Ramanujan s Lost Notebook. Part II

Transcription

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2 Ramanujan s Lost Notebook Part II

3 S. Ramanujan

4 George E. Andrews Bruce C. Berndt Ramanujan s Lost Notebook Part II 3

5 George E. Andrews Department of Mathematics Pennsylvania State University University Park, PA 680 USA Bruce C. Berndt Department of Mathematics University of Illinois at Urbana-Champaign Urbana, IL 680 USA ISBN e-isbn DOI 0.007/ Library of Congress Control Number: Mathematics Subject Classification 000: 33-0, 33D5, P8, P8, F, 05A7, 05A30 c Springer Science+Business Media, LLC 009 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher Springer Science+Business Media, LLC, 33 Spring Street, New York, NY 003, USA, except for brief excerpts 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 in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper springer.com

6 By the kindness of heaven, O lovely faced one, You stand before me, The darkness of delusion dispelled, By recollection of that which was lost. Verse 7. of Kalidasa s Sakuntala, 4th century A.D.

7 Preface This is the second of approximately four volumes that the authors plan to write in their examination of all the claims made by S. Ramanujan in The Lost Notebook and Other Unpublished Papers. This volume, published by Narosa in 988, contains the Lost Notebook, which was discovered by the first author in the spring of 976 at the library of Trinity College, Cambridge. Also included in this publication are other partial manuscripts, fragments, and letters that Ramanujan wrote to G.H. Hardy from nursing homes during The authors have attempted to organize this disparate material in chapters. This second volume contains 6 chapters comprising 34 entries, including some duplications and examples, with chapter totals ranging from a high of fifty-four entries in Chapter to a low of two entries in Chapter.

8 Contents Preface... vii Introduction... The Heine Transformation Introduction Heine smethod Ramanujan s Proof of the q-gauss Summation Theorem Corollaries of.. and Corollaries of..6 and Corollaries of..8,..9, and Corollaries of Section. and Auxiliary Results The Sears Thomae Transformation Introduction Direct Corollaries of.. and Extended Corollaries of.. and Bilateral Series Introduction Background The ψ Identity The ψ Identities Identities Arising from the Quintuple Product Identity Miscellaneous Bilateral Identities Well-Poised Series Introduction Applications of Applications of Bailey s Formulas

9 x Contents 5 Bailey s Lemma and Theta Expansions Introduction TheMainLemma Corollaries of Corollaries of 5..4 and Related Results Partial Theta Functions Introduction A General Identity Consequences of Theorem The function ψa, q Euler s Identity and Its Extensions TheWarnaarTheory Special Identities Introduction Generalized Modular Relations Extending Abel s Lemma InnocentsAbroad Theta Function Identities Introduction Cubic Identities Septic Identities Ramanujan s Cubic Class Invariant Introduction λ n and the Modular j-invariant λ n and the Class Invariant G n λ n and Modular Equations λ n and Modular Equations in the Theory of Signature λ n and Kronecker s Limit Formula The Remaining Five Values Some Modular Functions of Level Computations of λ n Using the Shimura Reciprocity Law Miscellaneous Results on Elliptic Functions and Theta Functions A Quasi-theta Product An Equivalent Formulation of 0.. in Terms of Hyperbolic Series Further Remarks on Ramanujan s Quasi-theta Product A Generalization of the Dedekind Eta Function Two Entries on Page A Continued Fraction Class Invariants

10 Contents xi Formulas for the Power Series Coefficients of Certain Quotients of Eisenstein Series Introduction The Key Theorem The Coefficients of /Qq The Coefficients of Qq/Rq The Coefficients of πpq/3/rq and πpq/3 /Rq The Coefficients of πpq/ 3/Qq Eight Identities for Eisenstein Series and Theta Functions The Coefficients of /Bq Formulas for the Coefficients of Further Eisenstein Series The Coefficients of /B q A Calculation from [76] Letters from Matlock House Introduction A Lower Bound An Upper Bound Eisenstein Series and Modular Equations Introduction Preliminary Results Quintic Identities: First Method Quintic Identities: Second Method Septic Identities Septic Differential Equations Series Representable in Terms of Eisenstein Series Introduction The Series T k q The Series U n q Eisenstein Series and Approximations to π Introduction Eisenstein Series and the Modular j-invariant Eisenstein Series and Equations in π:firstmethod Eisenstein Series and Equations in π: Second Method Page Ramanujan s Series for /π Miscellaneous Results on Eisenstein Series A generalization of Eisenstein Series Representations of Eisenstein Series in Terms of Elliptic Function Parameters Values of Certain Eisenstein Series Some Elementary Identities

11 xii Contents Location Guide...39 Provenance References...40 Index...45

12 Introduction This volume is the second of approximately four volumes that the authors plan to write on Ramanujan s lost notebook. We broadly interpret lost notebook to include all material published with Ramanujan s original lost notebook by Narosa in 988 [44]. Thus, when we write that a certain entry is found in the lost notebook, it may not actually be located in the original lost notebook discovered by the first author in the spring of 976 at Trinity College Library, Cambridge, but instead may be in a manuscript, fragment, or a letter of Ramanujan to G.H. Hardy published in [44]. We are attempting to arrange all this disparate material into chapters for each of the proposed volumes. For a history and general description of Ramanujan s lost notebook, readers are advised to read the introduction to our first book [3]. The Organization of Entries With the statement of each entry from Ramanujan s lost notebook, we provide the page numbers in the lost notebook on which the entry can be found. All of Ramanujan s claims are given the designation Entry. Results in this volume named theorems, corollaries, and lemmas are unless otherwise stated not due to Ramanujan. We emphasize that Ramanujan s claims always have page numbers from the lost notebook attached to them. We remark that in Chapter 9, which is devoted to establishing Ramanujan s values for an analogue λ n of the classical Ramanujan Weber class invariant G n,we have followed a slightly different convention. Indeed, we have listed all of Ramanujan s values for λ n in Entry 9.. with the page number indicated. Later, we establish these values as corollaries of theorems that we prove, and so we record Ramanujan s values of λ n again, listing them as corollaries with page numbers in the lost notebook attached to emphasize that these corollaries are due to Ramanujan. In view of the subject mentioned in the preceding paragraph, it may be prudent to make a remark here about Ramanujan s methods. As many read- G.E. Andrews, B.C. Berndt, Ramanujan s Lost Notebook: Part II, DOI 0.007/ , c Springer Science+Business Media, LLC 009

