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During the years 1903-1914, Ramanujan recorded many of his mathematical discoveries 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 Fundamental Research in Bombay in 1957. This book is the second of four volumes devoted to the editing of Ramanujan's Notebooks. Part I, published in 1985, contains an account of Chapters 1-9 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 proofs may be found; if a result is not known, we attempt to prove it. 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.

Inhaltsverzeichnis

Frontmatter

Introduction

Abstract
The quoted passages of Vivekananda, Klein, and St. Paul each point to a 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 series of 28 expository lectures and several contributed papers that traced Ramanujan’s influence to many areas of current research; see the conference Proceedings edited by Andrews et al. [1]. Thus, Ramanujan’s mathematics continues to generate a vast amount of research in a variety of areas. Second, in the sequel, we shall see many instances where Ramanujan made profound contributions but for which he probably did not have rigorous proofs; for example, see Entry 10 of Chapter 13. Third, although St. Paul’s passage is eschatological in nature, it points to the great need to learn how Ramanujan reasoned and made his discoveries. Perhaps we can 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 pages that follow.
Bruce C. Berndt

Chapter 10. Hypergeometric Series, I

Abstract
In 1923, Hardy published a paper [1],[7, pp. 505–516] providing an overview of the contents of Chapter 12 of the first notebook. This chapter which corresponds to Chapter 10 of the second notebook, is connected 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 second notebook possesses material not found in the first. Quite remarkably, Ramanujan independently discovered a great number of the primary classical theorems in the theory of hypergeometric series. In Particular, he rediscovered well-known theorems of Gauss, 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 first notebook is found after Entry 8, which is Gauss’s theorem. We shall present this argument of Ramanujan in the sequel.
Bruce C. Berndt

Chapter 11. Hypergeometric Series, II

Abstract
Much of Chapter 11 is contained in Chapters 13 and 15 of the first notebook, while some formulas from Chapter 11 may be found scattered among the “working pages” of the first notebook.
Bruce C. Berndt

Chapter 12. Continued Fractions

Abstract
In assessing the content of Ramanujan’s first letter, dated January 16, 1913, to him, Hardy [9, p. 9] remarked: “but (1.10)–(1.12) defeated me completely; I had never seen anything in the least like them before. A single look at them is enough to show that they could only be written down by a mathematician of the highest class. They must be true because, if they were not true, no one would have had the imagination to invent them.” These comments were directed at three continued fraction representations. Indeed, Ramanujan’s contributions to the continued fraction expansions of analytic functions are one of his most spectacular achievements. The three formulas that challenged Hardy’s acumen are not found in Chapter 12, but this chapter, which is almost entirely devoted to the study of continued fractions, contains many other beautiful and penetrating formulas. Unfortunately, Ramanujan left us no clues as to how he discovered these elegant continued fraction formulas. Especially enigmatic are the several representations for products and quotients of gamma functions. Three of the principal formulas involving gamma functions are Entries 34, 39, and 40. Entries 20 and 22, giving Gauss’s and Euler’s continued fractions, respectively, for a quotient of two hypergeometric functions, also play prominent roles. Several other formulas are dependent on these five entries, and it may be helpful to schematically indicate these connections among entries.
Bruce C. Berndt

Chapter 13. Integrals and Asymptotic Expansions

Abstract
In assessing the content of Ramanujan’s first letter to him, Hardy [9, p. 9] judged that “on the whole, the integral formulae seemed the least imprsessive.” Later he added that Ramanujan’s definite integral formulae “are still interesting and will repay a careful analysis” [9, p. 186]. Indeed, a dismissal of Ramanujan’s contributions to intergration would have beed decidedly premature. First, we might recall that this first letter contained several remarkable formulas on series and continued fractions. In evaluating infinite series and deriving series identities, Ramanujan had no peers, except for possibly Euler and Jacobi. Ramanujan’s work on continued fraction expansions of analytic functions ranks as one of his most brilliant achievements. Thus, if Ramanujan’s contributions to integrals dim slightly in comparison, it is only because the glitter of diamonds surpasses that of rubies. Indeed, there are many elegant and important integrals that bear Ramanujan’s name (See for example, Entry 22.)
Bruce C. Berndt

Chapter 14. Infinite Series

Abstract
Since Ramanujan’s death in 1920, there have perhaps been more published papers establishing results in Chapter 14 than in any of the remaining 20 chapters. In many cases, the authors were unaware that their discoveries are found in Ramanujan’s notebooks. In [6] and [7], the author showed that several results in Chapter 14, as well as many others as well, aries from a general transformation formula for a large class of analytic Eisenstein series. It should be emphasized, however, that Chapter 14 also contains many other types of results.
Bruce C. Berndt

Chapter 15. Asymptotic Expansions and Modular Forms

Abstract
The title of Chapter 15 does not entirely reflect its contents, because this chapter contains several diverse topics. Of the 21 chapters in the second notebook, Chapter 15 contains more disparate topics than the remaining chapters. Ramanujan appears to have collected here several “odds and ends.” While much of the materials is fascinating, a few part have little substance.
Bruce C. Berndt

Backmatter

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