Cuaderno de Ramanuyan, Tesis de Algoritmos Avanzados

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Ramanujan’s Notebooks

Part 1

S. Ramanujan, 1919 (From G. H. Hardy, Ramanujan, Twelue Lectures on Subjects Suggested by His Li&e and Work. Cambridge University Press, 1940.)

Bruce C. Berndt Department of Mathematics University of Illinois Urbana, IL 61801 U.S.A.

AMS Subject Classifications: 10-00, 10-03, OlA60, OlA75, lOAXX, 33-Xx

Library of Congress Cataloging in Publication Data Ramanujan Aiyangar, Srinivasa, 1887-l 920. Ramanujan’s notebooks. Bibliography: p. Includes index.

1. Mathematics-Collected works. 1. Berndt, Bruce C., 1939-.^ II.^ Title. QA3.R33 1985 510 8&

0 1985 by Springer-Verlag New York Inc. Al1 rights reserved. No part of this book may be translated or reproduced in any form without written permission from Springer-Verlag, 175 Fifth Avenue, New York, New York 10010, U.S.A.

Typeset by H. Charlesworth & CO. Ltd., Huddersfield, England. Printed and bound by R. R. Donnelley & Sons, Harrisonburg, Virginia. Printed in the United States of America.

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my wife Helen

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our children Kristin, Sonja, and Brooks

from the Uniuersity Library, Dundee

B. M. Wilson devoted much of his short career to Ramanujan’s work. Along with P. V. Seshu Aiyar and G. H. Hardy, he is one of the editors of Ramanujan’s Collected Papers. In 1929, Wilson and G. N. Watson began the task of editing Ramanujan’s notebooks. Partially due to Wilson% premature death in 1935 at the age of 38, the project was never completed. Wilson was in his second year as Professor of Mathematics at The University of St. Andrews in Dundee when he entered hospital in March, 1935 for routine surgery. A blood infection took his life two weeks later. A short account of Wilson’s life has been written by H. W. Turnbull [Il].

Preface

Ramanujan’s notebooks were compiled approximately in the years 1903-1914, prior to his departure for England. After Ramanujan’s death in 1920, many mathematicians, including G. H. Hardy, strongly urged that Ramanujan’s notebooks be edited and published. In fact, original plans called for the publishing of the notebooks along with Ramanujan’s Collected Papers in 1927, but financial considerations prevented this. In 1929, G. N. Watson and B. M. Wilson began the editing of the notebooks, but thetask was never completed. Finally, in 1957 an unedited photostat edition of Ramanujan’s notebooks was published. This volume is the first of three volumes devoted to the editing of Ramanujan’s notebooks. Many of the results found herein are very well known, but many are new. Some results are rather easy to prove, but others are established only with great difficulty. A glance at the contents indicates a wide diversity of topics examined by Ramanujan. Our goal has been to prove each of Ramanujan’s theorems. However, for results that are known, we generally refer to the literature where proofs may be found. We hope that this volume and succeeding volumes Will further enhance the reputation of Srinivasa Ramanujan, one of the truly great figures in the history of mathematics. In particular, Ramanujan’s notebooks contain new, interesting, and profound theorems that deserve the attention of the math- ematical public.