13 Introduction ers are aware from the work of the authors and others who have attempted to prove Ramanujan s theorems, we frequently have few or no clues about Ramanujan s methods. Many of the proofs of the values for G n that are given in [57] are almost certainly not those found by Ramanujan, for he would have needed knowledge of certain portions of mathematics that he likely did not know or that had not been discovered yet. Similar remarks can be made about our calculations of λ n in Chapter 9. In the last half of the chapter, we employ ideas that Ramanujan would not have known. So that readers can more readily find where a certain entry from the lost notebook is discussed, we place at the conclusion of each volume a Location Guide indicating where entries can be found in that particular volume. Thus, for example, if a reader wants to know whether a certain identity on page 79 of the Narosa edition [44] can be found in a particular volume, she can turn to this index and determine where in that volume identities on page 79 are discussed. Following the Location Guide, we provide a Provenance indicating the sources from which we have drawn in preparing significant portions of the given chapters. We emphasize that in the Provenance we do not list all papers in which results from a given chapter are established. For example, in Chapter 3, Ramanujan s famous ψ summation theorem, which is found in more than one version in the lost notebook, is discussed, but we do not refer to all papers on the ψ summation formula in the Location Guide, although in Chapter 3 itself, we have attempted to cite all relevant proofs of this celebrated formula. On the other hand, most chapters contain previously unpublished material. For example, each of the first four chapters contains previously unpublished proofs. This Volume on the Lost Notebook Two primary themes permeate our second volume on the lost notebook, namely, q-series and Eisenstein series. The first seven chapters are devoted to q-series identities from the core of the original lost notebook. These chapters are followed by three chapters on identities for the classical theta functions or related functions. The last six chapters feature Eisenstein series, with much of the material originating in letters to Hardy that Ramanujan wrote from Fitzroy House and Matlock House during his last two years in England. We now briefly describe the contents of the sixteen chapters in this volume. Heine s transformations have long been central to the theory of basic hypergeometric series. In Chapter, we examine several entries from the lost notebook that have their roots in Heine s first transformation or generalizations thereof. The Sears Thomae transformation is also a staple in the theory of basic hypergeometric series, and consequences of it form the content of Chapter. In Chapter 3, we consider identities arising from certain bilateral series identities, in particular the renowned ψ summation of Ramanujan and

14 Introduction 3 well-known identities due to W.N. Bailey. We have also placed in Chapter 3 some identities dependent upon the quintuple product identity. Watson s q- analogue of Whipple s theorem and two additional theorems of Bailey are the main ingredients for the proofs in Chapter 4 on well-poised series. Bailey s lemma is utilized to prove some identities in Chapter 5. Chapter 6, on partial theta functions, is one of the more difficult chapters in this volume. Chapter 7 contains entries from the lost notebook that are even more difficult to prove than those in Chapter 6. The entries in this chapter do not fall into any particular categories and bear further study, because several of them likely have yet-to-be discovered ramifications. Theta functions frequently appear in identities in the first seven chapters. However, in Chapters 8 0, theta functions are the focus. Chapter 8 is devoted to theta function identities. Chapter 9 focuses on one page in the lost notebook on values of an analogue of the classical Ramanujan Weber class invariants. The identities in Chapter 0 do not fit in any of the previous chapters and are among the most unusual identities we have seen in Ramanujan s work. As remarked above, the last six chapters in this volume feature Eisenstein series. Perhaps the most important chapter is Chapter, which contains proofs of results sent to Hardy from nursing homes, probably in 98. In these letters, Ramanujan offered formulas for the coefficients of certain quotients of Eisenstein series that are analogous to the Hardy Ramanujan Rademacher series representation for the partition function pn. The claims in these letters continue the work found in Hardy and Ramanujan s last joint paper [77], [4, pp. 30 3]. Chapter relates technical material on the number of terms that one needs to take from the aforementioned series in order to determine these coefficients precisely. In Chapter 3, the focus shifts to identities for Eisenstein series involving the Dedekind eta function. Chapter 4 gives formulas for certain series associated with the pentagonal number theorem in terms of Ramanujan s Eisenstein series P, Q, and R. These results are found on two pages of the lost notebook, and, although not deep, have recently generated several further papers. Chapter 5 is devoted primarily to a single page in the lost notebook demonstrating how Ramanujan employed Eisenstein series to approximate π. Three series for /π found in Ramanujan s epic paper [39], [4, pp. 3 39] are also found on page 370 of [44], and so it seems appropriate to prove them in this chapter, especially since, perhaps more so than other authors, we follow Ramanujan s hint in [39] and use Eisenstein series to establish these series representations for /π. This volume concludes with a few miscellaneous results on Eisenstein series in Chapter 6. Acknowledgments The second author is grateful to several of his former and current graduate students for their contributions to this volume, either solely or in collaboration with him. These include Paul Bialek, Heng Huat Chan, Song Heng Chan,

15 4 Introduction Sen-Shan Huang, Soon-Yi Kang, Byungchan Kim, Wen-Chin Liaw, Jaebum Sohn, Seung Hwan Son, Boon Pin Yeap, Hamza Yesilyurt, and Liang-Cheng Zhang. He also thanks Ae Ja Yee for her many collaborations on Eisenstein series during her three postdoctoral years at the University of Illinois, as well as Nayandeep Deka Baruah during his one postdoctoral year at the University of Illinois. A survey on Ramanujan s work on Eisenstein series, focusing especially on the claims in the lost notebook, has been written by the second author and Yee [76]. Heng Huat Chan was an indispensable coauthor of the papers on which Chapters 9 and 5 are based, and moreover, the last two sections of Chapter 9 were written by him. Consequently, he deserves special appreciation. The second author s colleague Alexandru Zaharescu has been a source of fruitful collaboration and advice. We are greatly indebted to Nayandeep Deka Baruah, Youn-Seo Choi, Wenchang Chu, Tim Huber, Andrew Sills, and Jaebum Sohn for carefully reading several chapters, uncovering misprints and more serious errors, and offering many useful suggestions. Others whom we thank for finding misprints and offering valuable suggestions include S. Bhargava, Soon-Yi Kang, Byungchan Kim, S. Ole Warnaar, and Ae Ja Yee. We are especially grateful to Michael Somos for his computer checking of many identities and for uncovering several mistakes. We also thank Tim Huber for his graphical expertise in Chapter and Han Duong for composing the index and uncovering a few misprints in the process. We thank Springer editor Mark Spencer for his patient guidance, and Springer copy editor David Kramer for a helpful and careful reading of our manuscript. The first author thanks the National Science Foundataion, and the second author thanks the National Security Agency and the University of Illinois Research Board for their financial support.

