240 '
5! (я - 5)!
-s(n + 1, n - 4)
(n + 1)!
96
and
s(n + \,n- 5)
6! (и - 6)!
5! (и - 6)!
63w5 - 315и3 - 224n2 + 140л + 96
= 4032 ¦
The coefficients of s{n + 1, n — 6) are not found in Jordan's book [1]. Thus, we
used F.6) and F.7) to calculate the needed coefficients. Therefore, the coefficient

516 Ramanujan's Notebooks, Part V
of a~s equals
- s(n + \,n -6)
= {135135
7! (л - 7)!
(n + 1)!
+ 540540
Си')
9и6 - 9и5 - 75и4 -
114n2 + 80л
П52 '
This completes the proof.
Although our first proof is a natural one, Ramanujan's formulas for the co-
coefficients of a~N indicate that another approach utilizing less calculation was
employed by him. In our second proof, calculations lead to coefficients in the
form given by Ramanujan, but the calculations are even more difficult to perform
by hand than in our first proof.
Second Proof. For n > 0 and a + к > 0,
(a + jfc)-"-2 = ! Г e-(a+k)'tn+'dt.
(и + 1)! Jo
Using also E.2) and inverting the order of summation and integration by absolute
convergence, we find that
_ r° (
Jo OT+1
= fVav
rdf
F.8)
where we have used the generalized binomial theorem and set
1 / t \"+l
Applying Watson's Lemma (Olver [1, p. 71]), we deduce that
*=o
as a tends to oo. Thus, it remains to calculate ^
. 0 < A < 7. Since
Я + 1
38. Approximations and Asymptotic Expansions 517
where Br denotes the rth Bernoulli number, we may readily calculate the coeffi-
coefficients. With the help of MACS YMA, we easily verified Ramanujan's coefficients
ofa'^.l <N<%.
Entry 7 (Formula E), pp. 272-273). Let a, p > 0. As p tends to oo,
vv/iere /*2„ : — P2n (/>), и > 1. iie polynomial in p of degree n — 1. /и particular,
and
5p2
30 10 9 54 '
5 5p lip1 35p3 35p4
691 691p 616n2 451p3 385p4 385p5
+ { + ^ + & + & +
2730 210 45 18 18 54 '
7 35p 7709p2 26026p3 2002p4 7007p5 5005p6
+ + + + + +
Moreover,forn > 1,
6000
- 2)(n - 3)(/i - 4)
240000
|(л-5)(и-6)(и-7)
-^(я - 5)(n - 6) + ^(n - 5) +
60001
G.2)

518 Ramanujan's Notebooks, Part V
Lastly, for л > 2 and n even,
*-^Ц>2
18
-Bn-
n2(n-\)
2)(л+3)(п
180
n(n- 1)(п-2)(я-3)
120
I (n + l)(n + 2)(и + 3)(я + 4)(я + 5)
36
п\п2-
2700
¦2)
о„
-S,,-2
270
л(п - \){n - 2)(w - 3)B3n - 25)
+ 5400 '
By, j > 0, denotes the jth Bernoulli number and where РгАр) has degree
n - 1, for n > 0.
Before proving Entry 7, we shall show that the case в^, - \ of Entry 4 follows
from Entry 7.
Completion of the Proof of Entry 4. Let
A{p)
/
pl2)
\-e-p
J.
In Entry 4, Ramanujan is claiming that, if the rational function within the large
parentheses of D.1) is expanded in powers of 1/p when в = |, then the first five
terms coincide with the asymptotic expansion of A(p) in powers of \/pasp tends
to oo. Expanding this rational function in powers of 1 / p, we therefore must prove
that
1 2 16 56 3712
D.4)
as p tends to oo.
Now let a = I and replace p by p/2 in G.1). Then, as p tends to oo,
4/3
p4\ + 2/p) рЦ1 + 2/pK
32C0 + T2
*- 2/PM
38. Approximations and Asymptotic Expansions 519
3712
_
1? + Jp1 3p4 + 4V
Comparing D.4) and D.5), we complete the proof.
Proof of Entry 7. From Entry 5 and F.8), forO < p < a,
ж (—2dV
S(a,p)= J2^-^-un(a)
я=0 П-
°G)
D.5)
l
G.4)
Referring to the definition of S(a, p) in G.1), we see that S(a, p) represents an
analytic function of a and p for Re a > 0 and Re p > 0. Likewise, the right side
of G.4) is analytic for Re a > 0 and Re p > 0. Thus, by analytic continuation,
G.4) is valid for all a and p with Re a > 0 and Re p > 0.
Multiplying both sides of G.4) by — e2'1, we see that G.1) is equivalent to the
asymptotic expansion
as p tends to oo. Since the first term on the right side of G.5) is equal to
1 /-00
2pJo
the asymptotic expansion G.5) is equivalent to
where
2t
n=\
where By, j > 2, denotes the yth Bernoulli number.
\t\ <2tz,
G.7)

520 Ramanujan's Notebooks, Part V
Let us now define the polynomials Р-щ (p), n > 1, by the expansion
2p
We therefore shall prove that the polynomials Pinip) have the properties enunci-
enunciated in Entry 7.
By employing MACSYMA, we verified that P2(p), fiU/O. • ¦ •, Puip) are in-
indeed given by the formulas displayed in Entry 7. Second, we remark that it is easy
to see from G.8) that the polynomial Ргп{р) has degree n - 1, n > 1.
Next, we prove G.3). From G.8),
f
!<Ч
л=1
In particular, by G.7),
hi Q»v.
<2л\
2л.
G.9)
G.10)
Equating coefficients of/2", и > 1, we find that /"^(O) = (-l)n"'B2n, as claimed
by Ramanujan in G.3).
Next, differentiating G.9) with respect to p and setting p = 0, we find that
Bn)!
to = w
4
Bn)!2!
) 2. =
12'
Л= I
Bn)! 3!
Bя)! 4!
' t2n — __
~ 48'
w
240'
and
2^ г2„-ц S
Bи)!5!
) 2, _
1440'
G.11)
G.12)
G.13)
G.14)
G.15)
Since the coefficient of pm, m > 0, in the Taylor series of Pn(p) about p - 0
equals Fn(m)@)//n!, we can calculate the coefficients of pm on the right side of
G.3), for 1 < m < 5, by equating coefficients of f2" on both sides in each of
the foregoing five equalities. These calculations are facilitated by observing from
G.7) that
w2 = 2w + 2tw' + t7.
Hence,
38. Approximations and Asymptotic Expansions 521
w3 = 2w2 +t(w2)' + wt2,
u»4 = 2ш3 + 2f (u»3/3)' + in2/2,
ш5=2ш4 + 2/(ш4/4)' + ц|3/2,
w6 = 2w5 + 2/(w5/5)' + ш4/2.
Using these five equalities in G.11 )-G.15), utilizing G.10), and employing MAC-
SYMA, we can readily verify that each of the coefficients given by Ramanujan in
G.3) is correct.
Lastly, we prove G.2). Replacing p by 1/p and / by t^/p in G.8), we find that
and
я=1
or
Bя)!
(-1)2,,(P)
Bл)!
/2п, G.16)
where Q2n(p) := p" ' Рщ^/р) is a polynomial in p of degree и - 1. Thus, the
coefficient of p""m in P2n(p) equals the coefficient of p'"~l in Qm(p)-
Now, by G.7),
?,/):=
In particular,
Thus, from G.16),
OO /
Bл)!
E
и@./)=-
-/ =
G.17)
G.18)
,2k
Equating coefficients of z2", n > 1, on both sides, we find that
„ „, Bл)!
2-6"и!'
as claimed by Ramanujan.
Next, by G.17),
t* /4
G.19)

522 Ramanujan's Notebooks, Part V
Thus, from G.16), G.18), and G.19),
2f^ (-1)" 6^@) 2, = у- и*-'@,0и,@.0 = у i-0j_
fa Bл)! ?- (fc-1)! ?f
Equating coefficients of t2n, n > 1, on both sides, we find that
Bn)! и(л -
- 1)!
2-6"-2360(/i-2)! 2-6"я! 10
which again agrees with Ramanujan.
Next, by G.17),
4jB6/6
63-35'
G.20)
Thus, from G.16) and G.18)-G.20),
) г» =
= ?(—
(Jk-2)! (fe-1)!
Thus, equating coefficients of t2", we find that
1 1
2!
_ Bя)! /
4 \6"
Bя)! /л(я
~ 2-6"n! \
C60J(/i-4)! ' 65(/i-3)!
- 2)(я - 3) n(n-l)(/i-2)
200
70
as claimed by Ramanujan.
The remaining two coefficients recorded by Ramanujan are similarly calculated,
and we omit the details. (We used MACSYMA to effect the calculations.)
We now return to the task of establishing G.6). We would like to employ Wat-
son'sLemma, but we cannot do so because exp(pu»(/))- 1 depends upon p. Thus,
a completely new procedure seems necessary. We prove a very general theorem
(Theorem 7.1 below) from a paper by the author and R. J. Evans [2] that includes
G.1) as a special case.
Define
T(x) :=
n=0
n\ (x + a + w)"
G.21)
where r is a fixed positive integer, and a and s are fixed complex numbers with
positive real parts. Note that when r = s = 1, Tilp) = S(a, p), where S(a, p)
is defined in G.1).
38. Approximations and Asymptotic Expansions 523
Theorem 7.1. Let N denote a positive integer. Then as x -* oo in the sector
| ArgJtl < \n - 8, where 8 > 0,
2'
r-l
,-t-j
/N-\
\m=0
Cm(x)
(a
ou-'-'-"
, G.22)
w/iere
G.23)
and where the Junctions Cm(x) (defined in G.32)) have the estimate
Cm(x) = O(x|m/2'-r), G.24)
as jc few/j to oo.
The arbitrarily small positive number 5 is fixed throughout the sequel. Observe
that G.22) is a genuine asymptotic expansion, in view of G.24). Note also that x
can be replaced by x + b in G.22), for any constant b. Thus, for example, if the
sign of a is reversed in the denominator of G.21), then Cm{x)/(a + jc/2)m+l is
replaced by 2m+1Cffl(je - 2a)/xm+].
If r = s = 1, the leading sum on the right side of G.22) equals 1 /x. Further-
Furthermore, assuming the validity of G.22), we conclude that
CmBp)
f 0
if m is odd,
if m = 2n is even.
Before beginning the proof of Theorem 7.1, we need to define several functions
and prove an auxiliary lemma.
Consider the confluent hypergeometric function
G.25)
This function is related to U(s, s + г; г), the confluent hypergeometric function
of the second kind, by
G.26)
where ^n < arg z < jtt (see, e.g., Olver's book [1, p. 257, eq. A0.09)] or N. N.
Lebedev's book [ 1, p. 270, eq. (9.12.4)]). In many lexts (e.g., Lebedev [ 1, p. 263]),
U is designated by 4*. As z -»¦ oo with | arg г| < \n ~ 8, we have the asymptotic
expansion (Olver [ 1, p. 256])
U(r,s+r;z)~Yl-
m!zm+r
G.27)

524 Ramanujan's Notebooks, Part V
Since r is a positive integer, U(s, s + r; z) can be expressed as a Laguerrc poly-
polynomial (Erdelyi [2, pp. 188-189, eqs. G), A4)]). Thus,
U(s,s +r;z) =
*=o
7k+.i
For brevity, write, for t > 0,
w := w(t) :=
1-е-''
so that by @.1) of Chapter 37,
OO D
W =
m=0
и»!
For r > 0 and Re x > 0, define
f(t,x) := е*(|
Finally, the functions Cm(x) in Theorem 7.1 arc defined by
С» (x) = /(m) @, x), Re x > 0,
where the superscript m denotes the mth derivative with respect to /.
We remark that in the case r = 1,
f(t,x)=x-'ex+"'2r(s,wx),
where V(s, z) denotes the incomplete gamma function
/oo
e-'ts-xdt, Res>0.
This follows from G.31) and the formula (Erdelyi [2, p. 136, eq. A5)])
Define, for each integer m > 0,
Г(т+Д+П)
G.28)
G.29)
G.30)
G.31)
G.32)
G.33)
Note that Um(a) generalizes the function un(a), defined in E.2). From Euler's
integral representation of the gamma function,
1
(a + n)m+s+r Г(т + s +
/•00
r)Jo
Thus,
_ Г(т+5) Г» +,+,_, А Г<« +,+/,) з
1 (,/Я -г S + T) Jo
n=O
38. Approximations and Asymptotic Expansions 525
where absolute convergence justifies the interchange of integration and summa-
summation. The sum on n in G.34) equals A - e~')~m~\ and so
Г(т-М)
Um =
f e-a'tr-xwm+sdt, G.35)
Jo
V(m+s
where ш is defined in G.29).
Recall that T(x) is defined in G.21) for Re x > 0. Assuming for the moment
that |x| < |o|, we find that
T(x) =
By G.33) and G.36),
1 +
-^
a +n/
+ n) / -x \"
+ 5) \« + и/
m=0
m!
G.36)
G.37)
where absolute convergence justifies the interchange of summation. Note that E.1)
in Entry 5 is the case r = s = 1 of G.37).
Put G.35) in G.37) to deduce that, for |x| < \a\,
m\
G.38)
m=0
where the interchange of integration and summation can be justified by absolute
convergence. By G.25) and G.38), for |x| < \a\,
T(x) =
+ r) Jo
Fl(s,s+r,-wx)dt. G.39)
As |jc| -> oo with |Arg д:| < jtt -<5, -шд: ->¦ oo with ^n +& < arg(-wx) <
|jr - 5. Thus, by G.26)-G.28), the integral in G.39) is convergent and analytic
in each variable a, x in the right half-plane. From G.21), T(x) is also seen to be
analytic in each of a, x in the right half-plane. Thus, G.39) holds for all x with
Re x > 0.
The proof of Lemma 7.2 below makes heavy use of Faa di Bruno's formula (J.
Riordan [1, p. 36], S. Roman [1])
у n\hk(g(t))
G.41)
where the sum is over all integers fc(, fc2,. ¦. ,к„ for which
n = k\ + 2кг H h nJtn, it, > 0, 1 < i < n.

526 Ramanujan's Notebooks, Part V
and where к = k{ + k2 H V kn,
hk(z) =—kh{z), and 8t := 8*0 :=—g(t). G.42)
Lemma 7.2. Fix an integer N > 1. As x -+ oo with \ Argx| < \n - 8,
= о к
\
G.43)
uniformly for 1 in [0,1].
Proof. Let 0 < t < 1 and n > 0. We shall obtain uniform estimates for the nth
derivatives of each factor (-f)'~\ ws, ?JrA~t"+'/2), and C/(r, s + r;vux)of f(t,x)
in G.31) and then combine them to deduce G.43) from Leibniz's rule.
First, for each n > 0,
~A-tY~x = 0A), G.44)
dt
since r is a positive integer. Next, by G.30), we find that, for each к > 0,
G.45)
G.46)
Consequently, by G.40) with /i(z) = zJ and g(t) = w,
d"
^-ws = O(\).
dt"
For |Arg z| < \n - 5, f/(r, л + r; z) is analytic (Olver [1, p. 257, eq. A0.04)]),
and so we can differentiate G.27) (Olver [1, pp. 9-10, Theorem 4.2)) to obtain,
for it > 0 and \z\ sufficiently large,
= O(z-"-r). G.47)
m=0
m!
Now apply G.40) with h (z) = U(r,s + r; z)andg(r) = wx to deduce from G.45)
and G.47) that, as x tends to oo with |Arg x| < \n - S,
— U(r,r+s;wx) = O(x '),
dt"
G.48)
uniformly for 0 < t < 1.
A final application of G.40) with h(z) = ezx and g(t) — 1 + t/2 - w yields
where the sum is over all integers fc, satisfying G.41), where the coefficients
i, Аг,..., kn) are independent of x, t, and where g, is defined by G.42). By
38. Approximations and Asymptotic Expansions 527
G.30),
and so
and
Since g(/) < 0 for 0 < t < 1,
By G.41),
Combining G.49)-G53), we see that
jn
?, = 0@, for all odd i > 1
for all j > 1.
G.50)
G.51)
G.52)
G.53)
,=o G.54)
The result now follows from G.44), G.46), G.48), G.54), and Leibniz's rule.
Proof of Theorem 7.1. By G.26) and G.39),
Г(х) - A(x) - B[x),
where
A(x)=— e-<"tr-\-wYU(s,s+r;-wx)dt
"(*") Jo
and
B(x)
= f e-a'(-t)r-[wse-wx
Jo
G.55)
G.56)
G.57)
where \n < arg(—wx) < |тг.
We first examine Л (л), which yields the dominant part of the asymptotic ex-
expansion of T(x). Using G.28) in G.56), we find that
A(x) =
-rh Гe-a'tr-\-wY{-wx)-s-kdt
х-ч

528 Ramanujan's Notebooks, Part V
G.58)
where we have expanded (e"f - 1)* by the binomial theorem. It follows easily
from G.58) that
k=0
in agreement with G.22) and G.23).
Now G.24) follows by putting t = 0 in G.43). Thus, by G.22), G.31), G.55),
and G.57), it remains to show that
dt =
G.59)
By G.27) and G.31),
f(t,x)«ex°-w+'n)tr-li
and so
e-'ia+xmf(t, x) « e-a'ex"-w)x-rts-*, G.60)
uniformly for ( > 1. Since, for t > 1, we know that 1 - w < - \, it follows from
G.60) that
f°° e-Ha+x/2)f{tx) d) <<; e-xl2x-r f e-i Re^Re.-l^, ^ g-*/2 (?61)
In view of G.59) and G.61), it remains to show that
Jo ?^0(a+ x/2)m
G.62)
Integrating by parts N times, we find that
Joe ' ~ ho
{a + x/2)™+]
+ (a +x/2)-N f e-'ia+x/2)fw(t, x) dt.
Jo
By Lemma 7.2,
Thus, to prove G.62), it remains to prove that
f e-'la+x/2)f{N4t,x)dt
Jo
38. Approximations and Asymptotic Expansions 529
Again, by Lemma 7.2,
Jo
y=0
= xNI2-' У ¦
UK)!
E
This completes the proof of Theorem 7.1.
We show now that Cm(x) possesses an asymptotic expansion in descending
powers of x.
As in the proof of Lemma 7.2, we shall estimate C,n(x) = /(m) @, л:) by combin-
combining Leibniz's rule with formulas for the nth derivatives of (-/)'"', ш*>е*и-и>+'/2\
and U(r, s + r; wx). The wth derivatives of (-t)r~[ and ws at / = 0 are constants.
Since, for the function g(t) in G.49), we have g@) = 0, the nth derivative of
e*(' +'/2-№> at / = 0 is, by G.49), a polynomial in x. It remains to show that the «th
derivative of U(r, s + r; wx) at t =0 has an asymptotic expansion in descending
powers of x. By G.40) with h(z) = U(r,s + r; z) and g(t) = wx, we find that
dt"
U(r,s + r; wx)
1=0 k=0
G.63)
for some constants E*. Using the asymptotic formula G.47) in G.63), we obtain
the desired result.
If s is a positive integer, we can deduce the stronger result that Cm(x) is a
Laurent polynomial. To see this, note that when s is an integer,
U(r,s +r;z) =
,/t+r
by G.28) with r and л interchanged. Thus, C/(r, s + r; z) and its derivatives with
respect to г are Laurent polynomials in г, and the result follows from G.63) as
before.
After stating G.3), Ramanujan provides what is evidently a hint for proving
G.3). However, we have been unable to use Ramanujan's advice in establishing
G.3). Correcting three misprints, we state Ramanujan's "hint" as a separate entry.