Urbana, Illinois June, 1984

X Contents

CHAPTER 8 Analogues of the Gamma Function

CHAPTER 9 Infinite Series Identities, Transformations, and Evaluations

Ramanujan’s Quarterly Reports

References

Index

181

232

295

337

353

Introduction

Srinivasa Ramanujan occupies a central but singular position in mathemat- ical history. The pathway to enduring, meaningful, creative mathematical research is by no means the same for any two individuals, but for Ramanujan, his path was strewn with the impediments of poverty, a lack of a university education, the absence of books and journals, working in isolation in his most creative years, and an early death at the age of 32. Few, if any, of his mathematical peers had to encounter SO many formidable obstacles. Ramanujan was born on December 22,1887, in Erode, a town in southern India. As was the custom at that time, he was born in the home of his materna1 grandparents. He grew up in Kumbakonam where his father was an accountant for a cloth merchant. Both Erode and Kumbakonam are in the state of Tamil Nadu with Kumbakonam a distance of 160 miles south- southwest of Madras and 30 miles from the Bay of Bengal. Erode lies 120 miles west of Kumbakonam. At the time of Ramanujan’s .birth, Kumba- konam had a population of approximately 53,000. Not too much is known about Ramanujan’s childhood, although some stories demonstrating his precocity survive. At the age of 12, he borrowed Loney’s Trigonometry [l] from an older student and completely mastered its contents. It should be mentioned that this book contains considerably more mathematics than is suggested by its title. Topics such as the exponential function, logarithm of a complex number, hyperbolic functions, infinite products, and infinite series, especially in regard to the expansions of trigonometric functions, are covered in some detail. But it was Car?s A Synopsis of Elementary Results in Pure Mathematics, now published under a different title [l], that was to have its greatest influence on Ramanujan. He borrowed this book from the local Government College library at the age of 15 and was thoroughly captivated by its contents. Carr was a tutor at

Introduction 3

in his foreword to Hardy’s book [17], that Ramanujan wrote two English mathematicians before he wrote G. H. Hardy. Snow does not reveal their identities, but A. Nandy [Il, p. 1471 claims that they are Baker and Popson. Nandy evidently obtained this information in a conversation with J. E. Littlewood. The first named mathematician is H. F. Baker, who was G. H. Hardy’s predecessor as Cayley Lecturer at Cambridge and a distinguished analyst and geometer. As Rankin [2] has indicated, the second named by Nandy is undoubtedly E. W. Hobson, a famous analyst and Sadlerian Professor of Mathematics at Cambridge. According to Nandy, Ramanujan’s letters were returned to him without comment. The many of us who have received letters from “crackpot” amateur mathematicians claiming to have proved Fermat’s last theorem or other famous conjectures cari certainly empathize with Baker and Hobson in their grievous errors. Ramanujan also wrote M. J. M. Hi11 through C. L. T. Griffith, an engineering professor at the Madras Engineering College who took a great interest in Ramanujan’s welfare. Rankin [l] has pointed out that Hi11 was undoubtedly Griffith’s mathematics instructor at University College, London, and this was obvi- ously why Ramanujan chose to Write Hill. Hi11 was more sympathetic to Ramanujan’s work, but other pressing matters prevented him from giving it a more scrutinized examination. Fortunately, Hill’s reply has been preserved; the full text may be found in a compilation edited by Srinivasan [l]. On January 16, 1913, Ramanujan wrote the famed English mathematician G. H. Hardy and “found a friend in you who views my labours sympatheti- cally” [15, p. xxvii]. Upon initially receiving this letter, Hardy dismissed it. But that evening, he and Littlewood retired to the chess room over the commons room at Trinity College. Before they entered the room, Hardy exclaimed that this Hindu correspondent was either a crank or a genius. After 29 hours, they emerged from the chess room with the verdict-“genius.” Some of the results contained in the letter were false, others were well known, but many were undoubtedly new and true. Hardy [20, p. 91 later concluded, about a few continued fraction formulae in Ramanujan’s first letter, “if they were not true, no one would have had the imagination to invent them. Finally (you must remember that 1 knew nothing whatever about Ramanujan, and had to think of every possibility), the writer must be completely honest, because great mathematicians are commoner than thieves or humbugs of such incredible skill.” Hardy replied without delay and urged Ramanujan to corne to Cambridge in order that his superb mathematical talents might corne to their fullest fruition. Because of strong Brahmin caste convictions and the refusa1 of his mother to grant permission, Ramanujan at first declined Hardy’s invitation. But there was perhaps still another reason why Ramanujan did not wish to sail for England. A letter from an English meteorologist, Sir Gilbert Walker, to the University of Madras helped procure Ramanujan’s first officia recognition; he obtained from the University of Madras a scholarship of 75 rupees per month beginning on May 1, 1913. Thus, finally, Ramanujan