16 The Heine Transformation. Introduction E. Heine [78], [79, pp. 97 5] was the first to generalize Gauss s hypergeometric series to q-hypergeometric series by defining, for q <, φ a, b c ; q, t : where t < and where, for each nonnegative integer n, a; q n b; q n q; q n c; q n t n,.. a n a; q n : a aq aq n,.. with the convention that a 0 a; q 0 :. If an entry and its proof involve only the base q and no confusion would arise, we use the notation at the left in.. and..4 below. If more than one base occurs in an entry and/or its proof, e.g., both q and q appear, then we use the second notation in.. and..4. Ramanujan s central theorem is a transformation for this series, now known as the Heine transformation, namely [79, p. 06, equation 50], φ a, b c ; q, t b; q at; q c; q t; q φ where t, b < and where c/b, t at ; q, b,..3 a a; q lim n a; q n, q <...4 His method of proof was surely known to Ramanujan, who recorded an equivalent formulation of..3 in Entry 6 of Chapter 6 in his second notebook [43], [54, p. 5]. Furthermore, numerous related identities can be proved using Heine s original idea. In Section., we prove several basic formulas based on Heine s method. In the remainder of the chapter we deduce 53 formulas found in the lost G.E. Andrews, B.C. Berndt, Ramanujan s Lost Notebook: Part II, DOI 0.007/ , c Springer Science+Business Media, LLC 009

17 6 The Heine Transformation notebook. In some instances, we call upon a result not listed in Section., but each identity that we prove relies primarily on results in Section.. In order to keep our proofs to manageable lengths, we invoke certain standard simplifications usually without mentioning them explicitly, such as q; q q; q,..5 a; q n a; q n a ; q n, 0 n<,..6 a; q n a; q aq n, ; q <n<...7 The identity..5 is a famous theorem of Euler, which we invoke numerous times in this book. Identity..7 can be regarded as the definition of a; q n when n is a negative integer.. Heine s Method In [6], Heine s method was encapsulated in a fundamental formula containing ten independent variables and a nontrivial root of unity. As a result, it is an almost unreadable formula. Consequently, we prove only special cases of this result here. In light of the fact that many of these results are not easily written in the notation.. of q-hypergeometric series, we record all our results in terms of infinite series. For further work connected with that of Andrews in [6], see Z. Cao s thesis [97] and a paper by W. Chu and W. Zhang [3]. We begin with a slightly generalized version of Heine s transformation [6], [7]. Theorem... If h is a positive integer, then, for t, b <, a; q h m b; q hm q h ; q h t m b; q at; q h c/b; q m t; q h m m0 m c; q hm c; q t; q h q; q m0 m at; q h b m. m.. Proof. We need the q-binomial theorem given by [54, p. 4, Entry ], [8, p. 7, Theorem.] a/b; q m b m a; q,.. q; q m b; q m0 where b <. Since we frequently need two special cases in the sequel, we state them here. If a 0 in.., then [8, p. 9, equation..5] m0 b m...3 q; q m b; q Letting b 0 in.., we find that [8, p. 9, equation..6]

18 . Heine s Method 7 a m q mm / a; q...4 q; q m m0 Upon two applications of.., we see that a; q h n b; q hn q h ; q h n c; q hn t n b; q c; q which is... b; q c; q b; q c; q b; q c; q m0 a; q h n q h ; q h n cq hn ; q bq hn ; q t n a; q h n q h ; q h n t n c/b; q m q; q m c/b; q m q; q m m0 b; q at; q h c; q t; q h m0 b m m0 c/b; q m b m q hmn q; q m a; q h n q h ; q h n tq hm n b m atqhm ; q h tq hm ; q h c/b; q m t; q h m q; q m at; q h m b m, Heine s transformation is the case h of Theorem.., and Theorem A 3 of [6] is the case h. The complete result appears in [7, Lemma ]. The next result is more intricate, but it is based again on Heine s idea; it is Theorem A of [6]. Theorem... For t, b <, Proof. a; q n b; q n q ; q n c; q n t n b; q at; q c; q t; q + b; q atq; q c; q tq; q Using.. twice, we find that a; q n b; q n q ; q t n b; q n c; q n c; q b; q a; q n c; q q ; q t n n b; q c; q + m0 a; q n q ; q n t n c/b; q n t; q n q; q n at; q b n..5 n c/b; q n+ tq; q n q; q n+ atq; q b n+. n c/b; q m+ b m+ q m+n q; q m+ a; q n q ; q n cq n ; q bq n ; q t n c/b; q m b m q mn q; q m0 m { c/b; q m b m q mn q; q m0 m }

19 8 The Heine Transformation b; q c; q m0 c/b; q m q; q m b m a; q n q ; q n tq m n + b; q c/b; q m+ b m+ a; q n c; q q; q m0 m+ q ; q tq m+ n n b; q c/b; q m b m atqm ; q c; q q; q m0 m tq m ; q + b; q c/b; q m+ b m+ atqm+ ; q c; q q; q m0 m+ tq m+ ; q b; q at; q c/b; q m t; q m c; q t; q q; q m at; q b m m m0 + b; q atq; q c; q tq; q m0 c/b; q m+ tq; q m q; q m+ atq; q m b m+. In addition to Theorems.. and.., we require two corollaries of Theorem... The first is also given in [7, equation I5]. Corollary... For t <, b; q n q ; q t n tb; q b; q n t n...6 n t; q q; q n tb; q n Proof. By.. with h,a c 0,andt replaced by t, we see that b; q n q ; q n t n b; q t ; q b; q t ; q t ; q n b n q; q n t; q n t; q n q; q n b; q t ; q t; q tb; q b; q b n b; q n q; q n tb; q n t n, by.. with t b and then h,a t, b t, andc 0.Upon simplification above, we deduce..6. The next result can be found in [7, equation I6]. Corollary... For b <, t; q n b n btq; q t; q n q; q n bq; q q ; q n btq; q b n...7 n

