"Ramanujan’s
Notebooks
Part II
Bust of Ratnanujan
by Paul Granlund
Bruce C. Berndt
RLamanujan’s Notebooks
Part II
Springer-Verlag New York Berlin Heidelberg London Paris Tokyo
Bruce C. Berndt Department of Mathematics University of Illinois Urbana, IL 61801 USA The following journals have published earlier versions of chapters in this book: LEnseignement Mathématique 26 (1980), l-65. Journal of the Indian Mathematical Society 46 (1982), 31-16. Bulletin London Mathematical Society 15 (1983), 273-320. Expositiones Marhematicae 2 (1984) 289-347. Journal fur die reine und angewandte Mathematik 361(1985), Rocky Mountain Journal of Mathematics 15 (1985), 235-310. Acta Arithmetica 47 (1986) 123-142.
118-134.
Mathematics
Subject
Classification
(1980):
1 l-03,
1 lP99
Library of Congress Cataloging-in-Publication Data (Revised for volume 2) Ramanujan Aiyangar, Srinivasa, 1887-1920. Ramanujan’s notebooks. Includes bibliographies and index. 1. Berndt. Bruce C., 19391. Mathematics. 11. Title. 510 84-20201 QA3.R33 1985 Printed on acid-free paper.
0 1989 by Springer-Verlag New York Inc. Al1 rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer-Verlag, 175 Fifth Avenue, New York, NY 10010, USA), except for briefexcerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use of general descriptive names, trade names, trademarks, etc. in this publication, even if the former are not especially identifïed, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone. Typeset Printed Printed 987654321 ISBN ISBN O-387-96794-X 3-540-96794-X Springer-Verlag Springer-Verlag New York Berlin Berlin Heidelberg Heidelberg New York by Asco Trade Typesetting Ltd., Hong Kong. and bound by R. R. Donnelley and Sons, Harrisonburg, in the United States of America.
Virginia.
to my mother Helen and the memory of my father Harvey
Dedicated
The relation between Hardy and Ramanujan is unparalleled in scientific history. Each had enormous respect for the abilities of the other. Mrs. Ramanujan told the author in 1984 of her husband’s deep admiration for Hardy. Although Ramanujan returned from England with a terminal illness, he never regretted accepting Hardy’s invitation to visit Cambridge.
Photograph reprinted with permission from Collected Papers of G. H. Hardy, Vol. 1, Oxford University Press, Oxford, 1969.
Preface
During the years 1903-1914, Ramanujan recorded many of his mathematical disco,veries in notebooks without providing proofs. Although many of his results were already in the literature, more were not. Almost a decade after Ramanujan’s death in 1920, G. N. Watson and B. M. Wilson began to edit his notebooks, but never completed the task. A photostat edition, with no editing, was published by the Tata Institute of F’undamental Research in Bombay in 1957. This book is the second of four volumes devoted to the editing of Ramanujan’s notebooks. Part 1, published in 1985, contains an account of Chapters l-9 in the second notebook as well as a description of Ramanujan’s quarterly reports. In this volume, we examine Chapters 10-15 in Ramanujan’s second notebook. If a result is known, we provide references in the literature where proofis may be found; if a result is not known, we attempt to prove it. Except in a few instances when Ramanujan’s intent is not clear, we have been able to establish each result in these six chapters. Chapters 10-15 are among the most interesting chapters in the notebooks. Not only are the results fascinating, but for the most part, Ramanujan’s methods remain a mystery. Much work still needs to be done. We hope readers Will strive to discover Ramanujan’s thoughts and further develop his beautiful ideas.
Urbana, Illinois Bruce C. Berndt
Novernber 1987
Contents
Preface Introduction
CHAPTER 10
ix
Hypergeometric
CHAI’TER 11
Series, 1
7
Hypergeometric
CHAI’TER 12
Series, II
48
Conlinued
CHAI’TER
Fractions
13
103
Integrals and Asymptotic Expansions
CHAI’TER 14
185
Inhite
CHAI’TER
Series
15
240 Forms
Asymptotic Expansions and Modular References Index
300 339 355
Introduction
We ta ke up something--we know it is fmite; but as soon as we begin to analyze it, it leads us beyond our reason, and we never find an end to all its qualities, its possibilities, its powers, its relations. It has become intïnite. Vivekananda In a certain sense, mathematics has been advanced most by those who are distinguished more for intuition than for rigorous methods of proof. Felix Klein For now we see through a glass, darkly; but then face to face: now 1 know in part; but then a,hall 1 know even as also 1 am known. First Corinthians 13 : 12
The quoted passages Vivekananda, Klein, and St. Paul each point to a of certain facet of Ramanujan’s work. First, on June 1-5, 1987, the centenary of Ramanujan’s birth was celebrated at the University of Illinois with a seriesof 28 expository lectures and several contributed papers that traced Ramanujan’s influence to many areas of current research; seethe conference Proceedings edited by Andrews et al. [l]. Thus, Ramanujan’s mathematics continues to generate a vast amount of researchin a variety of areas. Second, in the sequel, we shiallseemany instanceswhere Ramanujan made profound contributions but for which he probably did not have rigorous proofs; for example, seeEntry 10of Chapter 13.Third, although St. Paul’s passage eschatological in nature, is it points to the great need to learn how Ramanujan reasoned and made his discoveries. Perhaps we cari prove Ramanujan’s claims, but we may not know the well from which they sprung. These three aspects of Ramanujan’s work Will frequently be made manifest in the pagesthat follow.