4 Introduction

possessed a bona fide academic position that enabled him to devote a11 of his energy to the pursuit of the prolific mathematical ideas flowing from his creative genius. At the beginning of 1914, the Cambridge mathematician E. H. Neville sailed to India to lecture in the winter term at the University of Madras. One of Neville3 tasks was to convince Ramanujan that he should corne to Cambridge. Probably more important than the persuasions of Neville were the efforts of Sir Francis Spring, Sir Gilbert Walker, and Richard Littlehailes, Professor of Mathematics at Madras. Moreover, Ramanujan’s mother con- sented to her son’s wishes to journey to England. Thus, on March 17, 1914, Ramanujan boarded a ship in Madras and sailed for England. The next three years were happy and productive ones for Ramanujan despite his difficulties in adjusting to the English climate and in obtaining suitable vegetarian food. Hardy and Ramanujan profited immensely from each other’s ideas, and it was probably only with a little exaggeration that Hardy [20, p. 111 proclaimed “he was showing me half a dozen new ones (theorems) almost every day.” But after three years in England, Ramanujan contracted an illness that was to eventually take his life three years later. It was thought by some that Ramanujan was infected with tuberculosis, but as Rankin [l], [2] has pointed out, this diagnosis appears doubtful. Despite a loss of weight and energy, Ramanujan continued his mathematical activity as he attempted to regain his health in at least five sanatoria and nursing homes. The war prevented Ramanujan from returning to India. But finally it was deemed safe to travel, and on February 27, 1919, Ramanujan departed for home. Back in India, Ramanujan devoted his attention to q-series and produced what has been called his “lost notebook.” (See Andrews’ paper [2] for a description of this manuscript.) However, the more favorable climate and diet did not abate Ramanujan’s illness. On April 26, 1920, he passed away after spending his last month in considerable pain. It might be conjectured that Ramanujan regretted his journey to England where he contracted a terminal illness. However, he regarded his stay in England as the greatest experience of his life, and, in no way, did he blame his experience in England for the deterioration of his health. (For example, see Neville article [l, p. 295-J.) Our account of Ramanujan’s life has been brief. Other descriptions may be found in the obituary notices of P. V. Seshu Aiyar [l], R. Ramachandra Rao [l], Hardy [9], [lO], [ll], [21, pp. 702-7201, and P.V. Seshu Aiyar, R. Ramachandra Rao, and Hardy in Ramanujan’s Collected Papers [15]; the lecture of Hardy in his book Ramanujan [20, Chapter 11; the review by Morde11 [l]; an address by Neville [l]; the biographies by Ranganathan [l] and Ram [l]; and the reminiscences in a commemorative volume edited by Bharathi [ 11. When Ramanujan died, he left behind three notebooks, the aforemen- tioned “lost notebook” (in fact, a sheaf of approximately 100 loose pages), and other manuscripts. (See papers of Rankin [l] and K. G. Ramanathan [l] for

6 Introduction

contents. The reproduction is very clearly and faithfully executed. If one side of a page is left blank in the notebooks, it is left blank in the facsimile edition. Ramanujan’s scratch work is also faithfully reproduced. Thus, on one page we find only the fragment, “If I is positive.” The printing was done on heavy, oversized pages with generous margins. Since some pages of the original notebooks are frayed or faded, the photographie reproduction is especially admirable. Except for Chapter 1, which probably dates back to his school days, Ramanujan began to record his results in notebooks in about 1903. He probably continued this practice until 1914 when he left for England. From biographical accounts, it appears that other notebooks of Ramanujan once existed. It seems likely that these notebooks were preliminary versions of the three notebooks which survive. The first of Ramanujan’s notebooks was written in what Hardy called “a peculiar green ink.” The book has 16 chapters containing 134 pages. Following these 16 chapters are approximately 80 pages of heterogeneous unorganized material. At first, Ramanujan wrote on only one side of the page. However, he then began to use the reverse sides for “scratch work” and for recording additional discoveries, starting at the back of the notebook and proceeding forward. Most of the material on the reverse sides has been added to the second notebook in a more organized fashion. The chapters are somewhat organized into topics, but often there is no apparent connection between adjacent sections of material in the same chapter. The second notebook is, as mentioned earlier, a revised, enlarged edition of the first notebook. Twenty-one chapters comprising 252 pages are found in the second notebook. This material is followed by about 100 pages of disorganized results. In contrast to the first notebook, Ramanujan writes on both sides of each page in the second notebook. The third notebook contains 33 pages of mostly unorganized material. We shall now offer some general remarks about the contents of the notebooks. Because the second notebook supersedes the first, unless other- wise stated, a11 comments shah pertain to the second notebook. The papers of Watson [2] and Berndt [3] also give surveys of the contents. If one picks up a copy of the notebooks and casually thumbs through the pages, it becomes immediately clear that infinite series abound throughout the notebooks. If Ramanujan had any peers in the forma1 manipulation of infinite series, they were only Euler and Jacobi. Many of the series do not converge, but usually such series are asymptotic series. On only very rare occasions does Ramanujan state conditions for convergence or even indicate that a series converges or diverges. In some instances, Ramanujan indicates that a series (usually asymptotic) diverges by appending the words “nearly” or “very nearly.” It is doubtful that Ramanujan possessed a sound grasp of what an asymptotic series is. Perhaps he had never heard of the term “asymptotic.” In fact, it was not too many years earlier that the foundations of asymptotic series were laid by Poincaré and Stieltjes. But despite this possible shortcoming, some of Ramanujan’s deepest and most interesting