20 . Heine s Method 9 Proof. By.. with h anda c 0, we see that b; q t; q t; q n q; q n b n b; q n q ; q n t n bq; q n b; q n q ; q n t n b; q btq; q t; q t; q n q ; q n btq; q n b n, where we applied.. with q replaced by q, h,a bq, andc 0. Upon simplification, we complete the proof. Our next result comes from [9, Theorem 7]. Corollary..3. For t <, a; q n b; q n q; q n abt; q n t n at; q bt; q t; q abt; q a; q n b; q n q ; q n bt; q n tq n...8 Proof. In.., set h, interchange t with b, replace a by at, and then replace c by at. Upon simplification, we find that a; q n b; q n q; q n abt; q t n at; q b; q n t; q abt; q at; q b; q t; q abt; q t; q n tq; q n q ; q n atq; q n b n at; q b; q tq; q bt; q t; q abt; q atq; q b; q at; q n t; q n q ; q n at; q n b n a; q n b; q n q ; q n bt; q n tq n, where we invoked.. with h,q replaced by q, and the variables a, b, c, andt replaced by t, tq, atq, andb, respectively. Upon simplifying above, we deduce..8 to complete the proof. We also require the direct iteration of.. with h [9, Theorem 8]. This is often called the second Heine transformation. Corollary..4. For t, c/b <, a n b n t n c/b bt q n c n c t abt/c n b n q n bt n c n...9 b

21 0 The Heine Transformation Proof. By two applications of Theorem.. with h, the second with a, b, c, andt replaced by t, c/b, at, andb, respectively, we find that a n b n q n c n t n b at c t b at c t c/b n t n q n at n b n c/b bt at b abt/c n b n q n bt n c b n, which is the desired result. Finally, we need one more iteration of.. with h [8, p. 39, equation 3.3.3]. This is often called the q-analogue of Euler s transformation. Corollary..5. For t, abt/c <, a n b n q n c n t n abt/c t c/a n c/b n q n c n n abt...0 c Proof. Apply.. with h anda, b, c, andt replaced by b, abt/c, bt, and c/b, respectively. Consequently, abt/c n b n q n bt n c b n abt/c c bt c/b c/a n c/b n q n c n n abt... c Substituting the right-hand side of.. for the sum on the right-hand side of..9 and simplifying yields Ramanujan s Proof of the q-gauss Summation Theorem On pages in his lost notebook, Ramanujan sketches his proof of the q-gauss summation theorem, normally given in the form a n b n c n c/a c/b..3. c n q n ab c c/ab This theorem was first discovered in 847 by Heine [78], whose proof, which is the most frequently encountered proof in the literature, is based on Heine s transformation, Theorem.., with h. This proof can be found in the texts of Andrews [8, p. 0, Corollary.4], Andrews, R. Askey, and R. Roy [30, p. 5], and G. Gasper and M. Rahman [5, p. 0]. A second proof employs the q-analogue of Saalschütz s theorem and can be read in the texts of W.N. Bailey [44, p. 68] and L.J. Slater [63, p. 97]. Ramanujan s proof

22 .3 Ramanujan s Proof of the q-gauss Summation Theorem is different from these two proofs and was first published in full in a paper by Berndt and A.J. Yee [79]. Ramanujan s proof encompasses Lemma.3., Lemma.3., and Entry.3. below. After giving Ramanujan s proof, we prove a corollary of.3., which is found on page 370 in Ramanujan s lost notebook. Before providing Ramanujan s argument, we derive the q-analogue of the Chu Vandermonde theorem and record a special case that will be used in Chapter 6. If we set b q N, where N is a nonnegative integer, in.3. and simplify, we find that φ a, q N ; c; q, cq N /a c/a N c N,.3. which is the q-analogue of the Chu Vandermonde theorem. If we reverse the order of summation on the left-hand side of.3., we deduce an alternative form of the q-chu Vandermonde theorem, namely, φ a, q N ; c; q, q c/a N c N a N..3.3 Setting a q M.3.3 yields and c q M N, where M is a nonnegative integer, in φ q M,q N ; q M N ; q, q q N N q M N N q MN q M M q N N q M N M+N q MN q M q N q MM+/ NN+/ q MN q M+N q M+NM+N+/ q M q N q M+N..3.4 In this chapter, we are providing analytic proofs of many of Ramanujan s theorems on basic hypergeometric series. Another approach uses combinatorial arguments. In [78], Berndt and Yee provided partition-theoretic proofs of several identities in the lost notebook arising from the Rogers Fine identity; a few of these proofs were reproduced in [3, Chapter ]. In [79], the same authors gave a combinatorial proof of the q-gauss summation theorem. Other combinatorial proofs of this theorem based on overpartitions have been given by S. Corteel and J. Lovejoy [44], Corteel [43], and Yee [85]. Lemma.3.. If n is any nonnegative integer, then n a n k qn+ k k q kk / a k..3.5 q k k0 Lemma.3. is a restatement of the q-binomial theorem.. and can be found in [54, p. 4, Lemma.] or [8, p. 36, Theorem 3.3]. We now

23 The Heine Transformation use Lemma.3. to establish Lemma.3. below along the lines indicated by Ramanujan. Alternatively, Lemma.3. can be deduced from [5, p., equation.5.3] by setting c 0 and replacing q by /q there. Lemma.3.. If c 0and n is any nonnegative integer, then n c n c j /c j q n+ j j..3.6 q j j0 Proof. Denote the right side of.3.6 by gc and apply.3.5 with a /c and n j in the definition of gc to find that gc n j0 k0 j k qj+ k k q n+ j j q kk / c j k : q j q k The coefficient of c r, 0 r n, above is n a r c r. r0 n r a r k qr+ k q n+ r k r+k q kk /..3.7 q r+k q k k0 Now we can easily verify that and q r+ k q r+k q r q n+ r k r+k q n+ r k k q n+ r r. Using these last two equalities in.3.7, we find that a r qn+ r r q r qn+ r r q r n r n r k qn r+ k k q k k0 {, if r n, 0, otherwise, q kk / by.3.5. This therefore completes our proof of Lemma.3.. Entry.3. pp , q-gauss Summation Theorem. If abc < and bc 0, then ac abc a ab /b n /c n a n q n abc n..3.8 In Entry 4 of Chapter 6 in his second notebook [43], [54, p. 4], Ramanujan states the q-gauss summation theorem in precisely the same form as that given in.3.8.