2
Introduction
In this book, we examine Chapters 10-15 in Ramanujan’s second notebook. In many respects, these chapters contain some of Ramanujan’s most fascinating and enigmatic discoveries. Our goal has been to prove each claim made by Ramanujan. With a few possible exceptions where the meaning is obscure, we either give a proof or indicate where in the literature proofs cari be found. We emphasize that many (perhaps most) of our proofs are undoubtedly different from those found by Ramanujan. In particular, we have often employed the theory of functions of a complex variable, a subject with which Ramanujan had no familiarity. In no way should our proofs, or this book, be regarded as delïnitive. In many instances, more transparent proofs, especially those that might give insight into Ramanujan’s reasoning, should be sought. Each of Chapters 10-13 and 15 contains 12 pages, while Chapter 14 encompasses 14 pages in Ramanujan’s second notebook. The number of theorems, corollaries, and examples in each chapter is listed in the following table.
Chapter 10
11
Number of Results 116
103
12 13 14 15 Total
113 92 87 94 605
In the sequel, we have employed Ramanujan’s designations of corollary, example, and SOon, although the appellations may not be optimal. Generally, we have adhered to Ramanujan’s notation SO that the reader following our account with a copy of Ramanujan’s notebooks at hand Will have an easier task. At times, for clarity, we have changed notation, especially in Chapter 14 where we make heavy use of complex function theory. Except for some minor alterations, especially in Chapter 15, we have also preserved Ramanujan’s order of presentation. Many of the theorems communicated by Ramanujan in his famous letters to G. H. Hardy on January 16, 1913 and February 27,1913 may be found in Chapters 10-15. In the table below, we list these results.
Introduction
3
Location p. p. p. p. p. p. p. p. p. p. p. p. p. p.
P.
in Collected V, (2) V, (3) V, (4) V, (5) V, (6) VI, (3) VII, (2)
Papers
Location Chapter Chapter Chapter Chapter Chapter Chapter Chapter Chapter Chapter Chapter Chapter Chapter Chapter Chapter Chapter Chapter Chapter Chapter Chapter Chapter Chapter
in Notebooks 10, Section 7, Example 15 10, Section 7, Example 14 14, Section 13, Corollary (iii) 14, Entry 25(ii) 14, Entry 25(vii) 11, Section 20, Example 2 12, Entry 48, Corollary of Entry 48 13, Entry 6 13, Corollary (ii) of Entry 10 15, Section 2, Example (iv) 12, Section 25, Corollary 1 10, Equation (31.1) 11, Entry 29(i) 12, Entry 27 14, Entry 25(xi) 14, Entry 25(xii) 13, Corollary of Entry 21 13, Example for Corollary of Entry 21 12, Entry 34 10, Entry 29(b) 15, Section 2, Example (ii)
xxvi, xxvi, xxvi, xxvi, xxvi, xxvi, xxvi,
xxvi, VII, (3) xxvii, VII, (7) xxvii, IX, (1) xxviii, (3) xxviii, (10) xxix, (14) 349, v, (7) 349, V, (8)
formula in lïrst letter p. 352, penultimate paragraph of 3 p. 352, last paragraph of 3 p. 353, (16)
p. 350,VI, p. 350,VI, p. 350,IX, p. 35 1, last
(4) (5) (2)
Chapter 15, Section 2, Example (iv) Chapter 12, Corollary to Entry 34
Several of Ramanujan’s of the Indian Mathematical
published papersand problemsposedin the Journal Society have their origins in the notebooks. In most cases,only a small portion of the published paper is actually found in the notebooks. We list below those papers with their genesesin Chapters 10--l& together with the respective locations in the notebooks.
