Ramanujan's Lost Notebook

1. Front Matter

   Pages 1-10

   PDF

2. Introduction

   • Bruce C. Berndt, George E. Andrews

   Pages 1-4

3. The Heine Transformation

   • Bruce C. Berndt, George E. Andrews

   Pages 1-39

4. The Sears– Thomae Transformation

   • Bruce C. Berndt, George E. Andrews

   Pages 1-7

5. Bilateral Series

   • Bruce C. Berndt, George E. Andrews

   Pages 1-27

6. Well-Poised Series

   • Bruce C. Berndt, George E. Andrews

   Pages 1-16

7. Bailey’s Lemma and Theta Expansions

   • Bruce C. Berndt, George E. Andrews

   Pages 1-16

8. Partial Theta Functions

   • Bruce C. Berndt, George E. Andrews

   Pages 1-35

9. Special Identities

   • Bruce C. Berndt, George E. Andrews

   Pages 1-24

10. Theta Function Identities

    • Bruce C. Berndt, George E. Andrews

    Pages 1-22

11. Ramanujan’s Cubic Analogue of the Classical Ramanujan–Weber Class Invariants

    • Bruce C. Berndt, George E. Andrews

    Pages 1-30

12. Miscellaneous Results on Elliptic Functions and Theta Functions

    • Bruce C. Berndt, George E. Andrews

    Pages 1-18

13. Formulas for the Power Series Coefficients of Certain Quotients of Eisenstein Series

    • Bruce C. Berndt, George E. Andrews

    Pages 1-70

14. Two Letters on Eisenstein Series Written from Matlock House

    • Bruce C. Berndt, George E. Andrews

    Pages 1-13

15. Eisenstein Series and Modular Equations

    • Bruce C. Berndt, George E. Andrews

    Pages 1-28

16. Series Representable in Terms of Eisenstein Series

    • Bruce C. Berndt, George E. Andrews

    Pages 1-10

17. Eisenstein Series and Approximations to π

    • Bruce C. Berndt, George E. Andrews

    Pages 1-20

18. Miscellaneous Results on Eisenstein Series

    • Bruce C. Berndt, George E. Andrews

    Pages 1-6

19. Back Matter

    Pages 1-27

    PDF

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the authors plan to write in their examination of all the claims made by S. Ramanujan in The Lost Notebook and Other Unpublished Papers.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 1988, contains the “Lost Notebook,” which was discovered by the ?rst author in the spring of 1976 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 1917–1919. The authors have attempted to organize this disparate material in chapters. This second volume contains 16 chapters comprising 314 entries, including some duplications and examples, with chapter totals ranging from a high of ?fty-four entries in Chapter 1 to a low of two entries in Chapter 12. Contents Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 The Heine Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1. 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1. 2 Heine’s Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1. 3 Ramanujan’s Proof of the q-Gauss Summation Theorem . . . . . 10 1. 4 Corollaries of (1. 2. 1) and (1. 2. 5) . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1. 5 Corollaries of (1. 2. 6) and (1. 2. 7) . . . . . . . . . . . . . . . . . . . . . . . . . . 22 1. 6 Corollaries of (1. 2. 8), (1. 2. 9), and (1. 2. 10) . . . . . . . . . . . . . . . . . . 24 1. 7 Corollaries of Section 1. 2 and Auxiliary Results . . . . . . . . . . . . . 27 2 The Sears–Thomae Transformation . . . . . . . . . . . . . . . . . . . . . . . . 45 2. 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 2. 2 Direct Corollaries of (2. 1. 1) and (2. 1. 3) . . . . . . . . . . . . . . . . . . . . 45 2. 3 Extended Corollaries of (2. 1. 1) and (2. 1. 3) . . . . . . . . . . . . . . . . .

• Invariant
• approximation
• elliptic function
• equation
• function
• identity
• theta function
• transformation

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