24 Proof..3 Ramanujan s Proof of the q-gauss Summation Theorem 3 We rewrite the right side of.3.8 in the form aq j /b j /c j abc j.3.9 ab q j j0 and examine the coefficient of a n, n 0, on each side of.3.8. From.., with b replaced by ab and a replaced by aq j, we find that aq j q j /b k ab k..3.0 ab q k The coefficient of a n j in.3.0 is k0 q j /b n j q n j b n j, and so the coefficient of a n in.3.9 equals n j0 /b j /c j q j /b n j b n c j q j q n j /b nb n q n n j0 c j /c j q n+ j j q j /b nb n q n c n,.3. by Lemma.3.. But by.., with b replaced by abc and a replaced by ac, ac /b n abc n..3. abc q n So, the coefficient of a n in.3. is precisely that on the right side of.3.. Hence,.3.8 immediately follows, since the coefficients of a n, n 0, on both sides of.3.8 are equal. The proof of Entry.3. is therefore complete. Entry.3. p For any complex numbers a and b, aq b/a n a n q nn+/..3.3 bq q n bq n Proof. In.3.8, replace a by bq, c by a/b, andb by t to find that bqt aq /t n b/a n aqt n..3.4 bq aqt q n bq n If we let t 0 in.3.4, we immediately arrive at.3.3 to complete the proof. A combinatorial proof of Entry.3. in the case b has been given by S. Corteel and J. Lovejoy [45], but it can easily be extended to give a proof of Entry.3. in full generality. Another combinatorial proof can be found in a paper by Berndt, B. Kim, and A.J. Yee [73].

25 4 The Heine Transformation.4 Corollaries of.. and..5 Entry.4. p. 3. For 0 < aq, k <, aq; q cq; q bq; q kq ; q kq ; q n bq/a; q n cq; q a n q n n+ q; q n cq/k; q n aq; q n q ; q n bq; q n+ k n q n..4. Proof. In.., set h andt kq, and replace c by bq, a by cq/k, and b by aq. The resulting identity is equivalent to.4.. We note that no generality has been lost by the substitutions above; so Ramanujan had.. in full generality for h. Padmavathamma [5] has also given a proof of.4.. Entry.4. p. 3. For bq <, q; q aq; q bq; q q; q n bq; q n aq; q q n n+ q; q n aq; q n bq; q n+ q n. Proof. In.., set h,b q, andt q, and replace a by aq and c by bq. The result then reduces to the identity above upon simplification. Entry.4.3 p.. For aq, b <, a n q n q; q n bq; q n aq; q bq; q n aq; q n b n q n q ; q n. Proof. In.., set h,c 0,andt τ, and replace a by bq/τ and b by aq. Then let τ 0. The result easily simplifies to the identity above. Entry.4.4 p.. For a, b <, a n q n q ; q n bq; q n aq ; q bq; q n aq ; q n b n q nn+/ q; q n. Proof. In.., set h anda 0,letb 0, and then replace t by aq and c by bq. The previous two entries were also established by Padmavathamma [5]. The next result is a corrected version of Ramanujan s claim.

26 .4 Corollaries of.. and..5 5 Entry.4.5 p. 5, corrected. For any complex number a, aq; q n q; q n q n q; q aq; q q; q n q n aq; q n+. Proof. In.., set h,a 0,b q, c aq,andt q. Simplification yields Ramanujan s assertion. The next two entries specialize to instances of identities for fifth-order mock theta functions, as we shall see in our fourth volume on the lost notebook [33]. The first is a corrected version of Ramanujan s claim. Entry.4.6 p. 6, corrected. For any complex number a, aq; q q; q aq; q n q n q ; q n n a q ; q n q n q 4 ; q 4 n a n a q ; q n q nn+/ q; q n. Proof. The proof of this result is rather more intricate than the proofs of the previous entries in this section. In..5, replace t by q/a and let a to deduce that b; q n q n q ; q b; q q; q c/b n n c; q n c; q q; q n q; q b n n + b; q q ; q c/b n+ c; q q; q n+ q ; q b n+. n Now set c 0andb aq. If we multiply both sides of the resulting identity by aq; q / q; q, we arrive at aq; q q; q aq; q n q n q ; q a q ; q a n q n n q ; q q; q n q; q n + a q ; q a n+ q n+ q; q q; q n+ q ; q n : T + T..4. Next, in.. with h, replace q by q,seta q /t, b a q,and c 0, and let t 0. Noting that q ; q /q ; q 4, we deduce that n a q ; q n q n T q 4 ; q n

27 6 The Heine Transformation Finally, in.., set h,a 0,andc q, and let b 0. Then set t a q and multiply both sides of the resulting equality by / + q. We therefore find that a n q n q ; q n q; q n+ q; q a q ; q a q ; q n q nn+3/ q; q n. Upon multiplying both sides of this last identity by aq q; q a q ; q and noting that q; q /q; q, we obtain, after replacing q by q and replacing n by n on the right-hand side, T a a q ; q n q nn+/..4.4 q; q n n If we substitute.4.3 and.4.4 into.4., we obtain our desired identity to complete the proof. Entry.4.7 p. 6. If a is any complex number, then aq; q q; q aq; q n q nn+ q ; q n a q ; q n q nn+/ + a q; q n n a q ; q n q n +4n+ q 4 ; q 4 n. Proof. In..5, let t q /a and c 0. After letting a,setb aq. Multiplying both sides of the resulting identity by aq; q / q; q,we find that aq; q q; q aq; q n q nn+ q ; q n a q ; q q; q + a q ; q q ; q aq n q; q n q ; q n aq n+ q; q n+ q; q n+ : S + S..4.5 Now in.. with h,seta 0andc q, and let b tend to 0. Then set t a q. The result, after replacing q by q and simplifying, is given by a q ; q n q nn+/ a q ; q q; q aq n q; q n q ; q n S. q; q n.4.6 Next, in.., set h, replace q by q, and then set b a q, t q 6 /a, and c 0. After letting a and substantially simplifying, we find that