530 Ramanujan's Notebooks, Part V
Entry 8 (p. 273). Let n be a nonnegative integer and suppose that 0 < p < a.
Then
„2 2np3
,(a - p+n)n-
(a + p + п)пЛ
2(a+nL
1 /
(a + nJ\ Ч
I
(a + nJ
2np5
/ lap p2 2
\a+n + (a + nJ ~ 3(^
+
Ъ(а + яN
a + n
h 2p
4p3 f a3 + 2a
(a + nJ 3 \(a + nK
1
2(a+nJJ ' 3 | (а+п)л (а+иK
4p5 [ a5 + 10a3 + 23a/2 5a4 + 4
7j* (a+i
4p5 j a5 + 10a3 +
TF | (a+n)
¦}¦")¦
(8.1)
Proof. Write
(
/!)
"+'
(a + «J
(a + лJ
exp f(n - 1) log
\ a+n
\ a + nj
(n + 1) log
( \
expUn -!){-
- (л + 1)
a+n
a+n 2(a+ nJ Ъ(а + иK 4(a + иL
"У \ А
1
(a + nJ
CXP
a + n 2(a+nJ 3(a
2ap p2 2/ip3
i+n (a + nJ 3(a + nK
2np5 . p6
4(д+пL
2(a + яL 5(а + иM 3(a
This proves the first equality in (8.1).
If A(a, p, n) denotes the expression in large parentheses on the far right side
above, then
ехр(Л(я, p,n))
2ap
= 1 +
.
2np3
+
2np5
a + n (a+nJ 3(а + яK 2(a + nL 5(а + иM
) + 2 \a+n + (a + nJ ~~ 3(a + иK + 2(a + иL )
+ -
38. Approximations and Asymptotic Expansions
P2 2np3 , У
531
6 \a+n (a + nJ 3(a+nK
4
24\a+n (a+nJ
p2
(
120 \a + n
5
= 1 +
4a2/i
4a5
1
ra+n r (a+nJ ' 3
2/ I a4 + 5a2 + | 2a
3 j (a+nL ~ (a + nK
2(а+иJ
4p5 f a5 + 10a3 + 23a/2
+ T5~ I (a + лM
This concludes the proof.
_ 5a2 + 4 1
(a + wL J
The next result is somewhat enigmatic. Ramanujan offers an asymptotic series
for Yll=\ 'M as n tends to oo, but he expresses the asymptotic expansion in
powers of 1/m instead of l/и, where m — \n(n + 1). We cannot find a "natural"
method to produce such an asymptotic series. Therefore, we take Ramanujan's
expansion, convert it into powers of 1/n, and show that it agrees with Euler's
well-known asymptotic series for a partial sum of the harmonic series.
Entry 9 (Formula E), p. 276). Let m = ln(n+ I), where n is a positive integer.
Then as m approaches oo,
Y" - ~i logBm) + y + —
+
1
1
1
fef
12m 120m2 630m3
191 29 2833
360360m* 30030m7 1166880m8 17459442m9
where у denotes Euler's constant.
1680m4 2310m5
140051
(9.1)
Proof. We rewrite (9.1) in the form
- +
1
1
6n(w + 1) 30n2(/i -t- IJ

532 Ramanujan's Notebooks, Part V
4 1
16
315/i3(n+ IK 105/i4(/i + lL
8 191 64-29
U55n5(n+ IM
8•2833
45045n*(" + IN 150l5n7(n+lO 36465n8(n + I)8
256 140051
8729721п»(л + О9 +'" '
Using Mathematica, we expand log( 1 +1 /и) and (n +1 )~*, 1 < it < 9, in powers
of 1/n and collect coefficients of like powers of l//i. We then find that (9.2) can
be put in the shape
11
1
1
1
In \2пг 120и4 252лй
1 691 1
240/I8 132w'° 32760л12 12n14
3617 43867
(9.3)
8160w16 I4364n18
On the other hand, from a result of Euler found in Entry 2 of Chapter 8 (Part I
H, p. 182]),
" 1 °° Я
(9.4)
*=l
as n tends to oo, where #ь к > 1, denotes the fcth Bernoulli number. Calculating
the first 18 terms in the sum on the right side of (9.4), we find that they are in
agreement with those in (9.3).
D. W. DeTemple and S.-H. Wang [ 1 ] found an analogue of (9.4) with n replaced
by/i + i.
We quote Ramanujan in the next entry.
Entry 10 (Formula A3), p. 284). The property of the function
and the integral
f
Jo
tdt
'o (e2*1 - \)(t+x)'
Ramanujan did not inform us what property he had in mind. Since these two
functions are not equal, it would seem that he is claiming that they arc asymptot-
asymptotically equal as x tends to oo. However, as we shall demonstrate, this is not the
case.
38. Approximations and Asymptotic Expansions 533
Theorem 10.1. As x tends to oo,
л log*
2jc
(Ю.1)
Proof. By partial summation,
log л
*2tlogrq+l)
N2+x2 T\Z, (t2+x2J
Letting N tend to oo, we deduce that
Eiogn
, n2 + x2 =
dt.
=
dt,
since, by Stirling's formula,
logf(« + 1) ~ (и +
as и tends to oo. Using A0.3) in A0.2), we find that
dt
A0.2)
A0.3)
Fix) = 2
\0gt 2 Г
Jo t2+x2 Jo i
-dt + O(x~2)
tlogt
(t2 + x2)
= Л
:dt
(Ю.4)
A0.5)
say.
First, from elementary considerations,
к л л
2x 4x 4x
Next, from Gradshteyn and Ryzhik's Tables [1, p. 564, formula 4.231, no. 8],
2 r°log(xM) 2 /Triogx p logM \ ^logx
I) = - I —z -он = - I — h / — du = , A0.6)
x Jo u2 + 1 Л 2 Jo u2 + 1 ) x
{ = _ 2 y00 log(jc«)
logu
du
2x
and
(Ю.7)
A0.8)

534 Ramanujan's Notebooks, Part V
Putting A0.5МЮ-8) in A0.4), we complete the proof of A0.1).
So that we may find an asymptotic expansion of the integral in Entry 10, we
first establish an analogue of Watson's Lemma (Olver [ 1, p. 71 ]).
Lemma 10.2. Suppose that
f@
A0.9)
as t approaches 0. Then, as x tends to oo,
fit)
I
о e" - 1
-dt
Z_/ v-n+l
provided that the integral converges for x sufficiently large, where f (г) denotes
the Riemann zeta-function.
Proof. Let
/«¦@ :=/(/)-?«„/".
m-\
E-
Then
f
Jo
\апп\$(п+ 1)
A0.10)
The integral on the right side converges for x sufficiently large, because the cor-
corresponding integral with /„,(/) replaced by /(/) converges for all x sufficiently
large.
As t tends to 0, fm{t) = O{tm). Thus, for some positive constants km and Km,
!/»(/)! < Kmtm, 0 </<*«.
Hence,
/о
е
(шло
Let X be a value of л: for which the integral on the right side of A0.10) converges.
Now
Fm(t) :=
/f (it)
™ eXu -
38. Approximations and Asymptotic Expansions 535
is continuous and bounded on [fem, oo). Let Lm = sup^s,<oc | Fm(t)|. Then, for
x > X,
e»-
eX, _ 1 ext _
e —
- 1
<*> yoo
„.„'
^
An elementary calculation shows that, for x sufficiently large,
d (eXl -1\ л
— I I < 0,
dt \e*' -I/
t > km.
Thus,
- 1
Taking A0.10)-A0.12) together, we deduce that
Г fit) Jt ^ann\qn + \) / 1 \
as д: tends to oo, which completes the proof of the lemma.
Theorem 10.3. As x tends to oo,
f-00 tdt
A0.12)
G(x)
¦-Г
Jo
^(-О"(я + 1)!С(и+2)
A0.13)
Proof. Write
Since
tdt
-f
2 Jo (*" - 0«/Bя)
(-1)"
1/B*) + 1 ^
we may apply Lemma 10.2 with /(/) = t/(t/Bn) + 1) to deduce that, as x tends
toco,
G(x)
BтгJ ^
The proposed result is now immediate.

536 Ramanujan's Notebooks, Part V
In particular, the leading term in A0.13) is 1/B4jc). Since, in Theorem 10.1,
F(x) ~ n logjc/B*), F(x) and G(x) are asymptotically quite different. At x =
0, F(x) is analytic, while G(x) ~ - logx/B;r), as x tends to 0. In conclusion,
we have not been able to discern a property of F(x) and G(x) held in common.
We quote Ramanujan in the next entry.
Entry 11 (p. 307).
n=0
log a
(П.1)
Here, у denotes Euler's constant. Ramanujan evidently intends A1.1) to be an
asymptotic formula as x tends to 0 + . Clearly, a > 1.
In 1907, Hardy [1, p. 283], [4, p. 160] proved that
log* ,1 У ,-y (-1)"-'*"
2 loga f^ n\ (an - 1)
n=0
logo
1 °° /
— У n-
+
2nni\
log a )
r2nnif log a
A1.2)
(We have corrected a sign error in Hardy's formulation.) The first three terms on
the right side of A1.2) are identical to the right side of A1.1). By Stirling's formula,
the latter series on the right side of A1.2) converges absolutely for 0 < x < со.
Entry 12 (p. 307). Leta,b > 1. Suppose further that
2nni Imni
logo logfo '
for every pair of nonzero integers m, n. Then, for x > 0,
"" X
log2*
log
21oga
1 /logfc loga Д2 + 6y2 \ у ( 1
V + j 2 Vl
Г2
logal
6y2 \
ogfcj
|
7 4
n\(an -
n=—oo
n/0
г -
A2.1)
where у denotes Euler's constant.
38. Approximations and Asymptotic Expansions 537
Equality A2.1) is a double series analogue of Hardy's theorem A1.2). The three
series on the right side of A2.1) do not appear in Ramanujan's formulation. There
is one further discrepancy in that Ramanujan omitted the expression
on the right side of A2.1).
Proof. Set
Then, for a - Re s > 0,
ldx
/ f(x)x-ldx =
Jo
121ogalogb
-a"bmx
rn. n =0
By Mellin's inversion formula,
Let
a > 0. A2.2)
<1гз)
where C^. r >s a positively oriented rectangle with vertices at a ± i T and — M ± iT,
where Г > 0 and M — /V + |, where N is a positive integer. We choose 7" =
Т„,п > 1, tending to oo so that
and
-кл\ > л/3
\Tn\ogb-kn\ > л/3,
A2.4)
A2.5)
for every positive integer k.
We evaluate IM.r by the residue theorem. The integrand in A2.3) has a triple pole
at the origin and simple poles at s = —n, for each positive integer я. Furthermore,
there are simple poles at s — —Imtij logo and s = —ittiTci/ log b, where m and
и are nonzero integers. The latter two sets of poles are simple poles by hypothesis.

538 Ramanujan's Notebooks, Part V
To calculate the residue at 0, we use the expansions
\-a~$
1
- 1 5 logo 2 12
1 1 logfc ,
1 -b-s s logfc 2 12
and
A2.8)
A2.9)
The expansion A2.6) can be deduced from a well-known formula found in the
Tables of Gradshteyn and Ryzhik [ 1, p. 944, formula 8.321, no. 1 ]. After a lengthy
calculation, we find that
log2
2 log a
1
1
;a logfc 21oga 2\ogb/
±№ +
1Л-1
1 \
lfc/
\
12 \loga logfc logalogfc/ 2 \loga logfc/ 4'
A2.10)
The remaining residues are much easier to calculate. For each positive integer
" n! (a" - 1H»" - 1)'
For each nonzero integer я,
;/iog« —
and
logfl(l -fo^./
2nm\ 2nni/loeh
Hence, using A2.10H12.13) in the residue theorem, we find that
N (*)"
/_2птп
V log«
A2.11)
A2.12)
A2ЛЗ)
\ _
38. Approximations and Asymptotic Expansions 539
Г -
|x2i»ri/tog(.
logfc
A2.14)
By A2.2), A2.3), A2.10), and A2.14), if we can show that the integrals over the
two horizontal sides tend to 0 as T tends to oo and that the integral over the left
vertical side approaches 0 as M tends to oo, then A2.1) follows.
By A2.4) and A2.5), we see that, for s = a ± iT,
-a~s\ >
and
- b~
respectively. Recall Stirling's formula
uniformly for -M < a < a, as \t\ tends to oo. Hence,
Г((Т ± iT)x-(a±m
da =o(l),
as T tends to oo.
By the reflection formula for the gamma function,
Г(-М
Г(М+ 1 - it) cosh(nt)
~s\ > aM/2 and |1 - fc"v| > bM/2. Hence,
as M tends to oo. This completes the proof.
Entry 13 (p. 307). Let a, b,c > 1, and assume that no two of the numbers
2nni 2mni , 2kni
log a logp log с
are equal, where n,m, and к are nonzero integers. Then, for x > 0,
-а"Ьтс*х _
log3x
6 log a logfc log с
1
1
I a logfc logfc log с log с log a
log a log b log с
1
1
4 \ log a log
1
logb ' logc/
2 \ log a logfc log fo log с log с log с
1 / logc loga logfc \
12Vlogalogfc logblogc logcloga/

540 Ramanujan's Notebooks, Part V
.2 .
1
12 2 / logalogfclogc
1 /logfr + logc logc-f
24 \ loga logfc
+
it2 Y2\(
_ L (
12 V
logc
loga
logc
+
logalogb logblogc
lojgclo6a'
logaj
Г -
t/logfr
¦+-
г -
log с У
2nni/ logt-'\ '
A3.1)
5
у denotes Euler's constant, and <(.) denotes the Riemarm геШ-function.
The four infinite series on the right side of A3.1) do n** «PI*"in j'
formulation. Furthermore, the terms
;r2logjt
121ogalogblogc
and
24
are not found in Ramanujan's version. - j
The proof of Entry 13 follows along the same lines rfsUie proof of Enuy 1^2^to
particular, A2.6M12.9) are needed to calculate the rescue of the quadruple pole
at the origin. Therefore, we forego the proof.
In Entry 14 we quote Ramanujan.
38. Approximations and Asymptotic Expansions
Entry 14 (p. 314). Let perimeter of ellipse — к(а + b)(] + A), then
541
A4.1)
very nearly. According to the above approximation, the perimeter of a parabola
= 3.99944(a + b)for 4(a + b).
Proof. Set A. — (a — b)/(a + b). If L denotes the perimeter of the ellipse given
by x = acost,y = fcshw.O < t < 2n, then (Part HI [3, p. 146])
Hence,
F, f_i —I- I- 12\ —
l ><+ l ie , 25 8 49 l0
h).
441
From A4.1), we find that, according to Ramanujan,
ЗЛ4 + A6 - 4Х2)Л2 + (A.4 - 8Х2)Л + A.4 ¦
¦ 0.
A4.2)
A4.3)
A4.4)
Beginning with the approximate solution h = A.4/4, we solved A4.4) by the
method of successive approximations to deduce that
f
Comparing A4.3) and A4.5), we find that the two power series agree up to the
coefficient of A.12, where the difference is remarkably only 2Xl2/410. Thus, Ra-
Ramanujan's claim is certainly justified.
The second claim in Entry 14 remained an enigma to us for a few years before
R. J. Evans deciphered the proper interpretation.
Since the eccentricity for a parabola equals 1, set b = 0 in A4.1). Solving A4.1)
via Mathematica when the left side equals 1, we find that h = 0.2730576913.
Using this value of h in A4.2), we find that
L * n{a + b)(\
= 3.99943(a + b).
Taking h — 0.27306, we would obtain the approximation 3.99944(a +b) claimed
by Ramanujan.
Lastly, we know that
L = 4 / Va2 sin21 + b2 cos2 / dt.
Jo
lfb = O, then
rn/2
L = 4 I a sin t dt = Aa = 4(a + b).
Jo
This concludes the explanation of Ramanujan's strange claim.

542 Ramanujan's Notebooks, Part V
The work in the following section first appeared in a paper by the author and
Evans [1].
Before stating and proving Entry 15, we introduce some notation and offer a
preliminary lemma. Ramanujan defines the logarithmic integral Li(x) by
—,
Jo log'
Define the unique positive number д by
A5.1)
Jo log'
Ramanujan [9] and Soldner (N. Nielsen [1, p. 88]) numerically calculated д.
We used MACSYMA to also calculate fi. The table below summarizes the three
calculations:
Ramanujan
Soldner
MACSYMA
1.45136380
1.4513692346
1.4513692349
Lemma 15.1. Let ц, be defined by A5.1), and let у denote Euler's constant. Then,
for x > 1,
Li(x) = y + log log* -t-
log* x
Lemma 15.1 can be found in Ramanujan's notebooks on the same page as Entry
15, and a proof is given in Part IV [4, p. 126]. For another proof, see Nielsen's
book 11. pp. 3,11].
We first give Ramanujan's version of Entry 15: If
= log p,
A5.2)
thenn = (p + |)log/u. - \.
We might interpret Ramanujan's statement as giving an estimate for S when
n = [(p+^))og/;i_ij. with such an interpretation, the error made in the
approximation by log p is O(l/p), as p tends to oo. However, if p is chosen so
that n = (p + 5) log,tt - 5 is a positive integer, then, as stated in our version
of Entry 15 below, the error term is 0A/p2). Amazingly, Ramanujan found the
precise linear function of p that yields an error term of O(l/p2). Thus, if the
constant i log д - i in the definition of n is replaced by any other constant, the
error term is 0A/p).
Entry 15 (p. 318). Let S be defined by A5.2), and let n = (p + 0 log ц. - \ be
a positive integer. Then, as p tends to 00,
S = \ogp+O(p-2). O5-3)
38. Approximations and Asymptotic Expansions 543
Proof. Setting у = log//., applying Lemma 15.1, and using A5.1), we find that
y* P e' - 1
/ d A5.4)
P e' - 1
/ dt.
Jo '
Setting x = 1 + 1 /p and using the definition of S, a familiar estimate for a partial
sum of the harmonic series (Olver [1, p. 292]), and A5.4), we deduce that, as n
tends to oo,
A xk - 1 ^ 1
*=i
= ?^-i+logn + / + ^
A = l
-.(=К-Г^+в(?)- „,
A5.5)
Next, applying the Euler-Maclaurin summation formula from @.5) of Chapter
37, we deduce that, as p tends to 00,
««•
Employing A5.6) in A5.5), we find that
fnx'-\ fye'-\J 1 /n\ x" л/1\
S= / dt-f dt --\ogx+ log[-) + — +О I —)
Jo t Jo ' 2 \yj In \p2/
/@^--.ogx + .o
where f(t) = {e1 - [)/t. Now,
Jt"
A5.7)
and
X
In
as p tends to 00. Substituting these estimates in A5.7), we find that
fly)
as p tends to 00. Thus, it remains to prove that
f
Л Ю
i log л
A5.8)