Introduction 7

results are asymptotic expansions. Although Ramanujan rarely indicated that a series converged or diverged, it is undoubtedly true that Ramanujan generally knew when a series converged and when it did not. In Chapter 6 Ramanujan developed a theory of divergent series based upon the Euler-Maclaurin summation formula. It should be pointed out that Raman- ujan appeared to have little interest in other methods of summability, with a couple of examples in Chapter 6 being the only evidence of such interest. Besides basing his theory of divergent series on the Euler-Maclaurin formula, Ramanujan employed the Euler-Maclaurin formula in a variety of ways. See Chapters 7 and 8, in particular. The Euler-Maclaurin formula was truly one of Ramanujan’s favorite tools. Not surprisingly then, Bernoulli numbers appear in several of Ramanujan’s formulas. His love and affinity for Bernoulli numbers is corroborated by the fact that he chose this subject for his first published paper [4]. Although series appear with much greater frequency, integrals and continued fractions are plentiful in the notebooks. There are only a few continued fractions in the first nine chapters, but later chapters contain numerous continued fractions. Although Ramanujan is known primarily as a number theorist, the notebooks contain very little number theory. Ramanujan’s contributions to number theory in the notebooks are found chiefly in Chapter 5, in the heterogeneous material at the end of the second notebook, and in the third notebook. The notebooks were originally intended primarily for Ramanujan’s own persona1 use and not for publication. Inevitably then, they contain llaws and omissions. Thus, notation is sometimes not explained and must be deduced from the context, if possible. Theorems and formulas rarely have hypotheses attached to them, and only by constructing a proof are these hypotheses discernable in many cases. Some of Ramanujan’s incorrect “theorems” in number theory found in his letters to Hardy have been well publicized. Thus, perhaps some think that Ramanujan was prone to making errors. However, such thinking is erroneous. The notebooks contain scattered minor errors and misprints, but there are very few serious errors. Especially if one takes into account the roughly hewn nature of the material and his frequently forma1 arguments, Ramanujan’s accuracy is amazing. On the surface, several theorems in the notebooks appear to be incorrect. However, if proper interpretations are given to them, the proposed theorems generally are correct. Especially in Chapters 6 and 8, formulas need to be properly reinterpreted. We cite one example. Ramanujan offers several theorems about 1 1/x, where x is any positive real number. First, we must be aware that, in Ramanujan’s notation, 1 1/x = xnsx l/n. But further reinter- pretation is still needed, because Ramanujan frequently intends c 1/x to mean $(x + 1) + y, where $(x) = r’(x)/r(x) and y denotes Euler’s constant. Recall that if x is a positive integer, then $(x + 1) + y = xi= i l/n. But in other instances, 1 1/x may denote Log x + y. Recall that as x tends to CO, both $(x + 1) +Y and Cnsx l/n are asymptotic to Log x + y. The notebooks contain very few proofs, and those proofs that are given are