28 a n a q ; q n q n +4n+ q 4 ; q 4 n a q ; q q ; q.4 Corollaries of.. and..5 7 aq n+ q; q n+ q; q n+ S..4.7 If we substitute.4.7 and.4.6 into.4.5, we obtain the desired identity for this entry. Entry.4.8 p. 6. For arbitrary complex numbers a and b, aq; q aq; q n b n q n q ; q n bq; q + bq ; q aq n q; q n bq; q n aq n+ q; q n+ bq ; q n. Proof. This entry is a further special case of..5; replace a by bq/t, set c 0andb aq, and let t 0. In her thesis [5], Padmavathamma also proved Entry.4.8. For a combinatorial proof of Entry.4.8, see the paper by Berndt, Kim, and Yee [73]. The next entry is the first of several identities in this chapter that provide representations of theta functions or quotients of theta functions by basic hypergeometric series. We therefore review here Ramanujan s notations for theta functions and some basic facts about theta functions. Recall that the Jacobi triple product identity [8, p., Theorem.8], [54, p. 35, Entry 9] is given, for ab <, by fa, b : a nn+/ b nn / a; ab b; ab ab; ab..4.8 n Deducible from.4.8 are the product representations of the classical theta functions [8, p. 3, Corollary.0], [54, pp , Entry, equation.4], ϕ q :f q, q n q n q, q n.4.9 ψq :fq, q 3 q nn+/ q ; q q; q,.4.0 f q :f q, q n q n3n / q; q,.4. n where we have employed the notation used by Ramanujan throughout his notebooks. The last equality in.4. is known as Euler s pentagonal number theorem. We also need the elementary result [54, p. 34, Entry 8iii]

29 8 The Heine Transformation f,a0,.4. for any complex number a with a <. Later, we need the fundamental property [54, p. 34]: For ab < and each integer n, fa, b a nn+/ b nn / f aab n,bab n..4.3 Entry.4.9 p. 0. Let ϕ q be defined by.4.9 above. Then ϕ q q nn+/ q; q n n q nn+/ q ; q n..4.4 First Proof of Entry.4.9. In.., we set h,a q/τ, b τ, c q, and t τ. Letting τ tend to 0, we find that q nn+/ q n q q n q nn+/ q n q n..4.5 The desired result follows once we invoke the well-known product representation for ϕ q in.4.9. Second Proof of Entry.4.9. Our second proof is taken from the paper [73] by Berndt, Kim, and Yee. Multiplying both sides of.4.5 by q, we obtain the equivalent identity q nn+/ q n q n+ ; q n q nn+/ q n+ ; q,.4.6 q n since q ; q q; q q; q. The left side of.4.6 is a generating function for the pair of partitions π,ν, where π is a partition into n distinct parts and ν is a partition into distinct parts that are strictly larger than n and where the exponent of is the number of parts in ν. For a given partition pair π, ν generated by the left side of.4.6, let k be the number of parts in ν. Detach n from the each part of ν and attach k to each part of π. Then we obtain partition pairs σ, λ, such that σ is a partition into k distinct parts and λ is a partition into distinct parts that are strictly larger than k, and the exponent of is the number of parts in σ. These partitions are generated by the right side of.4.6. Since this process is easily reversible, our proof is complete. The series on the left-hand sides of.4.4 and.4.8 below are the generating functions for the enumeration of gradual stacks with summits and stacks with summits, respectively [3]. Another generating function for gradual stacks with summits was found by Watson [79, p. 59], [75, p. 38], who showed that

30 q nn+/ q; q n.4 Corollaries of.. and..5 9 q; q q nn+ q ; q n,.4.7 which is implicit in the work of Ramanujan in his lost notebook [44]. An elegant generalization of the concept of gradual stacks with summits has been devised by Yee, with her generating function generalizing that on the righthand side of.4.7 [86, Theorem 5.]. See Entry 6.3. for a significant generalization of Entry.4.0 involving two additional parameters. Entry.4.0 p. 0. q n q n q n q nn+/..4.8 Proof. In.., set h,t c q, anda 0, and then let b 0. Entry.4.0 follows immediately. Entry.4. p. 0. q n q n q + n q nn+/..4.9 Proof. In.., set h,a 0,c q, andt q.nowletb 0to deduce that q n q n q q n qn+ q nn+/ q q n q nn+/ + n+ q n+n+/ q + n q nn+/. n Observe that the sum on the right sides in Entries.4.0 and.4. is a false theta function in the sense of L.J. Rogers. Several other entries in the lost notebook involve this false theta function; see [3, pp. 7 3] for some of these entries. In providing a combinatorial proof of Entry.4., Kim [89] was led to a generalization for which he supplied a combinatorial proof. The following entry has been combinatorially proved by Berndt, Kim, and Yee [73]. Entry.4. p. 0. For a, b < and any positive integer n, bq n ; q n a m q mm+/ b m q nmm+/ q; q m bq n ; q n aq; q m q n ; q n. m aq; q nm m0 n m0.4.0