544 Ramanujan's Notebooks, Part V
Now,
2p
By the first mean value theorem for integrals and A5.9),
/00
logx
/@ dl = /(и)(> - n log*) =
( 1 \
\рг)
A5.9)
A5.10)
for some value и such that n logx < и < у. By the mean value theorem, there
exists a value v such that и < v < у and
/(и) = /(y) + (и -
= /00 +'
A5.11)
where the last equality follows from A5.9), since /' is bounded on (и, у). Using
A5.11) in A5.10), we complete the proof of A5.8).
So as not to interrupt the proof of Entry 16 below with two calculations, we
now set them aside in two lemmas.
Lemma 16.1. If a and в are positive and n is a nonnegalive integer, then
Proof. Setting и = z/VO and applying Cauchy's theorem, we find that
" J-
n лоо
I
an лоо
= * I u2"e-"'
у/Л J-oo
л/П Jo
2л — ил J
и е du
du
Lemma 16.2. If a and в are positive, and n is a nonnegative integer, then
¦ roo+ai
Jn := ' / е-*г">еB"^Чг = е-'"+1
у/IX 0 J—oo+ai
38. Approximations and Asymptotic Expansions
Proof. Setting z = uVO and applying Cauchy's theorem, we find that
545
Jn = ^= exp ( - (и -
у/Л J-<x
- (n
du
Entry 16 (p. 324). As t tends to 0+,
Л<Л+|)
A6.1)
The function on the left side of A6.1) is not a theta-function but is a false
theta-function in the sense of Rogers [3]. In fact, we shall obtain a more explicit
asymptotic expansion, which enables the calculation of further terms in the as-
asymptotic series. It is interesting that Ramanujan appended an abbreviation for
"asymptotically" after the series on the right side. We are unaware of any other
instance in the notebooks where Ramanujan used this word. Usually, he wrote
"nearly" or "very nearly."
Proof. Let
1 -t
\+t'
so that в is small and positive. If a > 0 and n is a nonnegative integer, by Lemma
16.2,
I
J-oo+ai
Multiply both sides by 2(-l)" and sum on n, 0 < n < oo. Upon inverting the
order of integration and summation, we see that the resulting series on the right side
converges absolutely and uniformly on (-00 + ai, oo + ai) and that the resulting
integral also converges absolutely, and so the inversion is justified. Hence,
п=0
/:
/-oo+ai cos г
Now recall that (Abramowitz and Stegun [1, p. 804])
-dz.
A6.2)
n=0

546 Ramanujan's Notebooks, Part V
where Es, j > 0, denotes the jth Euler number. Thus, for \z\ < я/4,
secz- ? ,„ ,. z~ <Cx\z\2N+2
я=0
where the positive constant C\ depends on N but not on z. If 0 < a < 1 and
|z| > jt/4 on the contour (-00 + ai, oo + ai), then
1
and
z2w+2
cosz z*"Ti zilVT' ^ Bn)!
are bounded functions of both a and z. In particular, observe that
cos(Bn - 1)я/2 + ia) = i(-D" sinha,
where n is an integer, and so a/ cos z remains bounded as a tends to 0. Hence, for
all points on the contour,
<C2,
for some positive constant C2, which is independent of both z and a.
Therefore, by A6.2) and Lemma 16.1,
Y (~ ° 2"
A6.3)
J Bn)!
^(-\)"Еь,в"
22"n\
A6.4)
n=0
where, by A6.3),
Г
since л2 + a2 does not exceed the larger of 2jc2 and 2a2. Setting и = л2/^ and
evaluating the resulting integrals in terms of gamma functions, we find that
38. Approximations and Asymptotic Expansions 547
We now choose a = VO. Hence,
A6.5)
In conclusion, by A6.4) and A6.5), we have obtained the asymptotic expansion
Using the values E^ = 1,-1, 5, -61, 1385, and -50521, where n = 0, 1, 2, 3,
4, and 5, respectively, and employing Mathemaiica, we calculated the coefficients
displayed on the right side of A6.1).
Using operator methods, D. Bradley found a shorter route to A6.6).
It is curious that each of the six coefficients on the right side of A6.1) is a positive
integer. Using Mathematica, we calculated the first 50 coefficients and found all of
them to be positive integers. Using his own multiple-precision integer arithmetic
package, Brent calculated the first 1000 coefficients to further substantiate the
conjecture that all coefficients are positive integers. Brent easily showed that the
coefficients are positive, while Galway [1] later established the more difficult
assertion that they are integral.
Entry 17 (p. 333). For n > 0 and x > 0, define
Then
n2
4
dun
dx
A7.1)
and
- )¦
Proof. We first prove A7.1). After a simple calculation,
n^ du^
4 "" Лс
_„ ^ (-1) (» + ^*с)(я + 1 )k-i e T ; '
= Г(л + 1)| (« +2)(я + i)e-<"+2>V4
V
/n2 (n+2kJ\

548 Ramanujan's Notebooks, Part V
(-l)*(n + 2 + 2k){n
= Г(п + 3)
= un+2.
\ k=\
Thus, A7.1) has been shown.
We are pleased to present M. L. Glasser's proof of A7.2). We induct on n.
First observe that
00
M0 = 1 +2 ?](-l)V*2jt = «?(-<?-*) = Л,@, ix/тг),
*=1
in the notation of Ramanujan (Part HI [3, p. 36]) and, e.g., H. Rademacher [1, p.
166], respectively. Now (Rademacher [1, p. 177]),
= 04@, ix/я) = . -Ы0, -n/{ix))
=Л t *-Hln
+i )*/<¦»*)
which agrees with A7.2) for n = 0.
Next recall that (Rademacher [1, p. 166])
and (Rademacher [1, p. 177])
A7.3)
A7.4)
Hence, by A7.3) and A7.4),
Ml =
л=0
1 d
2л dv
u=0
1 d
38. Approximations and Asymptotic Expansions 549
which agrees with A7.2) in the case n — 1.
Having proved A7.2) for n = 0,1, we assume that A7.2) is valid and that it is
also valid with n replaced by n +1. We shall then use these two equalities and A7.1)
to establish A7.2) with n replaced by n + 2. By straightforward differentiation,
n2 dun
Мл+2 = — Un + ——
4 dx
Bn
+1/2
n
4тг2
2 п(п - 1)
4х
-I
-t- 1
2х
«(я - 1)
h
^
-**2
9--)
which is precisely A7.2) with n replaced by n + 2.
The function и„ can be regarded as a modified theta-function or a modified
hypergeometric function. Recalling the proofs of A7.2) for л =0 and n = 1,
in general, we can regard A7.2) as an inversion formula for a modified theta-
function. It would be interesting to find further terms and even more interesting
to find an exact inversion formula. Entry 17 can also be regarded as an analogue
of the many asymptotic expansions for exponential scries found in the first seven
sections of Chapter 15 (Part II [2, pp. 303-314]). V. Kowalenko, N. E. Frankel,
M. L. Glasser, and T. Taucher [1] have considerably generalized and extended
these results. In particular, Ramanujan's results do not cover certain "exceptional
cases," and these are encompassed in their theorems.
The origins of the two families of approximations in the next two entries were
a mystery to us until Askey [1] explained their roots in Gaussian quadrature with
respect to a discrete measure. Thus, the following proofs are due to Askey, although
we have amplified the details.
Entry 18 (p. 349). Let
S(x, n):=22<p(x -n + \ + 2k).
Then S(x,n)/n has the following successive approximations:
A8.1)
A8.2)

550 Ramanujan's Notebooks, Part V
5(n2 - 1) { v I x +
n2 - 4)<»<jr)
6Cя2 - 7)
A8.3)
and
where
а = 3n2 - 13 aw/ /3 =
Proof. Let /(/) be a continuous function on [a, b], and let da(t) be a nonnegative
measure on [a, fc]. The problem of Gaussian quadrature is to approximate
/ f(t)da(t) A8.6)
Ja
by a finite sum which is exact for all polynomials of as high a degree as possible.
Let a < t\ < tj < ¦ • ¦ < h < b and set
к
.=1
wk(t)
and
Then
is a polynomial of degree at most к — 1 such that
L0j) = /Cy) !<
A8.7)
A8.8)
A8.9)
Now let /(») be a polynomial of degree not exceeding jfc-l. Then /(/) = L{(t),
since, by A8.9), these two polynomials of degree к - 1, or less, agree at к points.
Hence, by A8.8),
f(t) «fa@ = J2 fOj) [ WjMO da(O. A8.10)
; — i Jti
J *
Thus, we have an exact quadrature formula for polynomials of degree к — I.
38. Approximations and Asymptotic Expansions 551
We now want to keep the number of terms, k, constant and choose the points
fj, 1 < j: < k, so that A8.10) is exact for polynomials of degree as high as
possible. We will show that, if the points f,, 1 < j < k, are appropriately chosen,
A8.10) is exact for polynomials of degree 2k - l.Then
/@ - L[(t) = u»*@rt-i(/). A8.11)
where rk_ ((r) is a polynomial of degree к — 1.
Now let {u»*(/)}, 0 < к < со, be a set of polynomials with wk{t) of degree it,
which are orthogonal on [a, b] with respect to the measure da (I). Then, of course,
Wk(t) is orthogonal to all polynomials of degree < k. Then, by A8.11),
f f(l)da(t)- f Lfk(t)da(t)= f wk(t)rk.l0)da{t) = 0. A8.12)
Ja Ja Ja
if
then, by A8.10) and A8.12),
Ja
Vj.k(t)da(t),
A8.13)
f
Ja
;=<
and we have an exact quadrature formula of к terms for a polynomial of degree
2k— 1. In general, for an arbitrary continuous function /(/) on [a, b], the Gaussian
quadrature approximation of A8.6) equals
A8.14)
7=1
Other representations for A,, given by A8.13), exist; e.g., see G. Szego's book
[1, p. 48]. Two of these representations show immediately that kj > 0.
We now apply this theory to f(t) = <p(x — n 4-1 + 2t). Since S(x, n) is a sum
of n terms, we want da(t) to be a discrete measure weighted at the integral points
0, 1, 2,..., n — 1. This leads us to the Hahn polynomials, which were introduced
by P. L. Tchebychef [1] in 1875 and which are constant multiples of
For 0 < m,n < N, they satisfy an orthogonality relation of the form (A. F.
Nikiforov, S. K. Suslov, and V. B. Uvarov [1, p. 33])
,(« + !), (/3 + 1 )„_,
1=0
t\
(N-t)\
= с„

552 Ramanujan's Notebooks, Part V
for certain constants стя, 0 < m, n < oo.
In our application, we want a — (} = 0, so that the weights equal 1 at each
nonnegative integer k,Q < к < N := n - 1. Thus, wk(t) = ckQk(t; 0, 0, N)
for some constant ck, 0 < к < N. Although the value of ck is not needed in
applications to Gaussian quadrature, we see that, by A8.15),
Ck =
(-i)kBk)\(N-k)\
(k\JN\
We are now ready to calculate the four approximations A8.1)—A8.4) claimed
by Ramanujan.
Letfe= l.ByA8.7),u>,,,(/) = 1, and by A8.13),
r-1
A, = / da{t) = n.
= /"
By A8.15),
Hence, i\ = N/2. Hence, by A8.14), the first approximation to S(x, n) is
= n<p(x - n + 1 + (n - 1)) = n<p(x).
as claimed.
Second, let к = 2. Then, by A8.15),
6/
jW,;0.0.tf)=l--
We therefore want the roots of
6f2 -6tN + N(N - l) = 0,
and they are
Hence,
By A8.7), we find that
t — h t — t\
wi.2(f) = , л and W2.i(t) = --
38. Approximations and Asymptotic Expansions 553
Hence, by A8.13),
~t2)
. j=0
n
2'
and, by a similar calculation.
(t -tx)da(t) = -
Thus, by A8.14), the second approximation of S(x, n) equals
as claimed in A8.2).
Third, let к = 3. Then, by A8.15),
N N(N-\) N(N -\)(N -2)'
We therefore must solve
20/3 - ЗОЛ/;2 + D - 6N + 12/V2)/ - N(N - \)(N - 2) = 0.
These roots are found to be
2/i = n - 1 - ./ —-—, 2r2 - n - 1, and 2/3 = n - 1 +¦.
Thus,
f(tl) = <p\x-
, and/(r3) = <p I jt
Also, by A8.7),
f3n^+7
5
A8.16)
if 1.3О) =
(/ - f2)(f - f3)
(fi -'2)('i -h)
10 Л I
3n2-7

554 Ramanujan's Notebooks, Part V
and
(/-/1M/-/2)
10
3n2-7
Thus, by A8.13),
10
A i =
Г- / _
10 I (n- \)nBn - 1)
I
1 /Зи2 - 7 и(и - 1)
1
(л-1J + (л-1),
4
_ 5я(я2 - 1)
~ 6Cn2 - 7)'
By almost an identical calculation,
6Cn2 - 7)'
A8.17)
A8.18)
Lastly, by A8.13),
20
Зл2
20
\(n - l)nB« - 1) - i(
4я(л2 - 4)
3C«2-7)'
(±„2 - in + l)n)
К10 2 ^ 5/"/
A8.19)
38. Approximations and Asymptotic Expansions 555
Using A8.16>—A8.19) in A8.14), we obtain Ramanujan's approximation A8.3).
Fourth, let k = 4. Then, by A8.15),
n 1, a n ыл 1 20t j. 9O'(f-1) 140f</- 1H-2)
^,0,0,^ = 1-- + ^^-^- ДГ(ЛГ_1)(ДГ_2)
We used Mathematica to determine the roots of Q*(t; 0,0, N) — 0 and found that
l<*-p\
7/
A8.20)
where a and p are given in A8.5). For brevity, set A = (a + p)/7 and S =
(o-«/7. By A8.13),
f" (r-i
-fd)
-/4)'
dt
14 Г-1 . , 2 ч
= —t= / U - (h +h + h)i + ihh + hU + Uh)t - tzhh)da(t)
B-jA in
14 /n2(n-lJ /3(я-1) 1 j-\ (я- 1)иBя- 1)
/ i \
« - О2 - B)n
= U /n{n
~ /J V
24
и2
56
A8.21)
The calculations of X2, A.3, and A.4 are similar, and we find that
14 Z«2(n-1J /3(n-l) 1 ^-4 (n - 1)яBя -
л - I)
- B)n\
A8.22)

556 Ramanujan's Notebooks, Part V
14 (n2(n - lJ /3(л - 1) 1 r-\ (n - 1)яBи - 1)
f I \
A8.23)
and
14 /n2(n-lJ /3(в-1) 1
(и - 1)лBл -
Using A8.20H18.24) in A8.14), we obtain Ramanujan's last approximation
A8.4).
After his four approximations, Ramanujan illustrates his theorem with five ex-
examples. In all examples he uses a different notation from that of Entry 18.
Corollary 18.1 (p. 349). We have the following approximations:
Ёи**2(и2
26
13
E«*
k=\
22
13(М6
289
16lM2o),
A8.25)
A8.26)
A8.27)
A8.28)
38. Approximations and Asymptotic Expansions 557
and
21
55s E06(^B) +<PBO)) +931
ll +2^22/7)+^A1 - 2^22/7))) .
A8.29)
Proof. First, in the notation of Entry 18, set n = 7 and uk = <p(x - 8 + 2k), 1 <
к < 7. Then У(«2 — l)/3 = 4, <p(x - 4) = «2, and y)(x + 4) = u6. Hence,
A8.25) follows from A8.2).
Second,letn = 26mduk = <p{x-21+2k), 1 < к < 26.ТЬепУ(«2 - l)/3 =
,k ^(), У( )/
15, ^)(л - 15) = м6, and tp{x + 15) = u2\. Hence, A8.26) follows from A8.2).
Third,setn = 13andMt = ^>(x-14+2fe), 1 < к < 13. Then^/(Зп2 - 7)/5 =
10,
5(n2- I) _ 7 8(и2-4) _ 11
6(Зл2-7) ~ 25' 6Cn2-7) ~ 25'
f{x — 10) = м2, <p(.x + 10) = M12, and ^>(x) = и?. Hence, A8.27) follows from
A8.3).
17,
8.3).
Fourth,let« = 22andMA = <p(x-23+2k), 1 < к < 22.ThenУCп2 - 7)/5 =
161
8(я2 - 4) 128
6(Зл2-7) 2-289' 6(Зи2-7) 289'
(р(х — 17) = «з, Я>(х + 17) = иго, and <р(х) - м2з/2- Thus, A8.28) follows from
A8.3).
Fifth, let к = 21 and uk = <p(x - 22 + 2k), 1 < it < 21. Then a = 1310, p =
958,
1
4
n2-16
60
= 18,
506
3•958'
1
4+
«2-16
6K
931
3-958'
<p(x - 18) = M2, <p(x + 18) = иго- <p(x - 4V2277) = un_2jwl, and
^(x + 4V22/7) = W|i+2v/3277- Lastly, replace н„ by p(o). Then A8.29) follows
form A8.4).
To the best of our knowledge, with the exception of this and the following entry,
Ramanujan's notebooks, published papers, and unpublished papers give no indi-
indication that Ramanujan had any knowledge of Gaussian quadrature or orthogonal
polynomials. Thus, Entry 18 is very remarkable, for it shows that Ramanujan must
have derived some of the principal underlying ideas in these theories.

558 Ramanujan's Notebooks, Part V
Entry 19 (p. 352). Let <p(t) be continuous for t > 0. Then the function
has the successive approximations
and
/.=0
«»(*).
n!
1 / 1 + V" + 12jc\
--(* + ——)
A9.1)
A9.2)
A9.3)
As with the previous entry, Askey [1] first observed that the origin of these
approximations is found in Gaussian quadrature related to the discrete Charlier
polynomials. It is again remarkable that, with no apparent knowledge of orthogonal
polynomials, Ramanujan found these approximations.
Proof. We apply the general theory of Gaussian quadrature outlined in the proof
of Entry 18. Here f{t) = <p(t), and the interval [a, b] is replaced by [0, oo). The
discrete measure is realized at the nonnegative integers and is weighted by the
Poisson distribution
n\ '
0 < n < oo.
A9.4)
The corresponding polynomials are the Charlier [1] polynomials defined by (Niki-
forov, Suslov, and Uvarov [1, p. 35])
Cn(t;x) := 2'ro(-n.-'; -1/*), 0 < n < oo.
They satisfy an orthogonality relation
(=0
~«(-';*)y7 =CA
for certain constants c'mn, 0 < m, n < oo.
We now calculate the interpolation points t, and the coefficients X, in the ap-
approximation A8.14).
First,
C,(t;jc)= 2/4>(-l,-<;-!/*)= 1 -'/¦*.
38. Approximations and Asymptotic Expansions 559
which has t = x as its zero. Since w\.\(t) = 1, by A8.13),
Hence, our first approximation equals 1 <p(x), as claimed in A9.1). This approx-
approximation is exact for constant and linear polynomials.
Second,
2f
X XL
which has the zeros
So, by A8.7),
and
Hence,
A9.5)
U>1.2@ =
/ - x - i
Similarly,
A9.6)
A9.7)
2V1 + Ax
The approximation A9.2) now follows from A9.5)-A9.7). This approximation is
exact for all polynomials of degree 3 or less.
Next,
The three roots are very complicated, and upon examining A9.3), we see that
Ramanujan did not choose these roots as interpolation points. If he had done so,