Pape1
On question 330 of Prof. Sanjana Modular equations and approximations to ïl On the product fl ::[1+(&i)l] Some delïnite integrals
Location Chapter Chapter Chapter
in Notebooks 10, Section 13 14, Section 8, Example 13, Section 27 Entries 14, 15, lO(iii), of Entry 19, Entry 21, of Entry 21, Entry 22 Section 6 Entry 22(ii)
Some delïnite integrals connected with Gauss’s sums
Chapter 13, Corollary Corollary Chapter 14, Chapter 14,
4
Introduction
Paper On certain arithmetical functions
Location
in Notebooks
On certain trigonometrical sums and their applications in the theory of numbers Asymptotic formulae in combinatory analysis (with G. H. Hardy) A class of defïnite integrals Question 289 Question 294 Question Question Question Question 296 358 387 769
Chapter 15, Sections 9, 10, 12, 13, and 14 Chapter 14, Entry 13 Chapter 15, Section 2, Example (iv) Chapter Chapter (9, (ii) Chapter Chapter Chapter Chapter Chapter Chapter 13, Sections 23-25 12, Section 4, Examples 12, Section 48 13, Entry 6 13, Section 21, Example 14, Corollary of Entry 14 14, Section 8, Example 13, Entry 11 (iii)
We now provide brief summaries for each of Chapters 10-15. More detailed descriptions may be found at the beginning of each chapter. Of a11the topics examined by Ramanujan in his notebooks, only modular equations received more attention than hypergeometric series. Chapter 10 is the lïrst of two chapters devoted almost entirely to the latter subject. In 1923, Hardy [l], [7, pp. 505-5161 published a brief overview of the corresponding chapter in the lïrst notebook. Ramanujan rediscovered most of the classical formulas in the subject, including those attached to the names of Gauss, Kummer, Dougall, Dixon, and Saalschütz. Ramanujan possessed the uncanny ability for finding the most important examples of theorems, and Chapter 10 contains many elegant examples of infinite series summed in closed form. Ramanujan was the lïrst to discover identities for certain partial sums of hypergeometric series, and these may be found in the latter parts of Chapter 10. Ramanujan continues his study of hypergeometric series in Chapter 11. Two topics dominate the chapter. The lïrst concerns products of hypergeometric series, and most of these results are original with Ramanujan. Second, Ramanujan offers several beautiful asymptotic formulas for hypergeometric functions. By far, the most interesting is Corollary 2 in Section 24. Quadratic transformations of hypergeometric series are also featured in Chapter 11. Chapter 12 is almost entirely devoted to continued fractions and is one of the most fascinating chapters in the notebooks. Ramanujan’s published papers contain only one continued fraction! However, Ramanujan submitted some continued fractions as problems to the Journal of the Indian Mathematical Society, and his letters to Hardy contain some of his most beautiful theorems on continued fractions. Nonetheless, the great majority of the results in Chapter 12 are new. Perhaps the most exquisite theorems are the many
Introduction
5
continued fraction expansions for products and quotients of gamma functions. We have no idea how Ramanujan discovered these formulas. Especially awe inspiring is Entry 40 involving several parameters. Equally astonishing is Chapter 13. In the first 11 sections, one finds several beautiful, deep asymptotic expansions for integrals and series. Entries 7 and 10 are perhaps highlights. Ramanujan left us no clues of how he discovered these fascinating theorems. Are these results prototypes for further yet undiscovered theorems? Although we have given proofs, we do not have a firm understanding of how these wonderful theorems fit with the rest of mathematics. Those readers who are fascinated by elegant series evaluations and identities will take great pleasure in reading Chapter 14. Here, one cari find several series identities that have a symmetry that one often associates with certain applications of the Poisson summation formula, which, however, does not seem to be applicable in most cases here. Several closed form evaluations of series involving hyperbolic functions are given. Some of the results in this chapter cari be established by employing partial fraction decompositions. We have utilized two additional primary tools: contour integration and some: theorems of the author on transformations of Eisenstein series. Since neither of these techniques was in Ramanujan’s arsenal, we do not know how Ramanujan discovered most of the results in Chapter 14. C%apter 15 is the most unorganized of a11 the chapters in the second notebook. The first seven sections are primarily devoted to interesting asymptotic expansions of several series. Entry 8 offers an elegant transformation formula for a modified theta-function. In. the sequel, equation numbers refer to equations in the same chapter, unless another chapter is indicated. Unless otherwise stated, page numbers refer to pages in Ramanujan’s second notebook [ 151 in the pagination of the Tata Institute. Part 1 refers to the author’s account [9] of Chapters 1-9, and Part III refers to his account [l l] of Chapters 16-21. In what follows, the principal value of the logarithm is always denoted by Log. The set of a11(fïnite) complex