31 0 The Heine Transformation Proof. In.., set h n and let b tend to 0. Then set t bq n /a and let a tend to. Finally, replace c by aq. Entry.4.3 p.. For a <, aq n q ; q n bq; q n aq; q bq; q Proof. bq. aq; q n b n q n +n aq ; q bq; q q; q n aq ; q n b n+ q n +3n+ q; q n+. In..5, let a 0andletb 0. Then replace t by aq and c by In her doctoral dissertation [5], Padmavathamma gave another proof of Entry.4.3, and gave proofs of the following two entries as well. Entry.4.4 p.. For any complex number a, q ; q 4 aq ; q n q nn+/ aq 4 ; q 4 q; q n +aq ; q 4 aq ; q 4 n q 4n q ; q n aq 4 ; q 4 n q 4n +4n+ q ; q n+. Proof. In Entry.4.3, replace q by q and set b /q. This yields a n q n aq ; q 4 n q 4n q 4 ; q 4 n q; q n aq ; q 4 q; q q ; q n aq 4 ; q 4 n q 4n +4n+ + aq 4 ; q 4 q; q q ; q. n+ Consequently, in order to prove the desired result, we must show that aq ; q 4 q; q aq 4 ; q 4 q ; q 4 a n q n q 4 ; q 4 n q; q n aq ; q n q nn+/,.4. q; q n and this follows from... More precisely, let h,c q, anda 0, and let b tend to 0. Then put t aq and simplify. Entry.4.5 p.. If a is any complex number, then q ; q 4 aq ; q n q n+n+/ aq 4 ; q 4 aq ; q 4 n q 4n +4n+ q; q n q ; q n +aq ; q 4 aq 4 ; q 4 n q 4n +8n+4 q ; q n+.

32 .4 Corollaries of.. and..5 Proof. In Entry.4.3, replace q by q and set b q. Upon multiplication of both sides by q/ + q, we find that a n q n+ q 4 ; q 4 n q; q n+ aq ; q 4 q; q + aq 4 ; q 4 q; q aq ; q 4 n q 4n +4n+ q ; q n Consequently, in order to prove Entry.4.5, we must show that q; q aq ; q 4 aq 4 ; q 4 q ; q 4 aq 4 ; q 4 n q 4n +8n+4 q ; q n+. a n q n+ q 4 ; q 4 n q; q n+ aq ; q n q n+n+/ q; q n..4. This last identity follows from... Set h,c q,anda 0. Then let b 0. Setting t aq and multiplying both sides of the resulting identity by q/ + q, we complete the proof. Entry.4.6 p.. For any complex number a, q; q Proof. n aq; q n q nn+ q ; q n aq ; q aq; q Set b in Entry.4.3 to deduce that aq n q ; q n q; q n aq; q q; q aq; q n q n +n q; q n aq; q n q n +n aq ; q q; q aq ; q n q n +3n+ q; q n+. q; q n aq ; q n q n +3n+ q; q n+. Therefore, in order to complete the proof of Entry.4.6, we must prove that q; q aq; q aq ; q q; q aq n q ; q n q; q n n aq; q n q nn+ q ; q n,.4.3 and this follows from.. with h, first setting a q /t, then letting c and t tend to 0, and finally replacing b by aq.

33 The Heine Transformation Entry.4.7 p. 30. For each positive integer n, the series is symmetric in a and b. aq m0 b m q mm+/ q m aq nm Proof. By an application of the q-binomial theorem..4, aq m0 b m q mm+/ q m aq nm b m q mm+/ aq nm+ q m b m a j q mm+/+jj+/+nmj. q m q j m0 m,j0 This last series is obviously symmetric in a and b, and so the proof is complete. Berndt, Kim, and Yee [73] found a combinatorial proof of Entry.4.7. The next result from the top of page 7 of Ramanujan s lost notebook has lines drawn through it. Furthermore, the right-hand side has ellipses after the products forming the numerator and the denominator. If nothing is added, the result is clearly false. However, the following identity has the same left-hand side that Ramanujan gave, and the infinite products from the right-hand side of his proposed identity are isolated in front of our right-hand side. Entry.4.8 p. 7, corrected. For any complex numbers a and b with b 0, a/b; q n b n q nn+/ q; q n aq ; q n bq; q aq; q { a/b; q aq; q aq ; q } bq ; q n a n q ; q. n bq; q n b.4.4 Proof. In.., take h, and then replace a, c, andt by bq, bq, and a/b, respectively. Now let b 0. Simplification then yields the desired result..5 Corollaries of..6 and..7 The first two entries in this section were proved by G.N. Watson [78] and Andrews [7], with Berndt, Kim, and Yee [73] also providing a combinatorial proof of the former entry.

34 Entry.5. p. 4. If a is any complex number, then a n q n q; q n aq ; q aq; q.5 Corollaries of..6 and..7 3 a n q n q ; q n aq ; q n a n q n +n q ; q n aq; q n. Proof. The first line follows by setting t aq/b in..7 and letting b 0. The second line follows from the fact that each of the right-hand entries is equal to a m+n q n +m +m+mn q ; q m q ; q. n m, To verify this last claim, first apply..4 to aq n+ ; q. Secondly, apply..4 to aq n+ ; q and then switch the roles of m and n. then M. Somos has observed that if we set F a, b; q : bq; q a n q n q ; q n bq; q n, F a, b; q F b, a; q..5. Entry.5. then follows by taking b aq in.5.. To prove.5., return to Entry.4., set n, and replace q by q, a by a/q, andb by b/q. Then we easily see that.4.0 reduces to.5.. Entry.5. p. 4. If a is any complex number, then a n q 4n q 4 ; q 4 aq; q a n q n n q ; q n aq; q. n Proof. Replace q by q in..6. Then set b aq/t and let t 0. Entry.5.3 p. 6. q; q q n +n q ; q n n q nn+/ q ; q n. Proof. In Entry.5., replace q by q, and then set a q. Using Euler s identity, we find that the result simplifies to the equality above. Entry.5.4 p. 6. q; q q n n q ; q n n q nn+3/ q ; q n.

35 4 The Heine Transformation Proof. In..6, replace t by /b and let b. Hence, q n n n q nn / q ; q q; q + n q; q n q; q n q; q + q; q + + q; q q; q + n n n n n q nn / q n + q n q ; q n q n n q nn / q ; q n n q nn+/ + q n q ; q n n n q nn+/ q ; q n n q nn+/ q ; q n + n q nn+3/ q ; q n. n n q nn+3/ q ; q n By Euler s identity, this last identity is equivalent to that of Entry Corollaries of..8,..9, and..0 Entry.6. p. 36. If a and b are any complex numbers, then aq b n q n q n aq n n b/a n a n q nn+/ q n. Proof. Replace t by t/ab in..9 and let a and b tend to. This then yields the identity t n q n n n t/c n c n q nn /..6. q n c n c q n Replacing t by bq and c by aq in.6., we complete the proof. Entry.6. is identical to Entry 9 in Chapter 6 of Ramanujan s second notebook [43], [54, p. 8]. Earlier proofs of Entry.6. were given by V. Ramamani [34] and by Ramamani and K. Venkatachaliengar [35]. L. Carlitz [99] posed the special case a of Entry.6. as a problem. S. Bhargava and C. Adiga [8] proved a generalization of Entry.6., while H.M. Srivastava [68] later established an equivalent formulation of their result. Lastly, Berndt, Kim, and Yee [73] have devised a bijective proof of Entry.6..