560 Ramanujan's Notebooks, Part V
his approximations would have been exact for all polynomials of degree 5 or
less. To simplify the approximation, it is natural to choose t\ — x as one of the
interpolation points, because it is the expected value of the Poisson distribution.
Amazingly, Ramanujan found the proper interpolation points,
h, h =
so that his approximation is exact for all polynomials of degree 4 or less. It is a
tedious calculation to verify that A9.3) is exact for <p(x) — 1, x, x2, x3, x4, and
we resorted to Mathematica to check the right sides of A9.3) for x3 and x4. Thus,
A9.3) is not a Gaussian quadrature formula.
In his first approximation, Ramanujan actually writes
cxp
\n=\
= ex<p{x)
as the first approximation." Undoubtedly, D denotes a differential operator, and
so the latter equality is trivially true. However, we have no idea why Ramanujan
introduced this series of differential operators. Perhaps this provides a hint to
Ramanujan's derivations.
In Entry 10 of Chapter 3 (Part I [1, pp. 57-65]), Ramanujan provides an asymp-
asymptotic expansion for e~x J2T=a <p(n)x"/n\ as x tends to oo. As to be expected, the
form of this expansion is quite different from the approximations given in Entry
19.
On page 350, Ramanujan claims that
B0.d
"when a is very great. The above theorem is very useful to know." Prior to writing
B0.1), Ramanujan gives the special case when и = 2. As might be expected,
Ramanujan does not give the value of с or any hypotheses about <p. Although the
form B0.1) was perhaps convenient for applications that Ramanujan may have
had in mind, we shall make a simplification. Suppose we let /(x) = (<p(x)/a)n.
Next, rcintroduce a by replacing f(x) by (p(x)/a. After a change of variable in
the integral, we find that
B0.2)
k=\
which is simply the case n = 1 of B0.1). Thus, it is no loss of generality to assume
at the outset that n — ].
The following theorem is not as explicit as we would prefer, but its formulation,
due to J. L. Hafner, is better than the author's original version.
38. Approximations and Asymptotic Expansions 561
Entry 20 (p. 350). Assume that the product on the left side of B0,2) converges,
<p(x) is monotonically increasing, and <p@) > 0. Then B0.2) is valid, when с is
given (approximately) by B0.9) below.
Proof. Taking the logarithm of each side of B0.2), we find it suffices to show that
. B0.3)
Since 0?(O) > 0 and <p(x) is monotonically increasing,
a
/oo/ a \ °° / a \ f°° / a \
log(l + —-)</x<y>g(] + —- < / log (l + —-)dx.
By the intermediate value theorem, there exists a number xu, 0 < xa < 1, such
that
JO
<p(x)J /0
log
= /1-/2, B0.4)
say. By examining the inverse function of log(l +a/<p(.x)), we see that
-I
Settingx = l/(e> - 1), we find that
= Г 4L
J<p@)/a л(
'i = / r.. ^""\dx.
J<piO)/a
Thus, from B0.4),
x)
B0.5)
Comparing B0.5) with B0.3), we see that B0.3) has been proved with
logc= -/2+ jloga. B0.6)
We shall make the determination of с slightly more explicit.
For 0 < x < xa < 1,
as a tends to oo. Thus,
h = xa loga - f " \og<p(x) dx + O(l/«) = xa logo - /3 + O(l/a), B0.7)
Jo

562 Ramanujan's Notebooks, Part V
say. Since cp is increasing,
Xa lOg^(O) < h <Xa \Ogcp(xa),
so that by the second mean value theorem for integrals,
0 < /3 - xa log <p@)
= Г
Jo
log
= (xa - ?,) log
B0.8)
for some number ?a, 0 < fa <xa. Combining B0.6H20.8), we deduce that
c-^(fl—\ |^p] (l + 0(l/a)). B0.9)
This completes the proof.
It seems likely that, in many cases,
1 ( 1 \
xa = т + o\ ) and xa
2 \logfl/
as a tends to oo. In such cases, B0.9) yields
»A +*»(!)).
~ $a = 0A),
The next entry is recorded in Ramanujan's Quarterly Reports and is discussed
in detail in Part I [1, pp. 311-312].
Entry 21 (p. 351). Consider the equation
F(-x) := T
(-*3)*
tS C*)!
o-^X
= 0,
B1.1)
where со := ехрBлч'/3). Then there exist an infinite number of positive roots. They
are close to the zeros жBп + l)/\/3~ of cos(x*/3/2), where n is a nonnegative
integer. More precisely, if
h =
then these roots are given by
7гBл+1)
x —
49 ¦ 52 ¦ 57,
7!
(-1)"
9!
19-21
5! '
•¦)
37-39 43 ul ^ 61 -63-67-73 9
7!
9!
91 93 97 103 ¦ 111 .. \
+ h +¦¦¦).
B1.2)
Lastly, all roots of B1.1) are given by x, cox, and w2x, where x is given by B1.2)
and со = expB^i/3).
38. Approximations and Asymptotic Expansions 563
Entry 22 (p. 351). Let x0, x\,x2,... be the real roots of B1.1). Then
B21)
Proof. The real roots of F(x) - 0 are -x0, -xu -x2, — Then -х„, -xnw,
—х„а>2,0 < n < oo, constitute all the roots of F(x) = 0, where со — expBtfi'/3).
Now
{x + xn)(x + xnco)(x + х„а>2) - дг3 + x3.
We now apply the Weierstrass product formula. From the definition of F in B1.1)
and the fact that the infinite product in B2.1) converges, since х„ ~ лBп +1)/*Jb
as n tends to oo, we easily deduce that F{x) has the representation B2.1).
Entry 23 (p. 370). As x tends to I -,
„3n + l
r 3n+2
Proof. First,
oo
A
We apply the Euler-Maclaurin summation formula, @.5) of Chapter 37, to
— x)(i — x )x
Thus,
/W
- f /(„ « - i
Letx' = м. Since
fit) = fit)
we find that
¦ Г {' - [<] - j) /'@ dt
Jo
B3.2)
/ 4x2 4x 4x3 \
/
riogM1 1
_
4x
Лхъ \
~ 1 + к4*3/
du

564 Ramanujan's Notebooks, Part V
4л;2
х(з + ^-^ +
"Г
du
- м4х2 \+и4х ¦ 1 + и4
= 0A). B3.3)
as x tends to 1 -. Setting x' — и also in the first integral on the right side of B3.2)
and using B3.3), we find that, as x tends to 1 -,
„2/-1 _ ,Ar2\
x(l-x)
1 M2
U2(l -U4)
Using Mathematica, we find that
'¦«2A-«4), I_^tan-
- t| «<>gB -
B3.4)
V2
^- logB
Id
log(l
B3.5)
4 8V2 4V2
Using B3.5) in B3.4) and combining the result with B3.1), we complete the proof
of Ramanujan's asymptotic formula.
39
Miscellaneous Results in the First
Notebook
In this last chapter we collect together some miscellaneous results from the
unorganized portions of the first notebook. Most are from analysis, with some
pertaining to hypergeometric functions.
We use the familiar notation associated with hypergeometric functions; e.g., see
Part II [2, p. 8]. In particular, for each nonnegative integer n,
_ Г(а + л)
(a)" ~ Г {a) '
Page numbers after entries refer to the pagination of the Tata Institute's publi-
publication of the first notebook [9].
Entry 1 (p. 72). //
A.1)
then
A.2)
n=l
where f2(l) = Oand.forn > 1, J2(n) denotes the total number of prime factors
ofn counting multiplicities.
Proof. By A.1),
r=\ \d\r
A.3)

566 Ramanujan's Notebooks, Part V
It is easy to see that (-l)n(n( is completely multiplicative. Hence,
is multiplicative. For each prime p and positive integer n,
d\p"
Hence, by multiplicativity,
d\r
1,
0,
if n is even,
if л is odd.
if r is a perfect square,
otherwise.
Using the equality above in A.3), we deduce A.2).
Entry 2 (p. 94). Ifn is a positive integer,
focsin''x (-l)";r "'
Jo x l ' W
<-.ы--ад--,
where the plus sign is taken ifn is even, and the minus sign is chosen ifn is odd.
In fact, Ramanujan claimed different values for the integral and crossed out the
entry. The evaluation above appears to have been given first by O. Schlomilch [1]
in 1860, as pointed out by M. L. Glasser. In the book by D. S. Mitrinovic and J. D.
Keekic [1], the integral of Entry 2 is evaluated by contour integration. In 1980, E.
T. H. Wang [1] submitted Entry 2 as a problem; solutions by R. L. Young and T.
M. Apostol and several references to the problem's appearances in the literature
were given.
Entry 3 (p. 94). For 0 < x < 1,
2.i.
5' 2-
Г7-
1 +
and
(I, \; f; -,) =
Proof. From Entry 35(iii) of Chapter 11 (Part II [2, p. 99]) with x1 replaced by
—x,
(\ 2.1. VA
E,3, j,-л) =
2'ИЗ'З
cos E sin '('
C.3)
39. Miscellaneous Results in the First Notebook 567
Now,
cos E sin ' (i-Jx)) = cos I - log
- л/ж))
= 5 (VI + x +
+ \ y*J\ + x -
1/3
C.4)
Using C.4) in C.3), we deduce C.1).
Next, Entry 35(ii) of Chapter 11 (Part II [2, p. 99]), with x2 replaced by -x,
states that
2^1 C, f; |; -¦*) = —7= sin E sin (iy/x)).
The remainder of the proof of C.2) is similar to that for C.1).
Entry 4 (p. 104). Let x and у be complex numbers such that x/y is not purely
imaginary. Let <p(z) be an entire function such that
f(z) := \n sec (jttjcz) sech {{ityz) [<p(xyz) - <p(-xyz))
tends toOas z tends to 00. Then
\л sec (^лх) sech {{ny) {<p{xy) - ip(-xy)}
(_irsech(^Lll
{<p(Bn+l)y)-<pi-Bn-
n=(l
-<>?
(-l)"sech
п + ))x
2y
{(p(Bn + \)ix)-<p{-Bn
я=0
D.1)
Proof. Observe that /(г) has simple poles at г = Bл + 1)/л andz = Bл + l)/
for each integer я. The poles do not coalesce because x/y is not purely imaginary.
Letting Ra denote the residue of /(г) at a pole a, we easily find that
R(in+i)/x z— sech
and
2x
l)v
- «9(-Bл
)/дт) - <p{-Bn

568 Ramanujan's Notebooks, Part V
We thus find that the sum of the principal parts arising from the poles ±Bn +1 )/x,
n > 0, is
(-lX'^sechl 2x )
Bn + \J-x2z2
while the sum of the principal parts arising from the poles ±Bn + l)//y, n > 0,
equals
(-l)"yZisech
-4 ^ {<P(Bn + l)ix) - <p(-Bn + \)ix)).
Bfi + lJ + y2z2
Since /(г) tends to 0 as z tends to oo, we conclude from the Mittag-Leffler
theorem that
^(-irsech^2^1^
/ттBл + \)y\
( I
Bя+
n=0
» (-D"sech
/яBп +
\ 2y
,.=n
,2 i „2,2
1J+у2г
n + l)/jc) — ^(—Bя
Letting г = 1, we deduce D.1) to complete the proof.
Entry 5 (p. 108). Ifx is not an integer and 0 <в<2л, then
n 1 ~ cos(«0)
- (соЦтгл) cos@jc) + sin(&*)) = ^ ~2Z^ ~i J•
x x n=| n - x
Proof. By Entry 34(i) in Chapter 13 (Part II [2, p. 237J), for |d| < л,
n cos@x) 1 „^л (-I)" cos{n9)
+22
x sinGrjc)
n2-x2
E.2)
Replacing в by в - я in E.2), we readily deduce E.1) to complete the proof.
Entry б (p. 120). The maximum value ofax /Г(х + 1) is equal to
aa-\/2
Г(а
F.1)
"very nearly."
This entry is the same as Example (i) in Section 25 of Chapter 13 in the sec-
second notebook. However, after proving a slightly weaker version of the result in
Part II [2, p. 228], we unfortunately claimed that the appearance of 23.2e" in
39. Miscellaneous Results in the First Notebook 569
Ramanujan's expression F.1) is erroneous. We are very grateful to Richard Brent
for pointing out to us that Ramanujan, indeed, is correct. Moreover, he kindly
provided the proof below.
Proof. We shall prove more precisely that the maximum value equals
1 101
as a tends to oo. Since
1 101
6.2)
1152a1 414720a5
+ O — =
101
360a2
¦GO)
~ 1152a3 + 323.2a + 0A /a)'
this would confirm Ramanujan's claim.
Let /(a) := log Г (a + |). By Corollary 1 in Section 6 of Chapter 8 in the
second notebook (Part I [I, p. 184]),
F.3)
Further differentiations of the asymptotic expansion of /' (a) are valid by a general
theorem on the differentiation of asymptotic expansions (F. W. J. Olver [1, p. 21]),
and we have
a 12cr \аъ /
and
Now,
where
F.4)
F.5)
sup — = Гй(а),
x>o T(jc+ I) T(a + i)
F.6)
and where (see Part II [2, p. 228]), if the maximum is obtained at x = x(a),
F.7)

570 Ramanujan's Notebooks, Part V
as a tends to oo.
Using F.3H6.5), we find from F.6) that, as a tends to oo,
log ft(a) = e log a + f(a) - f(a + e)
= e log a - е/\а) - \t2f"{a) - ^3f'"(a) -
= -(—+-
\24a2 la
From F.7),
<? =
576д2 2560a4
and
as a tends to oo. Using these expansions and F.7) in F.8), we conclude that
1 101
log R{a) =
1152a3 414720a5
which is the required result in F.2).
Entry 7 (p. 138). Let n > m > 0 and let p be real. Put /,-(*) = cos;c,sinjc,/or
,/ = 1,2, respectively. Then, for j — 1,2,
-oo V* ~ c )
Г(т)Г(п - m)
"Й
tan

n — m + к
tan"
m +k
0
G.1)
Proof. It is not difficult to identify /, in terms of beta functions. More precisely
(A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev [1, p. 450, eq. 11]),
/i =
Г(я)
Re [Г(т + /р)Г(п - m - ip))
and
Г(п)
Im {Г(
л - m - ip)\.
G.2)
G.3)
39. Miscellaneous Results in the First Notebook 571
Now, by Euler's product formula for f(z),
V(m + ip)V(n — m — ip)
NmM>N\
= lim
"—oo (m + ip)(m + ip +!)¦•• (m + ip + N) (n — m — ip) ¦ ¦ ¦ (n — m — ip + N)
NmN\
= lim
(,v
-'E
ttn-
So m + k t=S n-m+kt
N -<* v/ra2 + p2 ¦ ¦ ¦ y/(m + NJ + p7/{n - mJ + p2 ¦ • • J(n - m + Л/K + P2
I -^ I P , P
Г(т)Г(п -ra)exp|i > {tan - Ian ' —^—
\frol n-m+k m+k
I)
G.4)
Using G.4) in G.2) and G.3), we readily deduce G.1).
We consider the next result as a formal identity.
Entry 8 (p. 158). //
fh л/л7
/ (p(x) cosBnx) dx = ijrin),
Jo 2
then
Proof. We have
лею
= / e-xl
Jo
I e~x ir{x)dx = — I e^ dx I <p(u) cosBxu) dи
Jo vn Jo Jo
= —= / ^)(и) du I e~*2 cosBjcu) dx
V Л" JO ^0
V я Jo 1
upon using a familiar integral evaluation (Gradshteyn and Ryzhik [1, p. 515,
formula 3.896, no. 4]). This completes the proof.
The next entry is somewhat obliquely stated by Ramanujan, who used a notation
peculiar to him (see, e.g., Part I [1, p. 138]).
Entry 9 (p. 184). //Re/3 > Re a, then
Г(л+о + 1) ,,, . n „
+ и + \))dx =

572 Ramanujan's Notebooks, Part V
where ^f(x) = V
Proof. We have
I, dx\
Г(дс + а + 1)
_
Г(а
Entry 10 (p. 194). For each nonnegative integer n.
Proof. This result follows in a straightforward manner from Entries 29(b), (c),
(d) of Chapter 10 of Ramanujan's second notebook (Part II [2, pp. 39-40]).
We are grateful to R. A. Askey for providing the proof of Entry 11 given below.
Entry 11 (p. 206). //0 < Ren < inf(Rea + 1, Re0 + 1), then
2Ft(a, p; a + P;-x) dx
x
Г(а-и+1)Г03-л+ 1)-^
/
Jo
Г(а + Р- п)
-п + к)(а + fi)kk\
(Ot)k(n)k
Г(о)
(П.1)
Proof. Let / denote the integral on the left side of A1.1). Using Pfaff's transfor-
transformation (Part II [2, p. 36, Entry 19]), we find that
I = г%"-'A+*)-'-" 2/
Making the change of variable l = x/(x + 1), we deduce that
~ Jo '
dx.
{a)\
Г (а)
f^ (or + *)(<* +1
39. Miscellaneous Results in the First Notebook
573
which proves the second equality in A1.1).
To prove the first equality in A1.1), we first establish a general transformation for
iF2(a, b,c;d,e; 1). Suppose that Re(d+e-a-b-c) > OandO < Re с < Ree.
Then, by inverting the order of integration and summation, we find that
J :=
2F\(a, b; d\ x) dx
^ (d)kk\ Г(е
Г(с)Г(*-с)
3F2(a,b,c; d, e; 1).
(П.2)
On the other hand, by a fundamental transformation for hypergcometric functions
(W. N. Bailey [1, p. 2]) and by the same argument as given above,
ч:
-iA _x)d+e-a-h-,-\ 2Fl(d-a,d-b;d;x)dx
1 - a)k(d - b)k Г(с + k)V(d Л-е-а-Ь-с)
¦ e — a — b + к)
V(d + e-a-b)
Combining A1.2) and A1.3), we deduce that
[a,b, c~\ Г(«?)Г(</ + e - a - b - c)
d,e
Г d-a,d-b,c Л
{d,d + e -a -fej '
-a,d-b,c
A1.3)
.....
AL4)
Now set a — a, b = n, с — a, d — a + p, and e = a + 1 in A1.4). Then
' =
Г(о
[а, а, я "I
Г(п)Г(« - n + 1) Г(а + 1)ГрЗ - и + 1)
Г(а
Г р,а+р-п,а
[a+Ba+P-n+
3^2
which establishes the first equality in A1.1).
One might surmise that Entry 11 can be found in integral tables. However, we
are unable to find these evaluations in the tables of Gradshteyn and Ryzhik [1] or
Pnidnikov, Brychkov, and Marichev [2], although on page 315 of the latter tables,
some similar evaluations are given.
Entry 12 (p. 208). Define
), n = 2m, where m is a nonnegative integer, by
F"\x)=x,
FB)(jc) =
B-

574 Ramanujan's Notebooks, Part V
andFw(x) = F(n\F™{x)). Then
F{n\x) =
4xn
{A
~^)"\
A2.1)
Proof. It is easily checked that A2.1) is valid for m = 0,1. Proceeding by induc-
induction, we find that
..2л
4
B - xJn
!(¦
l -
+ П-. 1
я 1 2
B - XJ
4x2n
{B - x + 2^/Г^х)" + B-х-
4л2"
which completes the induction.
Entry 13 (p. 265). If<p(x) is any polynomial, then formally
A3.1)
Proof. Let
V(x) = <pm{x) := jc(jc - l)(x - 2) • • • (x - m + 1),
where m is any nonnegative integer. Since [fm(x)}, 0 < m < j, form a basis for
the vector space of all polynomials of degree < j, it suffices to prove A3.1) for
fix) = fm(x). With this choice of <p(x), the proposed identity becomes
First,
j=0
A3.3)
;=0
39. Miscellaneous Results in the First Notebook 575
Second,
oo
(j)»
Comparing A3.3) and A3.4), we see that we have established A3.2), as desired.
Observe that the series in Entry 13 do not converge.
Entry 14 (p. 265). Leta,0 > 0 with a/S = ж2. Then
A4.1)
Proof. Let
Res>0.
я=0
It is well known that this Dirichlet L-function can be analytically continued to an
entire function.
We apply Entry 21(iii) of Chapter 14 (Part II [2, p. 277]) in the case n = 0 to
find that
where a, p > 0 and afi = ж2. From the functional equation (H. Davenport A, p.
L(s) = cos (±
ГA - s)L(l - s)
and the value L(\) = ж/4, found as C2.7) in Chapter 37, we easily deduce that
1@) = j. Thus, A4.2) may be rewritten in the form
A4.3)
But
n=l
oo
;=0
oo
Using A4.4) in A4.3), we deduce A4.1).
We consider the next result in a formal sense only.
A4.4)