Introduction 9

many of these theorems by working with divergent series. However, Ramanujan’s theorems cari be proved rigorously by manipulating the series where they converge and then using analytic continuation. Thus, just one concept outside of Ramanujan’s repertoire is needed to provide rigorous proofs for these beautiful theorems analogizing properties of the gamma function. TO be sure, there are undoubtedly some instances when Ramanujan did not have a proof of any type. For example, it is well known that Ramanujan discovered the now famous Rogers-Ramanujan identities in India but could not supply a proof until several years later after he found them in a paper of L. J. Rogers, As Littlewood [l], [2, p. 16041 wrote, “If a significant piece of reasoning occurred somewhere, and the total mixture of evidence and intuition gave him certainty, he looked no further.” In the sequel, we shall indicate Ramanujan’s proofs when we have been able to ascertain them from sketches provided by him or from the context in which the theorems appear. We emphasize, however, that for most of his work, we have no idea how Ramanujan made his discoveries. In an interview conducted by P. Nandy [l] in 1982 with Ramanujan’s widow S. Janaki, she remarked that her husband was always fearful that English mathematicians would steal his mathematical secrets while he was in England. It seems that not only did English mathematicians not steal his secrets, but generations of mathematicians since then have not discovered his secrets either. Hardy [20, p. 101 estimated that two-thirds of Ramanujan’s work in India consisted of rediscoveries. For the notebooks, this estimate appears to be too high. However, it would be difficult to precisely appraise the percentage of new results in the notebooks. It should also be remarked that some original results in the notebooks have since been rediscovered by others, usually without knowledge that their theorems are found in the notebooks. Chapter 1 has but 8 pages, while Chapters 2-9 contain either 12 or 14 pages per chapter. The number of theorems, corollaries, and examples in each chapter is listed in the following table.

Chapter Number of Results

1 43 2 68 3 86 4 50 5 94 6 61 1 110 8 108 9 139

Total 159

(^10) Introduction

In this book, we shall either prove each of these 759 results, or we shall provide references to the literature where proofs may be found. In a few instances, we were unable to interpret the intent of the entries. In the sequel, we have adhered to Ramanujan’s usage of such terms as “corollary” and “example.” However, often these designations are incorrect. For example, Ramanujan’s “corollary” may be a generalization of the preceding result. An “example” may be a theorem. SO that the reader with a copy of the photostat edition of the notebooks cari more easily follow our analysis, we have preserved Ramanujan’s notation as much as possible. However, in some instances, we have felt that a change in notation is advisable. Not surprisingly, several of the theorems that Ramanujan communicated in his two letters of January 16, 1913, and February 27, 1913, to Hardy are found in his notebooks. Altogether about 120 results were mailed to Hardy. Unfortunately, one page of the first letter was lost, but a11 of the remaining theorems have been printed with Ramanujan’s collected papers [15]. We have recorded below those results from the letters that are also found in Chapters l-9 of the second notebook or the Quarterly Reports. Considerably more theorems in Ramanujan’s letters were extracted from later chapters in the notebooks.

Location in Collected Papers

p. xxiv, (2), parts (b), (c) p. xxv, IV, (4) p. xxvi, VI, (1) p. 350, VII, (1) p. 351, lines 1, 3

Location in Notebooks or Reports

Chapter 5, Section 30, Corollary 2 First report, Example (d) Chapter 7, Section 18, Corollary Chapter 9, Section 27 Chapter 6, Section 1, Example 2

Many of Ramanujan’s papers have their geneses in the notebooks. In a cases, only a portion of the results from each paper are actually found in the notebooks. Also some of the problems that Ramanujan submitted to the Journal of the Indian Mathematical Society are ensconced in the notebooks. We list below those papers and problems with connections to Chapters l- or the Quarterly Reports. Complete bibliographie details are found in the list of references. A condensed summary of Chapters l-9 Will now be provided. More complete descriptions are given at the beginning of each chapter. Because each chapter contains several diverse topics, the chapter titles are only partially indicative of the chapters’ contents. Magie squares cari be traced back to the twelfth or thirteenth Century in India and have long been popular amongst Indian school boys. In contrast to the remainder of the notebooks, the opening chapter on magie squares evidently arises from Ramanujan’s early school days. Chapter 1 is quite elementary and contains no new insights on magie squares.