36 Entry.6. p. 8. For any complex number a,.6 Corollaries of..8,..9, and..0 5 a n q n + n q; q n a n q nn+/ aq ; q n. Proof. Subtracting from both sides of this entry, shifting the summation indices down by on each side, and dividing both sides by aq, we see that the identity above is equivalent to the identity a n q n +n q; q n a n q nn+3/ aq ; q n+. Now in..8, replace a by aq /t, setb q, and let t 0. The resulting identity then simplifies to that of Entry.6.. See the paper [73] by Berndt, Kim, and Yee for a combinatorial proof of Entry.6.. The next result does not properly belong under the heading of this section, but we have put it here because of its similarity to the previous entry. We note that the case a is Entry 9.3. of Part I [3, p. 9]. Entry.6.3 p. 8. For any complex number a, n a n q nn+/ n q; q n a n q nn+ aq; q n+. Proof. In.., let h, replace a by a q /t, then set b q and c aq, and let t 0. After simplification, we find that n q; q n a n q nn+ aq; q n+ q; q aq; q n a n q nn+/, q n q; q n aq; q n where we applied.. with h and replaced a, b, c, andt, respectively, by 0, 0, aq, andq. Entry.6.4 p. 38. For aq <, aq n aq ; q n n a n q nn+/..6. aq; q n Proof. In..8, set a 0andb q, and then replace t by a. Simplification yields

37 6 The Heine Transformation aq n aq ; q n +a q; q n a n n a n q nn+/ aq; q n, where the last line follows from..9, wherein we replaced a by q and b by q, then set t a, and let c 0. Entry.6.4 can also be derived from Entry 9..6 in our first book on the lost notebook [3, p. 6]; this entry is on page 30 of [44]. In fact, when Berndt and A.J. Yee gave a combinatorial proof of Entry 9..6 in [78], after some elementary manipulation and the replacement of a by aq in Entry 9..6, they derived.6., for which they gave a bijective proof. Berndt, Kim, and Yee [73] have recently found a simpler bijective proof of.6.. Entry.6.5 p. 38. For any complex number a, a m q mm+ q ; q m + aq m+ aq ; q a n q nn+/..6.3 aq; q n m0 First Proof of Entry.6.5. Expanding / + aq n+ in a geometric series, inverting the order of summation, and using..4, we find that, for aq <, a n q nn+ q ; q n + aq n+ m a n+m q nn++mn+ q ; q m0 n aq m a n q nn++m q ; q m0 n aq m aq +m ; q m0 aq ; q aq ; q m0 aq m aq ; q m a n q nn+/ aq; q n, where the last line follows from Entry.6.4. The desired result now follows by analytic continuation in a. Second Proof of Entry.6.5. Our second proof is taken from a paper by Berndt, Kim, and Yee [73]. By Entry.6.4, the identity.6.3 can be written in the equivalent form a m q mm+ q ; q m + aq m+ aq n aq n+ ; q. m0

38 .7 Corollaries of Section. and Auxiliary Results 7 Note that aq n+ ; q generates partitions into distinct even parts, each greater than or equal to n +, with the exponent of a denoting the number of parts. Let m be the number of parts generated by a partition arising from aq n+ ; q. Detach n from each of the m parts. Combining this with aq n, we obtain aq m+ n. However, note that, for n 0, all of these odd parts are generated by / + aq m+, and each part is weighted by a. The remaining parts, which are even, are generated by m0 a m q mm+ q ; q m. For these partitions into m distinct even parts, the exponent of a again denotes the number of parts. Entry.6.6 p. 35. Recall that ψq is defined by.4.0. Then n q n +n q ; q n q n+ ψq. Proof. Set a in Entry.6.5. Using..4, we find that n q n +n q ; q n q n+ q ; q q nn+/ q; q n q ; q q; q q ; q q; q ψq, by Euler s identity and.4.0. Entry.6.7 p. 40. For a <, a a n q n bq n a n b n q n q n bq n. Proof. In..0, let both a and b tend to 0. Then replace t by a and c by bq, and lastly multiply both sides by a..7 Corollaries of Section. and Auxiliary Results Up to now in this chapter, we have concentrated on results from the lost notebook that can be traced to pairs of antecedent formulas proved in Section.. In this section, we also often draw on several of the results from Section., but additionally we require other formulas that have appeared often in Ramanujan s work. The Rogers Fine identity [49, p. 5]

39 8 The Heine Transformation α n β n τ n α n ατq/β n β n τ n q n n ατq n β n τ n+.7. is needed in this chapter. Ramanujan frequently used this identity in the lost notebook; see Chapter 9 of [3], which is entirely devoted to formulas in the lost notebook derived from.7.. The next two results are, in fact, special cases of the q-binomial theorem,... However, it will be more convenient to invoke them using the q-binomial coefficients, which are defined by [ ] k l q [ ] k : l 0, if l<0orl>k, q k, q l q k l otherwise..7. For any complex numbers a, b, and any nonnegative integer n, n [ ] n j a j b j q jj / b/a j n..7.3 j0 Also, for z < and any nonnegative integer N, [ ] n + N z n..7.4 n z N+ Entry.7. p. 5. For any complex number a, a; q n+ q n+ q; q + a n q nn+/ n+ aq; q q; q a n q nn+/ aq; q n. Proof. In..5, set t q, b q, anda 0; then replace c by aq. We therefore deduce that q n q; q a; q n q n q; q n aq; q n aq; q q; q q ; q n q; q a; q n+ q n+ aq; q q ; q q; q..7.5 n+ In.., set h,a 0,andt q. Then, replacing c by aq and letting b tend to 0, we find that q n a n q nn+/..7.6 q; q n aq; q n q; q aq; q