576 Ramanujan's Notebooks, Part V
Entry 15 (p. 265). //
\lr(n) = I <p(x)cos(nx) dx, A5.1)
Jo
then, ifr is an even nonnegative integer,
f{r\n) - cosUnr) I x>(x)cos(rcx)dx @
Jo
and
-<р{г\п)=со&({лг) x>(x)cos(nx)</x. (ii)
2 Jo
Ramanujan put no restrictions on r. He also wrote f(n) and <pr(n) for i/r<r)(«)
and <fi(r)(n), respectively.
Proof. Formally differentiating A5.1) r times, we deduce (i).
By Fourier's inversion theorem (Titchmarsh [2, pp. 16-17], Part I [1, p. 333]),
-<p(n) = I f(x)cos(nx)dx.
2 Jo
Formally differentiating r times, we deduce (ii).
Entry 16 (p. 267). Let a, /3 > 0 with ар = л/4. Then
1. A6.1)
л=0
/i=0
Proof. We apply Poisson's summation formula for Fourier sine transforms (Titch-
(Titchmarsh B, p. 66], Part II [2, p. 236]) to the function x exp (-x2). Thus, if a, fi > 0
and ufi = 7Г/2, then
)П Г
A6.2)
But (Gradshteyn and Ryzhik [1, p. 529, formula 3.952, no. 1])
Гхе-'1 sin {Bn + \)px}dx = ^Bn + We'1™1'1'4.
Jo
Replace 0 by 20 and use the evaluation above in A6.2). Thus, for a, 0 > 0 with
afi = л/4,
Since Va = у/л/Bу/Р), we find that the proof of A6.1) is complete.
39. Miscellaneous Results in the First Notebook 577
Entry 17 (p. 339). ///„ denotes the Bessel Junction of imaginary argument of
order v, then, ifn is an integer,
/пBдс)-/_пBх) = 0.
A7.1)
Ramanujan's recording of Entry 17 is incomplete. He writes the difference of
two series on the left side but does not indicate what this difference equals. The
series are Bessel functions of imaginary argument, as we have indicated (Watson
[15, p. 77]), and A7.1) is a well-known, easily proved equality (Watson [15, p.
79]).
Entry 18 (p. 350).
„-2л
Most of Ramanujan's approximations for ехр(-ая) arise from modular equa-
equations or class invariants. However, it appears that this approximation to ехр(-2тг)
was empirically derived by some method of successive approximations.
Proof. Using Mathematica, we find that
540 ('+ I1
35 Г + 90 + 9^25})))
120 |" ' 100
= 0.001867442731726....
On the other hand,
e-2" = 0.001867442731707....
Thus, Ramanujan's approximation agrees to 13 decimal places.
We conclude by briefly mentioning two claims made by Ramanujan on page
112 that apparently have no precise meanings.
Entry 19.
Since Ramanujan does not specify the function <p or the constants с„, we are
unable to offer a definite interpretation of this formula. Possibly, Ramanujan ap-
applied the Euler-Maclaurin summation formula to the function <p(\ogx)/x on the
interval A, 00). A simple change of variable then gives the integral on the right
side above. The constants cn are therefore those that appear in the Euler-Maclaurin
summation formula, and the series on the right side should probably be interpreted
as an asymptotic series.

578 Ramanujan's Notebooks, Part V
Entry 20.
^ k" + k~" l_
Since the series on the left side diverges and since the meaning of the "dots" on
the right side is not divulged by Ramanujan, we are unable to offer a meaningful
interpretation of this formula. We think that it pertains to material in the first part
of Chapter 7 of the second notebook. In particular, see Part I [1, p. 161].
Location of Entries in the Unorganized
Portions of Ramanujan's First Notebook
In Part IV [4J we provided the locations in the second and third notebooks of all
entries in the 16 organized chapters of the first notebook. A small minority of these
entries cannot be found in the second or third notebooks, and so we provided proofs
for these results in [4]. Like the second notebook, the first notebook contains much
unorganized material, in fact, considerably more than in the second notebook.
The unorganized pages also contain a higher proportion of entries not found in
the second notebook than the organized part of the first notebook. In the sequel,
we indicate where proofs can be found for each correct result in the unorganized
portions of the first notebook. Page numbers of the first notebook are given in
boldface at the left margin. We assign numbers, in order, to each formula on each
page. If the result appears in the second or third notebooks, we indicate where in
these notebooks, and where in Parts I-V, it can be located. If an entry cannot be
found in the second or third notebooks, we inform readers where a proof can be
found in the present volume.
20
1. This is a version of Entry 1 of Chapter 3 [ 1, p. 45].
2.,3. These are Corollaries 1 and 2, respectively, in Section 2 of Chapter 3 [1,
p. 46].
26
1. This is contained in Entry 10 of Chapter 3 [1, pp. 57-59].
46
1. Entry 11(i) of Chapter 18 [3, p. 162].
2.,3. The two radical expressions are equivalent to the formulas for G,225 and
G441, respectively, given in the table of class invariants in Section 2 of Chapter
34.
48
l.,2. Entries 24 and 24(i), respectively, of Chapter 14 [2, pp. 291-292].
3. Entry 36, Chapter 13 [2, p. 238].

580 Ramanujan's Notebooks, Part V
50
l.,3. These are equivalent to Entries 36(i) and (ii), respectively, of Chapter 11
[2, p. 100].
2.,4. Entries 36(iii) and (iv), respectively, of Chapter 11 [2, pp. 100-101].
52
1. Entry 49, Chapter 12 [2, p. 184].
2. Corollary 2, Section 44, Chapter 12 [2, p. 170].
54
1..2. Entries 2(i), (ii), Chapter 13 [2, p. 188].
3. Entry 3, Chapter 13 [2, p. 188].
4.,5. Entry 4, Chapter 13 [2, p. 189].
6. Entry 5, Chapter 13 [2, p. 190].
56
1. Part (i) is the same as Entry 20, Chapter 5 [1, p. 123].
2. Part (ii) follows from Entries 19(i), (ii), Chapter 5 [ 1, pp. 122-123]. The word
"multiple" on page 56 should be replaced by "factor."
58
1. Although this formula is not in the second notebook, it is formula 14 of Table
1 in Ramanujan's paper [7], [10, p. 141].
2. The value of the Bernoulli number Bw can be found in Ramanujan's paper
[1], [10, p. 5] and on page 53 of his second notebook [9].
3. This deleted entry is a partial version of formula 15 in Table 1 mentioned
above.
60
1.-3., 5.-9. The values of the Bernoulli numbers Bn,n = 22,24,26,28,30,32,
34, 36 can be found in Ramanujan's paper [1], [10, p. 5] and second notebook [9,
p. 53].
4. This is formula 12 in Table 1, mentioned in our commentary on page 58.
62
1. This is a trivial statement about the factorization of polynomials.
2.-4. These three formulas are renditions of the same result. The first two contain
unexplained asterisks and are deleted by Ramanujan. The third version is imprecise
and contains an error term that is not completely specified. See Entry 27(ii) of
Chapter 7 [1, p. 178] for a correct version of these three formulas.
64
66
1 .,2. Entry 11, Chapter 11 [2, p. 54].
3. This geometrical figure can be found in Section 19 of Chapter 18 [3, p. 190].
4.,5. Entries 19(iv), (iii), respectively, of Chapter 18 [3, pp. 184, 181].
i
1. See Section 24 of Chapter 18 [3, p. 211].
2. See Section 19 of Chapter 18 [3, p. 194].
Location of Entries in the Unorganized Portions of Ramanujan's First Notebook 581
3. Part of the corollary in Section 3 of Chapter 18 [3, p. 151].
4. Part of Entry 3, Chapter 18 [3, p. 146].
68
l.,2. Corollaries (i), (ii), Section 19 of Chapter 18 [3, pp. 185, 190].
70
1. Entry 14, Chapter 4 [1, p. 107]. This is also a special case of Entry 27 of
Chapter 13 [2, p. 230].
2.,3. Both drawings appear to be versions of a figure of Chapter 18 [3, p. 194].
72
1. Entry 29, Chapter 5 [1, p. 130].
2. Entry 1 of Chapter 39.
3. Entry 6, Chapter 12 [2, p. 111].
74
1.-3. Entries 21(i), (iii), (iv), Chapter 18 [3, pp. 200-201].
78
1. Entry 4(i), Chapter 6 [1, p. 136].
2.,3. Examples 1,2 in Section 4 of Chapter 6 [1, p. 137].
4. See Section 4 of Chapter 6 [1, p. 137].
80
1 .-3. These three singular moduli are given in Theorem 9.9 of Chapter 34.
82
84
1 .-4. Examples 9.4 of Chapter 34.
5.-7. Examples 9.7 of Chapter 34.
8.-10. Theorems 9.6, 9.3, and 9.5, respectively, in Chapter 34.
1. Entry 14, Chapter 15 [2, p. 332].
86
1. Entry 1 l(xv), Chapter 20 [3, p. 385].
2.-10. Entries 45, 42, 43,44, 46, 48, 49, 47, 50, respectively, of Chapter 36.
88
1. Special case of Entry 11, Chapter 6 [1, p. 143].
2.,3. Two elementary abelian theorems for power series with no hypotheses.
See Titchmarsh 's treatise [ 1, pp. 229-231 ] for general abelian theorems.
4.,5. Entries 52 and 51, respectively, of Chapter 36.
90
1. Entry 5(xii), Chapter 19 [3, p. 231].
2. Entry 13(xiv), Chapter 19 [3, p. 282].
3. Entry 19(ix), Chapter 19 [3, p. 315].
4.-7. Entry 41, Chapter 36.

582 Ramanujan s Notebooks, Part V
92
1. Entry 5(xiii), Chapter 19 [3, p. 231].
2. Entry 13(xv), Chapter 19 [3, p. 282].
3. Entry 19(x), Chapter 19 C, p. 315].
94
1 .,2. See Entry 2 of Chapter 39 for comments on these deleted formulas.
3. This is a special case of Entry 35(iii), Chapter 11 [2, p. 99].
4.,6. See Entry 3 of Chapter 39.
5. This is a special instance of Entry 35(ii), Chapter 11 [2, p. 99].
At the bottom of the page, Ramanujan apparently indicates that he has found the
generalizations given in Entries 35<i>—(iii) of Chapter 11 of his second notebook
(Part II [2, p. 99]).
96
1. Theorem 2.12 of Chapter 33.
2.-5. These are contained in Theorem 8.7 of Chapter 33.
6. All of Section 12 of Chapter 33 is devoted to an examination of this claim.
7.-9. See Theorem 8.1 of Chapter 33.
98
l.,2. Entries 9(iv), (iii), respectively, of Chapter 19 [3, p. 258].
100
1. Entry 35, Chapter 9 [1, p. 294].
2. Entry 1 l(iii), Chapter 13 [2, p. 217].
3. Entry 22(ii), Chapter 14 [2, p. 278].
4. Corollary, Section 22, Chapter 14 [2, p. 279).
102
1. Entry 18, Chapter 14 [2, pp. 267-268].
2. Ramanujan, in essence, states a general formula for the multiplication of two
Laurent series.
3. Special case of Corollary 1, Section 18, Chapter 14 [2, p. 268].
104
1. Entry 19(iv), Chapter 14 [2, p. 273].
2. See Entry 4 of Chapter 39.
106
1. Entry 20(i), Chapter 14 [2, p. 274].
2. Entry 19(i), Chapter 14 [2, p. 271].
3. Corollary, Section 20, Chapter 14 [2, p. 274].
4. Entry 20(iv), Chapter 14 [2, p. 275].
5. Corollary of Entry 20(iv), Chapter 14 [2, p. 275].
6. Special case of Entries 29(i), (ii), Chapter 13 [2, p. 231].
108
1. Entry 35, Chapter 12 [2, pp. 156-157].
Location of Entries in the Unorganized Portions of Ramanujan's First Notebook 583
2. See Entry 5 of Chapter 39.
3. Entry 34(ii), Chapter 13 [2, p. 237].
110
1. Entry 40, Chapter 12 [2, p. 163].
112
l.,2. Entries 19 and 20, respectively, in Chapter 39.
3.-5. Entry 14, Corollaries 1, 2, Chapter 7 [1, pp. 166-168].
114
1. Entry 6, Chapter 13 [2, p. 193].
2. Entry 10, Chapter 13 [2, p. 207].
3. A more precise version of this entry can be found in Entry 9 of Chapter 11
[2,p.51J.
116
1..2. Entries 1 l(i), (ii), Chapter 13 [2, pp. 215-216].
118
1. The entry is deleted and is a forerunner of Entry 7 of Chapter 13 [2, p. 195].
120
1. The first statement
D
exp
(/H
is incorrect.
2. The second claim is a version of Example (i), Section 25, Chapter 13 [2, p.
228]. See also Entry 6 of Chapter 39 for a correction to our claim made in Part II
[2].
122
1. Entry 26(ii), Chapter 13 [2, p. 229].
2.,3. Entries 17(iii), (iv), respectively, in Chapter 18 [3, p. 176].
124
l.,2. Entries 11 and 12, respectively, in Chapter 16 [3, pp. 21, 24].
126
1. Entry 31, Chapter 10 [2, p. 41].
2. Entry 7, Chapter 16 [3, p. 16].
128
1. Entry 8, Chapter 16 [3, p. 17].
2. Entry 4, Chapter 16 [3, p. 14].
130
1. Version of the corollary in Section 36 of Chapter 13 [2, p. 239].
2. Entry 5, Chapter 16 [3, p. 14].

584 Ramanujan s Notebooks, Part V
132
1. Entry 6(vi), Chapter 9 [1, p. 247].
2. Entry 6, Chapter 16 [3, p. 15].
3. Entry 8, Chapter 13 [2, p. 202].
4. Corollary of Entry 48, Chapter 12 [2, p. 181].
134
1. Entry 9, Chapter 16 C, p. 18].
2. Entry 15, Chapter 16 [3, p. 30].
3. This entry is very vague. It is possibly a less specific version of Entry 16,
Chapter 16 [3, p. 31].
136
1. Corollary (i) in Section 9, Chapter 16 [3, p. 18].
2. Entry 17, Chapter 16 [3, p. 32].
138
1. Entry 21, Chapter 13 [2, p. 224].
2. See Entry 7 of Chapter 39 for a proof.
140
1..2. Both entries are versions of Entry 21 of Chapter 13 [2, p. 224].
142
1. Version of Entry 17, Chapter 13 [2, pp. 220-221].
2. Deleted by Ramanujan.
3. Entry 20, Chapter 13 [2, p. 224].
4.,5. Entry 16 (Second Part) (iii), (i), respectively, of Chapter 18 [3, p. 174].
144
1. Deleted by Ramanujan.
2.,3. Entries 17(ii), (i), Chapter 18 [3, p. 176].
4.,5. Entry 16 (Second Part) (iv), (ii), respectively, of Chapter 18 [3, p. 174].
146
1. Entry 24, Chapter 12 [2, p. 139].
2..3- Entries 45(i), (ii), Chapter 12 [2, p. 171].
4. Entry 16, Chapter 16 [3, p. 31].
5. Entry 7(vii), Chapter 17 [3, p. 106].
148
1. Entry 6(viii), Chapter 9 [ 1, p. 247].
2.-11. Entry 14, Chapter 18 [3, pp. 168-169].
12. Entry 18(iv), Chapter 18 [3,p. 179].
150
1. Corollary, Section 12, Chapter 18 [3, p. 164].
2. Special case of Entry 14, Chapter 12 [2, p. 1211.
3. Corollary, Section 34, Chapter 12 [2, p. 156].
Location of Entries in the Unorganized Portions of Ramanujan's First Notebook
152
l.,2. Entry 47, Chapter 12 [2, p. 179].
3. Entry I3(i), Chapter 18 [3, p. 165].
4. Entry 22(iii), Chapter 18 [3, p. 207].
5. Deleted by Ramanujan.
154
1. Example (iv), Section 2, Chapter 15 [2, p. 305].
2. Entry 9, Chapter 13 [2, p. 205].
3. Entry 48, Chapter 12 [2, p. 181J.
4. Entry 7, Chapter 13 [2, pp. 195-196].
156
1. Entry 12(ii), Chapter 18 [3, p. 163].
2. Entry 22(ii), Chapter 18 [3, p. 207].
3.,4. Entries 13(ii), (iii), Chapter 18 [3, p. 165].
158
1. Entry 39(i), Chapter 16 [3, p. 83].
2. Entry 38(iv), Chapter 16 C, p. 80].
3. Entry 1 l(iii), Chapter 19 [3, pp. 265-266].
4. See Entry 8 of Chapter 39.
160
1. See Part II [2, p. 147] for comments.
2. Corollary, Section 15, Chapter 16 [3, p. 30].
3. Entry 3, Chapter 16 [3, p. 14].
4.,5. Entries 38(i), (ii), Chapter 16 [3, p. 77].
162
1. Entry 7(iii), Chapter 17 [3, p. 105].
2. Corollary 3.4 of Chapter 33.
3. Theorem 9.9 of Chapter 33.
4. Theorem 11.6 of Chapter 33.
5. Theorem 9.10 of Chapter 33.
6. Corollary 3.5 of Chapter 33.
7. Theorem 4.4 of Chapter 33.
8. Theorem 4.5 of Chapter 33.
164
1.-3. See Section 21, Chapter 13 [2, p. 225].
4. Entry 13, Chapter 16 [3, p. 27].
5. See Section 13, Chapter 16 [3, p. 28].
166
l.,2. Entries 31(i), (ii), Chapter 13 [2, pp. 233-234].
168
1. Entry 23, Chapter 12 [2, pp. 137-138].
585

586 Ramanujan's Notebooks, Part V
2. Entry 7(viii), parts (a), (c), and (d), Chapter 17 [3, p. 107].
170
1. Entry 7(ii), Chapter 17 [3, p. 105].
2. Entry 7(ix), Chapter 17 [3, p. 108].
3. Entry 7(x), Chapter 17 [3, p. 110].
4. Entry 12(i), Chapter 18 [3, p. 163].
5. Entry 22(i), Chapter 18 [3, p. 206].
6. Entry 7(i), Chapter 17 [3, p. 104].
172
1. Entry 7(xii), Chapter 17 [3, p. 112].
2.,3. See Entries 79 and 80, respectively, of Chapter 36.
4. Entry 7(xiii), Chapter 17 [3, p. 113].
5. Entry 7(iv), Chapter 17 [3, p. 105].
174
1. Deleted by Ramanujan.
2. Entry 6, Chapter 10 [2, p. 12].
176
l.,2. Deleted by Ramanujan.
3. Entry 7(vi), Chapter 17 [3, p. 106].
4. See Entry 1 of Chapter 36.
178
1. Entry 14, Chapter 13 [2, p. 219].
2. Quarterly Reports [1, p. 331].
3. Special case of Entry 32(i), Chapter 13 [2, p. 235].
4. Entry 16, Chapter 15 [2, p. 338].
180
1. Entry 15, Chapter 13 [2, p.220].
2. Entry 22(i), Chapter 13 [2, p. 225].
3. Entry 13, Chapter 13 [2, p. 219].
4. Corollary of Entry 13, Chapter 13 [2, p. 219].
5. Corollary of Entry 21, Chapter 13 [2, p. 224].
182
1. Quarterly Reports [1, p. 298].
2. Entry 23, Chapter 13 [2, p. 226].
3. Entry 14, Chapter 16 [3, p. 29].
4. Entry 22(ii), Chapter 13 [2, p. 225].
184
1. See Entry 9 of Chapter 39.
2. See Part II [2, eq. A7.3). p. 265].
3.-5. Examples (i), (ii), (iii), Section 30, Chapter 13 [2, p. 233].
Location of Entries in the Unorganized Portions of Ramanujan's First Notebook 587
186
1,2. Entries 36(i), (ii), Chapter 16 [3, pp. 65-66].
188
1. Entry 34, Chapter 12 [2, p. 156].
2. Entry 39, Chapter 12 [2, p. 159].
3.,4. Entry 38, Chapter 12 [2, p. 158].
190
1. Entry 20, Chapter 10 [2, pp. 36-37].
2. Part of second part of Entry 35, Chapter 12 [2, pp. 156-157].
3. Entry 27. Chapter 12 [2, p. 146].
192
1. First part of Entry 35, Chapter 12 [2, pp. 156-157].
2.,3. Entries 36 and 37, Chapter 12 [2, p. 158].
194
1. See Entry 10 of Chapter 39.
2. Entry 29(c), Chapter 10 [2, p. 40].
3. Entry 32, Chapter 10 [2, p. 41].
196
1. Entry 8, Chapter 11 [2, p. 51].
2. Entry 22, Chapter 11 [2, pp. 64-65].
3. Corollary, Section 22, Chapter 11 [2, p. 68].
4. Example, Section 17, Chapter 12 [2, p. 131].
198
For the meaning of the geometrical figure, see Section 7 of Chapter 19 [3, pp.
244-245].
1. Entry 22, Chapter 12 [2, p. 136].
200
For the significance of the geometrical figure, see Section 7 of Chapter 19 [3,
p. 243].
1. Entry 17, Chapter 12 [2, pp. 124-125].
2.,3. Corollaries (i), (ii), Section 17, Chapter 12 [2, pp. 130-131].
202
1. In essence, Entry 28, Chapter 13 [2, p. 231].
2.,4.-6. See Section 24 of Chapter 13 [2, pp. 226-227].
3. Entry 20(iii), Chapter 18 [3, p. 197].
204
1. Entry 21, Chapter 11 [2, p. 64].
2. See Theorem 7.3 of Chapter 33.
206
l.,2. See Entry 11 of Chapter 39.