numbers is denoted by %Z. residue of a The function f at an isolated singularity a Will be denoted by R(a). (The identity off ,will always be clear.) A small portion of this book has been aided by notes left by G. N. Watson and B. M. Wilson in their efforts to edit Ramanujan’s notebooks. We are grateful to the Master and Fellows of Trinity College, Cambridge, for providing a copy of these notes and for permission to use this material in this book. We sincerely appreciate the collaboration of Robert L. Lamphere on Chapter 12 and Ronald J. Evans on Chapters 13 and 15. Because of their efforts, our accounts of these chapters are decidedly better than what we would bave: accomplished without their help. Most of the material in this book appeared in previously published versions of these chapters. We are grateful
6
Introduction
for the cooperation shown by each of the journals publishing our earlier accounts. A table below indicates the bibliographie data for the original publications. (Portions of Chapter 15 were published in two parts.) Chapter 10 11
12 13 14 15 15
Coauthors
Publication J. Indian Math. Soc. 46 (1982), 31-76
Bull. London Math. Soc. 15 (1983), 273-320 Rocky Mt. J. Math. 15 (1985), 235-310 Expos. Math. 2 (1984), 289-347 L’Enseign. Math. 26 (1980), l-65 J. Reine Angew. Math. 361 (1985), 118-134 Acta Arith. 47 (1986), 123-142
R. L. Lamphere, B. M. Wilson R. J. Evans
R. J. Evans R. J. Evans
Although only one author is listed on the caver of this book, several mathematicians have made valuable contributions. We are very grateful to George Andrews,Richard Askey, Henri Cohen, Ronald Evans, Jerry Fields, P. Flajolet, M. L. Glasser, Mourad Ismail, Lisa Jacobsen, Robert Lamphere, David Masser, F. W. J. Olver, R. Sitaramachandrarao, and Don Zagier for the many proofs and suggestions that they have contributed. In particular, Askey, Evans, and Jacobsen have each supplied several proofs and offered many helpful comments, and we are especially indebted to them. Others, not named, have made helpful comments, and we publicly offer them our thanks as well. The author bears the responsibility for a11 errors and would like to be notified of such, whether they be minor or serious. The manuscript was typed by the three best technical typists in ChampaignUrbana-Melody Armstrong, Hilda Britt, and Dee Wrather. We thank them for the superb quality of their typing. Lastly, we express our deep gratitude to James Vaughn and the Vaughn Foundation for the generous funding that they have given the author during summers. This book could not have been completed without the support of the Vaughn Foundation.
CHAPTER
10
Hypergeometric
Series, 1
In 1923, Hardy published a paper [l], [7, pp. 505-5163 providing an overview of the contents of Chapter 12 of the lïrst notebook. This chapter, which corresponds to Chapter 10 of the second notebook, is concerned primarily with hypergeometric series. It should be emphasized that Hardy gave only a brief survey of Chapter 12; this chapter contains many interesting results not mentioned by Hardy, and Chapter 10 of the Se#cond notebook possesses material not found in the first. Quite remarkably, Kamanujan independently discovered a great number of the primary classical theorems in the theory of hypergeometric series. In particular, he rediscovered well-known theorems of Gams, Kummer, Dougall, Dixon, Saalschütz, and ‘Thomae, as well as special cases of Whipple’s transformation. Unfortunately, Ramanujan left us little knowledge as to how he made his beautiful discoveries about hypergeometric series. The lïrst notebook contains a few brief sketches of proofs, but the only sketch in the second notebook is found after E,ntry 8, which is Gauss’s theorem. We shall present this argument of Ramanujan in the sequel. As the reader Will see, this chapter contains a wealth of beautiful evaluations of hypergeometric functions, usually at the argument + 1 or - 1. In this connection, we mention the recent work of R. Wm. Gosper, 1. Gessel, and D. Stanton. By employing “splitting functions” a.nd the computer algebra system MACSYMA, Gosper discovered many new hypergeometric function evaluations. Most of these, in the terminating cases, were ingeniously proved by Gesse1 and Stanton [l]. Two conjectures of Gosper were established by P. W. Karlsson [l]. Many elegant and useful binomial coefficient. sums cari be evaluated, usually quite simply, by employing the theorems of Gauss, Dixon, Saalschütz, Kummer, and others. See the paper by R. Roy [2] for many illustrations.
8 We now offer several remarks (4 about notation. r(a + k) = r(a)-7
10. Hypergeometric Series, 1 As usual, we put
where k is any complex number. The generalized hypergeometric defined by 2, ..., pFq ;:: ;,, . ..) zix
series pFq is
1
(0.1)
where p and q are nonnegative integers and c(r, CI~, . . , c(~ and &, BZ, . . . , B, are complex numbers. If the number of parameters is “small,” we may sometimes use the notation pF&~l, c(*, . . . , clp; Pr, &, . . , &; x) in place of the notation on the left side of (0.1). In this chapter, we are concerned only with the cases w..."
|
You need to upgrade your Flash Player , or try to enable javascript in order see this document properly.
|
|
Comments