588 Ramanujan's Notebooks, Part V
3. Example 2, Section 12, Chapter 11 [2, p. 58].
208
Except for one example, which we establish in Entry 12 of Chapter 39, all results
on this page are found in Section 15 of Chapter 15 [2, pp. 335-337].
210
1. The first five lines on the page continue material from page 208 and can be
found in Section 15 of Chapter 15.
2. Theorem 11.5 of Chapter 33.
3.,4. Theorems 4.2 and 4.3 of Chapter 33.
5.,6. Theorems 9.5 and 9.6 of Chapter 33.
212
l.,2. Entries 23(i), (ii), Chapter 18 [3, p. 208].
3. See Example (iii) in Section 17 of Chapter 9 [2, p. 266].
4. See Theorem 6.4 of Chapter 33.
214
1. This is a definition.
2.,3.,5. These are, respectively, Theorems 10.1, 10.3, and 10.2 of Chapter 33.
4. Theorem 9.11 of Chapter 33
6.,7. Part of Theorem 9.2 of Chapter 34.
8. Corollary 2.4 of Chapter 33.
9. Theorem 10.4 of Chapter 33.
216
1. Theorem 11.4 of Chapter 33.
2. Theorem 11.1 of Chapter 33.
3. Theorem 5.6 of Chapter 33.
218
1. Theorem 6.1 of Chapter 33.
2.-6. Theorem 7.1 of Chapter 33.
7.,8. Theorems 7.6 and 7.8 of Chapter 33.
9. Deleted by Ramanujan.
220
1. This follows from B.6) and Theorem 2.10 in Chapter 33.
2. This is an incorrect version of part of Entry 3(i) of Chapter 21 [3, p. 460].
3. This is a misstatement of the first part of Entry 5(i) of Chapter 21 [3, p. 467].
4.,5. Theorem 2.13 of Chapter 33.
222
1.-4. Entries 8(i)-(iv), Chapter 19 [3, p. 249].
5. See Entry 14 of Chapter 36.
224
1.-3. Corollary, Section 37, Chapter 16 [3, p. 74].
Location of Entries in the Unorganized Portions of Ramanujan's First Notebook 589
4.,5. Entries 4(i), (ii) Chapter 19 [3, p. 226].
226
1 .,3. Deleted by Ramanujan.
2. Part of Entry 34(iii), Chapter 11 [2, p. 97].
4. Entry 7(xi), Chapter 17 [3, p. 111].
228
1. This result is essentially equivalent to the example in Section 37 of Chapter
16 and can be found explicitly in C7.7) on page 76 of Part III [3].
2. Entry 37(iii), Chapter 16 [3, p. 73].
3. This result follows from Entries 37(i), (ii), Chapter 16 [3, p. 73J.
4. Entry 32(v), Chapter 11 [2, p. 93].
230
1.-3. Entries 3(iii), (iv), (i), Chapter 19 [3, p. 223].
4.,8. Entries 17(i), (ii), Chapter 19 [3, p. 302].
5. Entry 9(i), Chapter 20 [3, p. 377].
6. See Entry 2 of Chapter 36.
7. Entry 29 of Chapter 36.
232
1. An incomplete version of Entry 28, Chapter 11 [2, p. 83].
2. Deleted by Ramanujan.
234
1. A deleted version of Entry 30, Chapter 11 [2, p. 87].
2. See the table in Section 2 of Chapter 34.
236
1. Corollary, Section 28, Chapter 11 [2, p. 85].
2. Special case of Entry 30, Chapter 11 [2, p. 87].
3. Entry 31(ii), Chapter 11 [2, p. 88].
238
1. Entry 30, Chapter 11 [2, p. 87].
2. Entry 14, Chapter 11 [2, p. 59].
240
1. Part of Corollary (ii), Section 31, Chapter 16 [3, p. 49].
2. Trivial algebraic identity.
3. Example (i), Section 31, Chapter 16 [3, p. 50].
4.,5. Corollary, Section 28, Chapter 16 [3, p. 44].
6. With the use of Entry 22(ii) and B2.4) in Chapter 16, it can easily be shown
that this entry is equivalent to Example (v), Section 31, Chapter 16 [3, p. 51].
242
1. See Theorem 9.3 of Chapter 33.
2. Entry 9(v), Chapter 19 [3, p. 258].

590 Ramanujan's Notebooks, Pan V
3. Entry 31, Chapter 16 {3, p. 48].
4. Corollary, Section 30, Chapter 16 [3, p. 47].
244
1. This result is easily seen to be equivalent to the first part of Example (iv)
in Section 31 of Chapter 16, although Ramanujan, in the numerator, neglected to
write the factors (*3;.*8)oo(*5;*8)oo(*8;.*8)oc [3, p. 51].
2. An incomplete version of the second part of Example (iv), Section 31, Chapter
16[3,p.5l].
3. See Entry 81 of Chapter 36.
4. This is equivalent to the example in Section 9 of Chapter 17 [3, p. 122].
5. See Entry 82 of Chapter 36.
6.,7. Example (ii), Section 31, Chapter 16 [3, p. 50].
246
l.,2. See Entry 22 of Chapter 36.
3.-11. Entry 10, Chapter 17 [3, p. 122].
12.-17. Entries 1 l(i), (iii), (iv), (v), (vi), (viii), Chapter 17 [3, p. 123].
248
I. Example (i), Section 6, Chapter 17 [3, p. 103].
3. Example (iii), Section 6, Chapter 17 [3, p. 104].
1.-16. See Entry 1 of Chapter 35.
250
1.-9. See Entry 2 of Chapter 35.
10. Entry 27, Chapter 11 [2, p. 80].
I1. Deleted by Ramanujan.
252
1. Entry 7, Chapter 15 [2, p. 313].
2. Special case of Theorem 6.1 of Chapter 15 [2, p. 310].
254
l.,2. See Entries 3 and 4 of Chapter 36.
3.-5. Entries 13(ii)-(iv), Chapter 15 [2, p. 330].
256
1.-3. Entries 12(ii)-(iv), Chapter 15 [2, p. 326].
4. Entry 13<i), Chapter 15 [2, p. 330].
258
1.-4. Entries 12(v)-(viii), Chapter 15 [2, p. 326].
260
1..2. Entries 35(ii), (i), Chapter 16 [3, p. 63, 61].
3. Entry 33(i), Chapter 16 [3, p. 52].
262
1. Entry 34(i), Chapter 16 [3, p. 54].
Location of Entries in the Unorganized Portions of Ramanujan's First Notebook 591
2. Entry 33(ii), Chapter 16 [3, p. 53].
3. Corollary (i), Section 34, Chapter 16 [3, pp. 57-58].
4. Corollary (ii), Section 34, Chapter 16 [3, pp. 59-60].
264
1.-3. Entries 32(i)-("i). Chapter 16 [3, pp. 51-52].
4. Entry 12(ix), Chapter 15 [2, p. 326].
5.,6. See Entries 38 and 39 of Chapter 36.
265
We use the numbering given by Ramanujan.
1. Entry 31, Chapter 12 [2, p. 150].
2. See Entry 13 of Chapter 39.
3.,4. Entries 8(i), (ii), Chapter 17 [3, p. 114].
5. See Entry 14 of Chapter 39.
6. Entry 120), Chapter 4; Quarterly Reports [1, pp. 107,321].
7(i). Special case of the preceding entry.
7(ii). Entry 12(ii), Chapter 4 [1, p. 107]; special case of the corollary of Entry
2lofChapterl3[2,p. 224].
8. See Entry 15 of Chapter 39.
266
1.-4. Entries 8(ix), (xi), (xii), (x), Chapter 17 [3, pp. 114-115].
5. See Entry 31 of Chapter 36.
6. Special case of Entry 18(i), Chapter 13 [2, p. 221].
7. We have found no meaningful interpretation of this equality.
8. See Entry 5 of Chapter 36.
9. Entry 9, Chapter 19 [3, p. 257].
267
The numbering is continued from page 265.
9..10. Entries 8(iii), (iv), Chapter 17 [3, p. 114].
11. See Entry 32 of Chapter 36.
12. Entry 24(ii), Chapter 16 [3, p. 39].
13. See Entry 16 of Chapter 39.
14. Special case of Entry 31(i), Chapter 13 [2, p. 233].
15.-17. Entries 16(i)-(iii), Chapter 13 [2, p. 220].
18. Special case of Theorem II in Ramanujan's Quarterly Reports [1, p. 313].
268
1. Entry 9(i), Chapter 17 [3, p. 120].
2. This is an incorrect version of Entry 9(iv) in Chapter 17 [3, p. 120].
3.,4. Entries 13(iii), (iv), Chapter 17 [3, p. 127].
5. Principle of duplication, Chapter 17 [3, p. 125J.
269
l.,2. Entries 13(i), (ii), Chapter 17 [3, p. 126].
3.-6. Entries 14(i)-(iv), Chapter 17 [3, pp. 129-130].

592 Ramanujan's Notebooks, Pan V
7. Entry 13(viii), Chapter 17 [3, p. 127].
270
1.-5. Entries 17(i)-(v), Chapter 17 [3, p. 138],
6. Entry 11, Chapter 14 [2, p. 258].
271
1.-4. Entries 17(vi)-(ix), Chapter 17 [3, p. 138].
5. See Entry 40 of Chapter 36.
6. See Entry 83 of Chapter 36.
272
1.-4. Entries 14(v)-(viii), Chapter 17 [3, p. 130].
5.-8. Entries 15(iH«v), Chapter 17 [3, p. 132].
273
l.-4.,8.,9. Entries I3(viii), (x), (xi), (ix), (v), (vi), Chapter 17 [3, p. 127].
5.-7. Entries 14(ix)-(xi), Chapter 17 [3, p. 130].
274
1. Entry 21(iii), Chapter 14 B, p. 277].
2. Entry 34<ii), Chapter 16 [3, p. 56].
3. See Entry 33 of Chapter 36.
275
1. Entry 33(iii), Chapter 16 C, p. 53].
2.-6. Entries 16(i)-(v), Chapter 17 [3, p. 134].
276
1. Entry 14, Chapter 14 [2, p. 262].
2. Corollary, Section 12, Chapter 14 [2, p. 260].
3. Entry 12, Chapter 14 [2, p. 260].
277
l.,2. Entry 15, Chapter 14 [2, p. 262].
3. Entry 16(vii), Chapter 17 [3, p. 134].
4. Entry I2(iii), Chapter 17 [3, p. 124].
5. Entry 9(iv), Chapter 17 [3, p. 120].
278
1.-4. Entries 15(ix)-(xii), Chapter 17 [3, pp. 132-133].
5.-7. Entries 8(viii), (vi), (vii), Chapter 17 [3, p. 114].
279
1. Entry 11, Chapter 15 [2, p. 323].
2. Entry 21(ii), Chapter 14 [2, p. 276].
280
1. Entry 6, Chapter 18 [3, p. 153].
2.,3. See Entries 77 and 78 of Chapter 36.
Location of Entries in the Unorganized Portions of Ramanujan's First Notebook 593
4. Entry 7, Chapter 18 [3, pp. 154-155].
281
1. Example, Section 7, Chapter 18 [3, p. 156].
2. See Entry 76 of Chapter 36.
3. This is a formal application of the "change of sign" process.
4. The principal of "change of sign," as described in Section 13 of Chapter 17
[3,p. 126].
282
1. Entry 8(i), Chapter 17 [3, p. 114].
2.,3. Entries 4(iii), (iv), Chapter 19 [3, pp. 226-227].
4. This is equivalent to the second part of Entry 5(iii) of Chapter 19 [3, p. 230].
In particular, see the first equality in the proof of (iii) [3, p. 232].
5. This is equivalent to the first part of Entry 5(i) of Chapter 19 [3, p. 230]. In
particular, see the equality at the top of page 232 [3], where <p*(-q6) should be
replaced by <p(-q6).
6. This is equivalent to the formula for Gj in the table in Section 2 of Chapter
34.
7.-9. Entries 5(i), (iv), Chapter 19 [3, p. 230].
10. Sec Entry 6 of Chapter 36.
283
1 .,2. See Entry 7 of Chapter 36.
3. This is equivalent to Entry 5(vi) of Chapter 19 [3, p. 230], with a replaced
by 1 — >3 and /3 replaced by 1 — a.
4. In essence, this is contained in Entry 5(vi) of Chapter 19 [3, p. 230]; in
particular, see E.4) on page 233 [3].
5.,6. See Entries 8,9 of Chapter 36.
284
1. Equivalent to the formula for G9 in the table in Section 2 of Chapter 34.
2. See Entry 4 of Chapter 35.
3. Deleted by Ramanujan.
4.-6. See Entries 34-36 of Chapter 36.
7. See Entry 15 of Chapter 36.
8. Second part of Entry 5(i), Chapter 19 [3, p. 230].
285
1. Corollary, Section 23, Chapter 18 [3, p. 209].
2.,4. Examples, Section 23, Chapter 18 [3, pp. 209-210].
3. Ramanujan gives the value l/<p4(e~") — 0.71777, which is correct and can
be verified using the evaluation of <p(e") given in Example (i) of Section 6 in
Chapter 17 [3, p. 103].
5. See Entry 37 of Chapter 36.
6. See Entry 16 of Chapter 36.
7. Entry 13(iii), Chapter 19 [3, p. 280].

594 Ramanujan's Notebooks, Part V
286
1. Entry 13(ii), Chapter 19 [3, p. 280].
2. See Entry 27 of Chapter 36.
3. See Entry 17 of Chapter 36.
287
1. Multiply the two equalities of Entry 5(iii), Chapter 19 [3, p. 230].
2. Multiply the two equalities of Entry 13(iv), Chapter 19 [3, p. 281].
3.-5. Equivalent to formulas for G9, G25, and C5 given in the table in Section
2 of Chapter 34.
6. The value of on is found in Theorem 9.9 of Chapter 34.
7. Entry 14(v), Chapter 19 [3, p. 289].
8. Entry 5 of Chapter 35.
9. Entry 14(iv), Chapter 19 [3, p. 289].
288
1 .-4. Deleted by Ramanujan.
5. See Theorem 9.2 in Chapter 34.
289
1. We have not been able to discern the meaning of this entry.
2. Deleted by Ramanujan.
3. Equivalent to the formula for G\$ in the table in Section 2 of Chapter 34.
4.,5. See Theorem 9.2 of Chapter 34.
6. Definition of a modular equation of degree 7.
7. Entry 19(i), Chapter 19 [3, p. 314].
8. Entry 5(ii), Chapter 19 [3, p. 230].
9. This modular equation of degree 1 is trivial since a — ft.
10. Ramanujan evidently intended to write a modular equation here, but there
is no equality sign. Furthermore, the degree is not given. The proposed modular
equation has the unusual feature that, in the first term, a and /J appear with no
radical signs about them.
290
1. Part of Entry 5(viii), Chapter 19 [3, p. 231].
2. Part of Entry 5(v), Chapter 19 [3, p. 230].
3. Entry 19(ii), Chapter 19 [3, p. 314].
4. Entry 13(v), Chapter 19 [3, p. 281].
5.,7. See Entry 10 of Chapter 36.
6. This modular equation of degree 7 follows from Entry 19(iii) of Chapter 19
[3, p. 314] by dividing the first part of Entry 19(iii) by the second part of the same
theorem.
8. This modular equation of degree 7 follows from Entry 19(ii) of Chapter 19
[3, p. 314] by dividing the first part of Entry 19(ii) by the second part of the same
theorem.
Location of Entries in the Unorganized Portions of Ramanujan's First Notebook
595
291
1. This modular equation of degree 3 follows from Entry 5(v) of Chapter 19
[3, p. 230] by dividing the first part of Entry 5(v) by the second part of the same
theorem.
2. Entry 7(i), Chapter 20 [3, p. 363].
3. Entry 15(i), Chapter 20 [3, p. 411].
4. Entry 13(i), Chapter 19 [3, p. 280].
5. Part of Entry 5(vii), Chapter 19 [3, p. 230].
6. Part of Entry 13(xii), Chapter 19 [3. p. 281].
7. Entry 3(x), Chapter 20 [3, p. 352].
8. Entry 15(i), Chapter 19 [3, p. 291].
9. Entry 8(iii), Chapter 20 [3, p. 376].
10. Part of Entry 19(v), Chapter 19 [3, p. 314].
11. Entry 1 l(vi), Chapter 20 [3, p. 384].
292
1. Entry 5(viii), Chapter 19 [3, p. 231].
2. Part of Entry 13(vii), Chapter 19 [3, p. 281].
3. Entry 22(i), Chapter 20 [3, p. 439).
4. Equivalent to a formula for С13 given in the table in Section 2 of Chapter 34.
5. Theorem 9.14 of Chapter 34.
6. Entry 3(i), Chapter 20 [3, p. 352].
7. Part of Entry 5(ix), Chapter 19 [3, p. 231].
8. Part of Entry 13(x), Chapter 19 [3, p. 281].
9. Entry 5(xiv), Chapter 19 [3, p. 231].
10. Entry 14(i), Chapter 19 [3, p. 288].
11. This entry,
sinB«) =
3sin2(±i)),
|y (),
is difficult to read. Although not apparent, by using elementary trigonometry, it
can be shown that this modular equation of degree 7 is equivalent to Entry 19(xi)
of Chapter 19 [3, p. 315].
293
1.-3. Equivalent, respectively, to formulas for G2\, G49, and G225 given in the
table in Section 2 of Chapter 34.
4. Part of Entry 1 l(ii), Chapter 20 [3, p. 383].
5. Entry 2(i), Chapter 20 [3, p. 349].
6.-10. Entries 3(iv), (vi), (v), (ii), (iii), Chapter 20 [3, p. 353].
294
l.,3. Entries 2(ii), (iii), Chapter 20 [3, p. 349].
2.,4. Entries 9(v), (vi), Chapter 20 C, p. 377].
5.,6. Equivalent to Entry 9(vii), Chapter 20 [3, p. 377].
7. Equivalent to the formula for Gl69 in the table in Section 2 of Chapter 34;
for more details, see Watson's paper [7, p. 195].

596 Ramanujan's Notebooks, Part V
8. Equivalent to the formula for G i2\ in the table of Chapter 34; for more details,
see Watson's paper [7, p. 190].
9. This entry is difficult to read and is evidently incomplete. We offer some
comments on it at the end of Section 8 of Chapter 34.
295
1.-3. See Entries 18,19, and 11, respectively, of Chapter 36.
4. Deleted by Ramanujan.
5.-8. Equivalent, respectively, to formulas for G45, Сц, G23, and G19 in the
table in Section 2 of Chapter 34.
296
1. Equivalent to the formula for G31 in the table of Chapter 34.
2. Entry 12(iii), Chapter 20 [3, p. 397].
3. Equivalent to the formula for Gp in the table of Chapter 34.
4. Deleted by Ramanujan.
5.,6. Entries 28 and 26, respectively, of Chapter 36.
7. Deleted by Ramanujan.
297
1.-4. See Entries 12,20, 13, and 25, respectively, of Chapter 36.
5. See Entry 7 of Chapter 35.
6. Entry 9(iii), Chapter 20 [3, p. 377].
7.-11. Entries 1 l(ib(v), Chapter 20 [3, pp. 383-384].
12. Part of Entry 19(iv), Chapter 19 [3, p. 314].
298
1. Entry 18(i), Chapter 20 [3, p. 423].
2. This is equivalent to Entry 12(iii) of Chapter 20 [3, p. 397]; the right-hand
side, K, in the formulation given on page 298 is y/m, in the customary notation
for the multiplier m.
3. See Entry 75 of Chapter 36.
4. Entry 13(viii) of Chapter 19 [3, p. 281].
5. See Entry 30 of Chapter 36.
6. Part of Entry 19(vii), Chapter 19C, p. 314].
299
1. Entry 18(ii), Chapter 20 [3, p. 423].
2. Entry 4(iv), Chapter 20 [3, p. 359].
3.,4. Entries 17(i), (ii), Chapter 20 [3, p. 417].
5.-7. Entries 19(i), (iii), (ii), Chapter 20 [3, p. 426].
8.,9. Entries 17(iii), (iv), Chapter 20 [3, p. 417].
300
1. Part of Entry 5(ix), Chapter 19 [3, p. 231).
2. Part of Entry 13(x), Chapter 19 [3, p. 281].
3..4. Part of Entry 5(x), Chapter 19 [3, p. 231].
5..6..10. Entry 24(ii), Chapter 18 C, p. 214].
Location of Entries in the Unorganized Portions of Ramanujan's First Notebook 597
7..8..11. Entry 24(iii), Chapter 18 [3, pp. 214-215].
9.,12. See Entries 23 and 24 of Chapter 36.
13.-15. See, respectively, Theorems 7.2,7.7, and 7.5 of Chapter 33.
301
1.-6. See Theorems 10.5-10.10, respectively, of Chapter 33.
7. Theorem 9.4 of Chapter 33.
8. Theorem 9.15 of Chapter 33.
9. This is an incorrect version of Entry 3(iii), Chapter 21 [3, p. 460].
10. This is an incorrect version of Entry 5(iii), Chapter 21 [3, p. 468].
302
1. Entry 19(i), Chapter 19 [3, p. 314].
2. Entry 150). Chapter 20 [3, p. 411].
3.,4. Deleted by Ramanujan.
5.-9., 12-14. See Entries 56, 53, 54, 55, 57, 58, 60, and 59, respectively, in
Chapter 36.
10. Entry 5(ii) of Chapter 19 [3, p. 230].
11. Another version of Entry 7(i) of Chapter 20 [3, p. 363].
303
A discussion of the material on this page is given in the last part of Section 7
of Chapter 36.
304
1. The top left corner of the page in the original notebook has been torn away,
and so the degTee of the first modular equation is unknown. However, the form of
the equation is exactly that of the modular equation of degree 23 in Entry 15(i) of
Chapter 20 [3, p. 411].
2.-6. Entries 62,63,64,65, and 61, respectively, of Chapter 36.
305
I. Incomplete version of Entry 4(ii) of Chapter 21 [3, p. 464], and is deleted by
Ramanujan.
2.-5.,7. Equivalent to formulas for G2i, G37, G39, G97, and G63, respectively,
in the table in Section 2 of Chapter 34.
6. Deleted by Ramanujan.
306
1.-3. Deleted by Ramanujan.
4. Part of Entry 19(vi), Chapter 19 [3, p. 314].
5. Entry 7(iv), Chapter 20 [3, p. 363].
6. Part of Entry 5(xi), Chapter 19 [3, p. 231].
7. Part of Entry 13(xiii), Chapter 19 [3, p. 282].
8. Entry 19(viii), Chapter 19 [3, p. 315].
9. Entry 7(ii). Chapter 20 [3, p. 363].
10. Entry 19(xi), Chapter 19 [3, p. 315].
I1. Part of Entry 19(i), Chapter 19 [3, p. 314].

598 Ramanujan s Notebooks, Pan V
307
1.-3. Entries 1 l(ix), (viii), (xiv), Chapter 20 [3, pp. 384-385].
4. Entry 15(v), Chapter 19 [3, p. 291].
5.,6. Entry 19(iv), Chapter 20 [3, p. 426).
7.,8. Entries 3(xii), (xiii), Chapter 20 [3, pp. 352-353].
308
l.,2. Entries 13(i), (ii), Chapter 20 [3, p. 401].
3. Entry 5(ii), Chapter 20 [3, p. 360].
4. Deleted by Ramanujan.
5. Entry 15(iv), Chapter 19 [3, p. 291].
6. Deleted by Ramanujan.
309
1. Entry 15(iii), Chapter 19 [3, p. 291].
2. Entry 5(i), Chapter 20 [3, p. 360].
3..4. Entries 69 and 68, respectively, of Chapter 36.
5. Entry 73 of Chapter 36.
6. Entry 13(iii) of Chapter 20 [3, p. 401].
7. Entry 14(i) of Chapter 20 [3, p. 408].
8. Entry 1 l(x) of Chapter 20 [3, p. 384].
9. Entry 13(i) of Chapter 20 [3, p. 401]. (There is a sign error; replace —4 by
4.)
10.-13. Entries 72,74, 71, and 70, respectively, of Chapter 36.
310
1. See Entry 66 of Chapter 36.
2.,3. See Theorems 7.10 and 7.9, respectively, of Chapter 33.
4.-11. These singular moduli are given in Theorem 9.2 of Chapter 34.
12., 13. Theorem 9.16 in Chapter 34.
311
1. Equivalent to the formula for G33 given in the table in Section 2 of Chapter
34.
2.,3. See Entry 8.1 of Chapter 34.
4.,5. Entry 9.14 of Chapter 34.
6.-9. Entries 7-10, respectively, of Chapter 32.
312
1. See Entry 6 of Chapter 35.
2.-12. See Section 10 of Chapter 34.
13., 14. Deleted by Ramanujan.
15., 16. Part of Theorem 9.2 of Chapter 34.
313
1. Equivalent to the formula for G73 in the table in Section 2 of Chapter 34.
2. Deleted by Ramanujan.
3. Equivalent to the formula for G« in the table of Chapter 34.
Location of Entries in the Unorganized Portions of Ramanujan's First Notebook 599
4. Entry 7(iii), Chapter 21 [3, p. 475].
5.-7. Part of Theorem 9.2 of Chapter 34.
, G363, G217, G205.
314
1.-9. Equivalent to formulas for C69, Сц7,(?ззз, Ggi.
and G265, respectively, in the table of Chapter 34.
315
1.-15. A table of values for Gn, n = 57,93,177,85,133,55.65,253,145,117,
333,153, 77,69,213. See the table of Chapter 34.
316
1. This is the definition of gn.
2.-13. These twelve values for gn, n = 2,6,10,16,18,22, 30,58,70,46, 142,
82, are found in the table of Chapter 34.
14..15. See Entry 8.2 of Chapter 34.
16. Deleted by Ramanujan.
17.-22. These six values for g,,,n =42,78,102,130,190,34, are found in the
table of Chapter 34.
317
1.-9. The nine values of Gn,n =289, 121, 169, 105,165,345,385,273,357,
are given in the table of Chapter 34.
318
l.-3.,5.,6.,8.-10. These eight values for gn, n =98,90,198,522,630,50, 126,
26, can be found in the table of Chapter 34.
4. See Entry 3.2 of Chapter 34.
7. This formula for ?ц7о has been deleted by Ramanujan.
319
l.,2.,5.,8.,9. Five values for gn, n = 66,138,238, 154, 310, are given. These
can be found in the table of Chapter 34.
3. This formula for gi54 has been deleted by Ramanujan, but it is correct, except
for one misprint; see the table of Chapter 34.
4. This formula for g{ 14 is incorrect; see the table of Chapter 34 for a correct
version.
6.,7.,10. These three values for gn,n = 62,94,158, are also given in the table
of Chapter 34, but the formulations are somewhat different.
11.,12.,13. Three values for Gn, n = 465,777,1353, are given. Some calcula-
calculations are needed to show that the formula for G1353 given here is equivalent to that
in the table of Chapter 34.
320
1.-6. The values for Gn,n = 1645, 897, 1677, 141, 445, 553, are given in the
table of Chapter 34.
7.,8. The values for g,,,n = 210, 330, arc given in the table of Chapter 34.
9. See Theorem 9.1 of Chapter 34.

600 Ramanujan's Notebooks, Part V
10. See the introduction to Section 9 of Chapter 34.
321
1..2. Entry 1 l(iii). Chapter 19 [3, pp. 265-266].
3.-6. Parts of Entries l(iv), (v), (ii), (iii), Chapter 20 [3, pp. 345-346].
7. Entry 1, Chapter 18 [3, p. 144].
8. Entry 2, Chapter 18 [3, p. 145].
9. Entry 8(i), Chapter 18 [3, p. 157].
10. Entry 9, Chapter 18 [3, p. 159].
322
1. Entry 18(vi), Chapter 19 [3, p. 306].
2.,3. Part of Entry 3(i), Chapter 21 [3, p. 460].
4.,5. Entry 3(ii), Chapter 21 [3, p. 460].
6.,7. Entry 4(i), Chapter 21 [3, p. 463].
8. Entry 4(ii), Chapter 21 [3, p. 464].
9. Part of Entry 5(i), Chapter 21 [3, p. 467].
323
1. Entry 5(ii), Chapter 21 [3, p. 468].
2. Part of Entry 7(i), Chapter 21 [3, p. 475].
3. Entry 7(ii), Chapter 21 [3, p. 475].
4.-6. Entries 2(vii), (v), (viii), Chapter 20 [3, p. 349].
7.-9. Entries 8(i), (ii), (iii), Chapter 21 [3, p. 480].
10. Entry 9(i), Chapter 21 [3, p. 481].
324
1..2. Entries 9(ii), (iii), Chapter 21 [3, p. 481].
3..4. Entry 4(iii), Chapter 21 [3, p. 464].
5.-7. Entry l(i), Chapter 20 [3, p. 345].
8.-10. Entry l(ii), Chapter 19 [3, p. 221].
325
1. The formula for G301 can be found in the table of Chapter 34.
2.,3.,6.,7. Entries 10(i)-(iv) of Chapter 20 [3, p. 379].
4.,5. Entry 21 of Chapter 36.
8. Entry 1, Chapter 19 [3, p. 221].
9. Entry 18(i), Chapter 19 [3, p. 305].
10. Entry 6(iii), Chapter 20 [3, p. 363].
326
1 .-4. Entry 18(i), Chapter 19 [3, p. 305]. Note that the definitions of u, v, and
w are different in the first notebook.
5.-10. Entry 8(ii), Chapter 20 [3, pp. 372-373].
327
1.-4. Entry 12(i), Chapter 20 [3, p. 397).
5. Entry 5(iv), Chapter 20 [3, p. 360].
Location of Entries in the Unorganized Portions of Ramanujan 's First Notebook 601
6.,7. Entry 18(v), Chapter 20 [3, p. 423].
328
1. Theorem 7.11 of Chapter 33.
2.-4. Entries 8-10, Chapter 30 [4, pp. 374-377].
329
1.-3. Entries 62,69, and 63, respectively, of Chapter 25 [4, pp. 221,236, 223).
4.,5. Entries 25 and 26 of Chapter 32.
330
1.-4. Entries 51, 49,52, and 50, respectively, of Chapter 32.
331
1. Entry 21, Chapter 28 [4, p. 309].
2.-6. Entries 48, 14, 19,20, and 21, respectively, of Chapter 32.
332
1. Entry 23 of Chapter 32.
2. Deleted by Ramanujan.
333
1 .,2. Entries 6 and 7 of Chapter 30 [4, pp. 364, 370].
334
1. Entry 5 of Chapter 30 [4, p. 363].
2. An incomplete version of Entry 34 of Chapter 37.
3. Deleted by Ramanujan but proved in Entry 31 of Chapter 37.
4. Entry 20, Chapter 28 [4, p. 309].
5. See Entry 32 of Chapter 37.
335
1. See Entry 33 of Chapter 37.
2..3. Entries 6 and 4, respectively, of Chapter 29 [4, pp. 338,336].
336
1 .,2. Entries 5 and 3, respectively, of Chapter 29 [4, pp. 337,336].
3. Entry 18, Chapter 28 [4, p. 307].
4. See Entry 27 of Chapter 37.
337
1 .,2. Entry 29 of Chapter 37.
3. Entry 30 of Chapter 37.
4. Entry 17, Chapter 28 [4, p. 306].
5. Deleted by Ramanujan; for a correct version see the tables of Gradshteyn and
Ryzhik [ 1, p. 546, formula 4.113, no. 4].
338
The results on this page comprise the contents of Section 3 of Chapter 35.

602 Ramanujan's Notebooks, Part V
339
1.-3. These results are also contained in Section 3 of Chapter 35.
4. A definition.
5. A triviality.
6.-8. These results are essentially Entries 4, 3, and 7, respectively, of Chapter
35.
9.-11. Entries 23,24, and 22, respectively, of Chapter 25 [4, pp. 154,155,153].
12. See Entry 17 of Chapter 39.
340
1. Entry 12 of Chapter 35.
2. Entry 3, Chapter 30 [4, p. 359].
3.-8. Entries 1-6, Chapter 26 [4, pp. 245-255].
341
1.-3. Entries 7-9, Chapter 26 [4, pp. 255-257].
4. Entry 4, Chapter 30 [4, p. 360].
5. Entry 24, Chapter 32.
6. Entry 20 of Chapter 37.
342
1 -3. Deleted by Ramanujan.
4. Entry 23 of Chapter 37.
5. Entry 10, Chapter 26 [4, p. 258].
6. Entry 1, Chapter 29 [4, p. 335].
343
1.-3. Entries 24-26, respectively, of Chapter 37.
4. Entry 22 of Chapter 37.
5. An incomplete entry. We offer some comments on it at the end of Section 8
of Chapter 34.
6. The value for Gn$ is given in the table in Section 2 of Chapter 34.
344
1. Entry 6 of Chapter 32.
2. The value of G505 is given in the table of Chapter 34.
3.-5. Monic irreducible cubic polynomials satisfied by gn, n — 38,26, and 50.
See the table in Section 2 of Chapter 34.
6. Entry 13 of Chapter 32.
345
1 .-7. Monic irreducible cubic polynomials satisfied by С„, n = 23, 31,11,19,
27, 43, and 67. See the table in Section 2 of Chapter 34.
8.-16. Factors of singular moduli ^5^, n = 3, 5, 7, 9, 13, 15, 17, 21, and 25.
However, for n = 21, Ramanujan fails to record any factors. See Theorem 9.9 of
Chapter 34.
Location of Entries in the Unorganized Portions of Ramanujan's First Notebook 603
346
1.-4. Factors of singular moduli <Ja^, л = 7,15, 39, and 55, but for л = ?9,
Ramanujan left a blank space. See Theorem 9.9 of Chapter 34.
347-349
All the results on these three pages are found in Ramanujan's paper on Bernoulli
numbers [1], [10, pp. 1-14].
350
For comments on Ramanujan's notes at the top of the page, see Bemdt and
Rankin'sbook [l,p. 10].
1. See Entry 18 of Chapter 39.
351
1. Ramanujan gives the first 22 digits of sfl.
2.-10. These are irreducible polynomials for the reciprocals of the invariants
Gn, n = 3,7,11,19,23,27,31,43, and 67. See the table of Chapter 34. Ramanujian
had evidently intended to calculate several further polynomials, as indicated by
vacant spaces beside certain other values of n.

References
Abramowitz, M. and Stegun, I. A., editors
[1] Handbook of Mathematical Functions, Dover, New York, 1965.
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350-372.
[4] Question 571, J. Indian Math. Soc. 7 A915), 32.
[5] On the integral/^ (tan t)/t dt,J. Indian Math. Soc. 7 A915), 93-96.
[6] On the sum of the square roots of the first n natural numbers, J. Indian Math. Soc. 7
A915), 173-175.
[7] On certain arithmetical functions, Trans. Cambridge Philos. Soc. 22 A916), 159-
184.
[8] Proof of certain identities in combinatory analysis, Proc. Cambridge Philos. Soc. 19
A919), 214-216.
[9] Notebooks B volumes), Tata Institute of Fundamental Research, Bombay, 1957.
[10] Collected Papers, Chelsea, New York, 1962.
[II] The Lost Notebook and Other Unpublished Papers, Narosa, New Delhi, 1988.
Rankin, R. A.
[ 1 ] Modular Forms and Functions, Cambridge University Press, Cambridge, 1977.
Rao, M. B. and Aiyar, M. V.
[1] On some infinite series and products. Part I, J. Indian Math. Soc. 15 A923-24),
150-162.
[2] On some infinite products and series, Part II, J. Indian Math. Soc. 15 A923-24),
233-247.
Riordan,J.
[1] An Introduction to Combinatorial Analysis, Princeton Universiy Press, Princeton,
NJ, 1978.
Rogers, L. J.
[1] Second memoir on the expansion of certain infinite products, Proc. London Math.
Soc. 25 A894), 318-343.
[2] On the representation of certain asymptotic series as convergent continued fractions,
Proc. London Math. Soc. B) 4 A907), 72-89.
[3] On two theorems of combinatory analysis and some allied identities, Proc. London
Math. Soc. B) 16 A917), 315-336.
[4] On a type of modular relation, Proc. London Math. Soc. 19 A921), 387-397.
Roman, S.
[ 1 ] The formula of Faa di Bruno, Amer. Math. Monthly 87 A980), 805-809.
Rotman, J.
[ 1 ] Group Theory, Springer-Verlag, New York, 1990.
Russell, R.
[1] ОпкХ-к'Х1 modular equations, Proc. London Math. Soc. 19A887-88), 90-111.
[2] On modular equations, Proc. London Math. Soc. 21 A889-90), 351-395.
Sansone, G. and Gerretsen, J.
[ 1 ] Lectures on the Theory of Functions of a Complex Variable, Noordhoff, Groningen,
1960.

616 Ramanujan's Notebooks, Part V
Schlafli. L.
[1] Beweis der Hermiteschen Verwandlungstafeln fur elliptischen Modularfunclionen,
J. Reine Angew. Math. 72 A870), 360-369.
Schlomilch, O.
[1] Ueberdas bestimmte Integral /0°° ^dx, Zeii. Math. Phys. 5 A860), 286-292.
Schoeneberg, B.
[ 1 ] Elliptic Modular Functions, Springer-Verlag, Berlin, 1974.
Schur, I.
[1] Ein Beitrag zur additiven Zahlentheorie und zur Theorie der Kettenbriiche, Sitz.
Preus. Mood. Wiss., Phys.-Math. Kl. A917), 302-321.
[2] Cesammelte Abhandlungen, Band П, Springer-Verlag, Berlin, 1973.
Selberg, A.
[ 1 ] Uber einige arithmetische Identitaten, Avh. Norske Vid. -Akad. Oslo I. Mat. -Naturv.
Kl. A936). 2-23.
[2] Collected Papers, Vol. 1, Springer-Verlag, Berlin, 1989.
Shen, L.-C.
[ 1 ] On an identity of Ramanujan based on thehypergeometric series 2Fi(j, f; \\x),J.
Number Theory, to appear.
Siegel, С L.
[ 1 ] Advanced Analytic Number Theory, Tata Institute of Fundamental Research, Bom-
Bombay, 1980.
Sitaramachandrarao, R.
[ 1 ] Some formulae of Ramanujan, Indian J. Pure Appl. Math. 18 A987), 678-684.
Stark, H. M.
[1] Class-numbers of complex quadratic fields, in Modular Functions of One Variable
I, W. Kuyk, ed., Lecture Notes in Mathematics No. 320, Springer-Verlag, Berlin,
1973, pp. 153-174.
Stieltjes, T. J.
[ 1 ] Surquelques integrates definies et leur developpement en fractions continues, Quart.
J. Math. 24 A890), 370-382.
[2] Recherches sur les fractions continues, Ann. Fac. Sci. Toulouse 8 A894), J, 1-122;
9A894), A, 1-47.
[3] Oeuvres Completes, I. 2, Noordhoff, Groningen, 1918.
Stolz, O.
[ 1 ] Vorlesungen iiber Allgemeine Arithmetik, Teubner, Leipzig, 1886.
Szasz, O.
[1] Uber die Erhaltung der Konvergenz unendlicher Kettenbruche bei independenter
Veranderlichkeit aller ihrer Elemente, У. Reine Angew. Math. 147 A917), 132-160.
Szego, G.
[1] Orthogonal Polynomials, 4th ed., American Mathematical Society, Providence, RI,
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Tchebychef, P. L.
[1] Sur Г interpolation des valeurs equidistantes, in Oeuvres, t. 2, Chelsea, New York,
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References 617
Thiele, T. N.
[ I ] Bemaerkningerom periodiske kjaedebwkers konvergens, Tidsskr. Math. D) 3 A879),
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Thiruvenkatachar, V. R. and Venkatachaliengar, K.
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618 Ramanujan's Notebooks, Part V
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A987), 313-324.
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11 ] <7-difference equations and Ramanujan-Selberg continued fractions, Acta Arith. 57
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pp. 817-838.
[3] Kronecker's limit formula, class invariants and modular equations (III), Acta Arith.,
to appear.
Zucker, I. J.
[1] The evaluation in terms of Г-functions of the periods of elliptic curves admitting
complex multiplication, Math. Proc. Cambridge Philos. Soc. 82 A977), 111-118.
Index
Abel-Plana summation formula, 409,
411,416,419-422
Abel-Plana summation theorem, 417
Adiga, C, 6, 37
Aiyar, M.V., 424, 435
Alladi, K., 35, 49
analogue of K(k) in signature 3, 133
Ananda Rau, K., 321
Anderson, N., 6
Andrews, G.E., 2, 5, 6,10, 37, 39, 46-48,
50
Apostol, T.M., 566
arithmetic-geometric mean, 90
Askey, R.A., 3,6, 549, 558, 572
Atkin, A.O.L., 136,322,455
Bachman, G., 3, 6, 410, 491, 500
Bailey, W.N., 73, 80, 106, 487, 506, 514,
573
Bailey's 6^6 summation, 138
Barrow, D.F., 490, 491
base, 91
basic hypergeometric function, 43
Bauer-Muir transformation, 33, 37,
55-57,77
Bernoulli numbers, 12, 66, 67, 410, 423,
425-427,444, 445, 466, 478,488,
489, 513,517-519,532, 580, 603
Bernoulli polynomials, 411
Berwick, WE., 311,312, 321
Bessel function, 69, 71, 577
Bhargava, S., 2, 6, 37, 133, 180
Biagioli, A.J., 6
Bialek, P., 3, 6,427
Birch, B.J., 258,259, 322
Boas, R.P., Jr., 505
Borevich, Z.I., 224, 263
Borwein, J.M., 6, 89,90,93, 94, 97. 108,
111,135-137,143,152,180,187,
205, 281-284, 324, 330, 334, 451
cubic theta-function, 327
cubic theta-function identity, 93
Borwein, P.B., 6, 89, 90, 93,94, 97, 108,
111, 143, 152,187,205,281-284,
330, 334
cubic theta-function, 327
cubic theta-function identity, 93
Bowman, D., 48
Bradley, D., 3,6, 27, 451,547
Brent, R., 3, 470,505, 547, 569
Brillhart, J., 188, 189, 276
Brychkov, Yu.A., 570,573
Buell, D.A., 225
Cardan's method, 271, 272
Can, G.S., 81,208,209
Catalan's constant, 458
Cauchy, A., 430, 440, 442,463
Chan, H.H., 1-3, 6, 10,20,29,30, 46,50,
97, 180, 183, 186, 208, 259, 277,
317,327, 336-338,341,343, 346,
351,362
Chandrasekharan, K., 309
Charlier polynomials, 504, 558

620 Index
Chen, I., 322
Chrystal,G..81
Chudnovsky, D.V., 90
Chudnovsky, G.V., 90
class field theory, 186, 257-265
class invariants, 21, 22, 183, 324
class number, 217
Clausen's formula, 90,177, 178
Cohen, H., 6, 39, 224
complementary modulus, 323
complete elliptic integral of the first kind,
90,185,323
Comtet, L., 508, 509
confluent hypergeometric function, 523
confluent hypergeometric function of the
second kind, 523
continued fractions, 9-88
correspondence, 68,72
modified convergence, 35
Copson, E.T., 69
Cox, D.A., 183, 258-260, 262-264, 309,
310
cubic analogue of the arithmetic-
geometric mean, 97
cubic analogue of the Jacobian functions,
133
cubic continued fraction, 208
cubic theta-function identity, 96
cubic transformation, 96,97, 101, 123
Davenport, H., 575
Dedekind eta-funciion, 217, 323
Dedekind zeta-funclion, 217
Dedekind zeta-function for an ideal class,
217
degree, 91,185
denesting, 208
denesting theorem, 284
DeTemple, D.W., 532
Deuring.M., 183,259
dimidiation, 101
Dirichlet, P.G.L., 436
Dirichlet L-function, 575
discriminant, 217
duplication, 101
Eisenstein series, 97,105, 376-378, 409,
426, 484
elliptic integrals, 355-356, 403-405
empirical method, 205
empirical process, 184-186, 257
Epstein zeta-function, 216
Erdelyi,80,161, 162,524
eta-function identity, 20
Euler, L., 47,81,410, 439, 490,496, 531,
532
Euler numbers, 503,546
Euler (p-function, 131
Euler constant, 50, 53, 165, 217, 445,
451,531,536,540,542
Euler continued fraction, 77, 79
Euler identity, 47, 49, 187
Euler pentagonal number theorem, 92
Euler-Maclaurin summation formula,
410, 477,504. 513, 543,563, 577
Evans, R.J., 3,6,503, 522, 541,542
Faa di Bruno's formula, 525
false theta-function, 503, 545
Farkas, H.F., 180
Ferrari's method, 331
Fiedler, E., 386
Fine, N.J., 136
Fitzroy House, 93
Frank, E., 79
Frankel, N.E., 549
fundamental unit, 217, 317
Galway, W.,451,503, 547
gamma function, 50-66
Garvan, F.G., 2,6,94, 111,135,136,180,
181
Gasper, G., 138
Gauss, C.F., 90
Gauss genus character, 218
Gauss's continued fraction, 76
Gauss's theorem, 487
Gaussian binomial coefficient, 30
Gaussian quadrature, 549, 558
Gelfond-Schneider theorem, 165
general convergence, 88
genus field, 259
genus group, 218
Gerretsen, J., 421
Gill, J., 82
Glaisher, J.W.L., 427, 430
Glasser, M.L., 3. 6, 548, 549,566
Gordon, В., 46, 49, 50
Index 621
Goursat, E., 97
Greenhill, A.G., 281,312, 313, 316, 321,
353
Gross, B.H., 322
Grosswald, E., 5, 424
group of units, 218
Grobner bases, 253
Hafner, J.L., 3,6, 445, 476, 560
Hahn polynomials, 504, 551
Halberstam, H., 6
Hall, H.S., 271, 331, 359
Hanna, M., 386,389
Hardy, G.H., 4,9, 10, 20, 37,48, 67, 74,
76, 93, 184, 277, 424, 428, 434,
467, 470, 505, 536, 537
harmonic series, 531, 543
Heine, E., 43
Henrici, P., 144, 411
Hilbert class field, 257
Hildebrand, A., 3, 6, 482, 500
Hirschhorn, M.D., 37, 46, 48, 135, 136,
180
Hua, L.K., 442
Huang, S.-S., 36, 50, 183,259
Huard, J.G., 324
Hurwitz, A., 427
Hurwitz zeta-function, 513
hypergeometric functions, 72,73, 89, 90,
323, 327-328, 503, 566, 572
Ideal class character, 217
ideal class group, 217
incomplete gamma function, 524
infinite exponential, 410, 490
Institute for Advanced Study, 5
inversion formula, 92,93,99
iterated logarithm, 491
./-invariant, 186
Jacobi, C.G.J., 377
Jacobi triple product identity, 323
Jacobi's identity for fourth powers, 93
Jacobsen, L., 2,12, 33, 35,38, 51, 62,65,
72, 85. 88
Janusz, J., 262
Jones, W.B., 12,34,68,70,72,81, 85-88
Jordan, C, 514
Joshi, P., 6
Journal of the Indian Mathematical
Society, 440, 461
Joyce, G.S., 324, 327
К -periodic continued fraction, 81, 83,84
Kaltofen, E., 322
Kaplan, P., 324
Kaufmann-Buhler, W., 6
Keckic,J.D.,566
Khovanskii, A.N., 68, 83
Knight, S.R., 271, 331, 359
Knoebel, A., 490
Knopp, M.I., 5, 130
Kopeliovich, Y., 180
Kortum, R., 225
Kowalenko, V, 549
Kra.I., 180
Krishnaiah, P.V., 467
Kronecker, L., 218
Kronecker limit formula, 341
Kronecker's Limit Formula, 216-243
Kronecker's limit formula, 185
Lagrange inversion formula, 490
Laguerre polynomial, 524
Lambert series, 373-376, 504
Lambert. J.H., 70
Lamphere, R.L., 2,6
Landau, S., 284
Landen's transformation, 101
Lebedev, N.N., 523
Legendre, A.M., 302
Lewin, L., 40
limit A-periodic continued fraction, 81,
85
limit periodic continued fraction, 82
Lindemann, I., 6
Littlewood, J.E., 467, 470
logarithmic integral, 542
Lorentzen, L., 6
lost notebook, 10,36, 37, 39, 41,48, 50
MACSYMA, 515, 517,520-522, 542
MAPLE, 96, 144, 174, 175, 180
Marichev, O.I., 570, 573
Mathematica, 129, 131, 132, 145, 179,
254,255,257, 269,270,301,310,
315,316,335, 336, 343,369,383,

622 Index
Mathematica (com.)
384, 393-395,397,406, 469,532,
541,547,555,560,564,577
Mclntosh, R., 6
McNiel, G., 225
method of successive approximations,
541
Meyer, J., 383,393
Mitrinovic, D.S.. 566
modified hypergeometric function, 549
modified theta-function, 549
modular /-invariant, 309-322
modular equations, 49, 91, 92,94, 185,
243-257, 324, 353-407
of degree n, 185
of degree 3, 121-124, 153-154,
156-157,222,302,354-363
of degree 5, 124-126, 154-155,
157-158,223,303,363-367
of degree 7, 125, 155, 158, 223, 368,
369
of degree 8, 128-133,368
of degree 9, 123-125, 158, 370-372,
387, 394
of degree 11, 125-129, 155-156, 372
of degree 13, 159-160,387
of degree 14, 128-132
of degree 15,385
of degree 16,368
of degree 17,388
of degree 19,386
of degree 20, 128-132
ofdegrec21,380
of degree 25, 160-161,396
ofdegree27,386,394
of degree 29,388
ofdegree31,385
of degree 33, 380, 392
of degree 35,380, 381, 386, 392
of degree 37, 388
of degree 39, 381
of degree 47, 385
Ofdegree49,396,397
ofdegree51,392
ofdegree55,380,392
of degree 59,386
of degree 65, 381
ofdegree71,385
of degree 95, 385
of Weber type, 132
of degree, 120
in the theory of signature 3, 120-133
modulus, 89, 90,183, 323
Mollin,R., 219,222
Montgomery, H.L., 428, 436
Mordell, L.J., 89
Morton, P., 188, 189,276
multiplier, 13,91,92,324
Mdbius function, 409, 467
National Science Foundation, 6
National Security Agency, 6
Nielsen, N.. 542
Nikiforov, A.F.,551,558
Niven, I., 428, 436
nome, 91
norm, 209
norm of an ideal, 217
Olver, F.W.J., 410, 425, 516, 523, 526,
534, 543, 569
orthogonal polynomials, 557, 558
Pentagonal number theorem, 131, 132
perimeter of ellipse, 541
Perron, O., 37, 58, 76, 81-83
Pfaff's transformation, 302, 572
Pohst, M., 225
Poisson distribution, 558
Poisson summation formula, 409, 447,
466,467
Poisson summation formula for Fourier
sine transforms, 449, 576
Preece, СТ., 73, 74, 76, 424
Princeton University, 5
principal genus, 218
Prudnikov, A.P., 570, 573
Purtilo, J.M., 6
^-continued fractions, 45-50
quintuple product identity, 18
Rademacher, H., 548
Rahman, M., 138
Ramanathan, K.G., 10, 20, 22, 44, 46,
48-50, 76, 204, 220, 221, 277,
281-284,339,354,379,386
Ramanujan's cubic continued fraction, 28
Index 623
Ramanujan's cubic transformation, 93,
332
Ramanujan's Quarterly Reports, 562
Rankin, R.A., 5, 128,172, 321, 382, 429.
455,603
Rao, M.B., 424, 435
representations of и as a sum of four
squares, 377
Riemann hypothesis, 470
Riemann zeta-function, 56, 218, 410,
423, 471, 472,476, 477, 503, 534,
540
functional equation, 410
Riordan, J., 525
rising factorial, 411
Rogers, L.J., 14, 17, 20, 31, 45, 70, 545
Rogers-Ramanujan continued fraction, 9,
10, 12-45
Rogers-Ramanujan identities, 31
Roman, S., 525
Root, W, 510
Rotman, J., 259
Russell, R., 3, 254, 275, 313, 321, 353,
354, 385-389
5-fraction, 68
Sansone.G., 421
Schlafli, L., 3, 353, 354. 378, 379
Schlomilch, O., 566
Schoeneberg, В., 131,438
Schur, I., 35
Selberg.A., 10,46,49,50
Selberg-Chowla formula, 324
Shafarevich, I.R., 224, 263
Shen, L.-C, 145
Siegel,C.L., 217, 218,220
signature, 92
singular modulus, 183,186, 277
Sitaramachandrarao, R., 467, 474
Sloan Foundation, 6
Smart, J.R., 5
Soldner, 542
Somashekara, D.D., 37
Stark, H.M., 317
Stieltjes, T.J., 55, 68, 83
Stirling numbers of the first kind, 508,
513
Stirling's formula, 539
Stolz, О..81
Strzebonski, A., 253, 256
Suslov,S.K.,55I,558
Swinnerton-Dyer, P., 136
Szego,G.,551
Szasz, O., 84, 85
Taucher, Т., 549
Tchebychef, P.L.,551
Tennyson, Alfred, Lord, 1
theta functions, 12.92
transformation formulas, 20, 330, 409
values of theta-functions, 323-351
Thiele,T.N.,81
Thiele oscillation, 81, 84-86
Thiruvenkatachar, V.R., 275
Thron, W.J., 12, 34, 52, 68, 70, 72, 81,
85-88,410,490,491
Titchmarsh, E.C., 410, 423, 426, 447,
449,466,470,471,576,581
transfer principle, 108,114
transformation formula, 245, 365
trimidiation, 101-103, 116
Trinity College, Cambridge, 5, 184
triplication, 101-104, 107
Trott, M., 256
United Kingdom, 505
University of Glasgow, 5
University of Illinois, 5,6
Uvarov,V.B.,551,558
Valence formula, 128, 131, 132
Van Vleck, E.B., 82
Vandermonde's theorem, 506, 514
Vaughn, J., 6
Vaughn Foundation, 6
Venkatachaliengar, K., 90. 180, 275
Viskovatoff's algorithm, 68
von Pidoll, M., 85
Wall, H.S., 11,55,68-70, 78
Wang, E.T.H., 532, 566
Watson, G.N., 1,5, 10,20, 29, 31, 46, 67,
68, 71,72,92. 133, 184-187, 189,
257, 275,277,280, 324, 333, 335,
379, 445, 458,577, 595, 596
Watson's Lemma, 69, 425, 516, 522, 534
Weber, H., 3, 132, 183, 184. 188, 189,
209, 211,213,215, 216, 264,270,

624 Index
Weber (com.) Yui,N.,322
273,275,276, 282,321, 353, 354,
391 392
Ё
Whittaker, Ё.Т., 68. 92, 133, 187, 324, Za8ier' DB- 6' 10' 37' 39' <°- 43' 322
445 458 Zassenhaus, H., 225
Williams, K.S., 6, 324 Zhang, L.-C., 1-3,6.10,30, 46, 49, 125,
Wilson, B.M., 1,5, 184 185,208,219,222,337.338,341.
Wright, E.M., 428,434 343,346,351
Zucker, I.J., 324, 327
Young,}., 566 Zuckerman, H.S., 428, 436

From about 1903 to 1914, Ramanujan recorded without proofs over 3000
of his mathematical discoveries in notebooks. This volume is the fifth and
final volume dedicated to proving all of the results in Ramanujan's three
notebooks. Since the second notebook is a revised, enlarged edition of
the first, most of our efforts have concentrated on the second notebook.
In the first three volumes, we examined all of the results in the 21 orga-
organized chapters of the second notebook. In the fourth book, theorems
from the 100 unorganized pages in the second notebook and the short,
33-page third notebook were our focus. In this last volume, we continue
to explore these 133 pages, but we also investigate the claims made by
Ramanujan in the unorganized parts of the first notebook that were not
recorded by him in the second or third notebooks.
Topics examined in this volume include continued fractions, Ramanujan's
theories of elliptic functions to alternative bases, class invariants, singu-
singular moduli, explicit values of theta-functions, modular equations, infinite
series, asymptotic expansions, and approximations.
These five volumes should not be regarded as a closing chapter on
Ramanujan's notebooks. For most of Ramanujan's discoveries, we have
not ascertained his proofs or how Ramanujan was led to his theorems.
Continued efforts should be made to discern Ramanujan's thinking.
Moreover, many of Ramanujan's beautiful theorems will undoubtedly lead
to further research.
I
Bruce C. Bern
Ramanujai
Notebook
PartV
UNIVERSITY OF
311U7 02
ISBN 0-387-94941-0
J#>5