Ramanujan's Lost Notebook: Part II

ISBN-10: 1441926666

ISBN-13: 9781441926661

Pub. Date: 10/29/2010

Publisher: Springer New York

This volume is the second of approximately four volumes that the authors plan to write on Ramanujan's lost notebook, which is broadly interpreted to include all material published in The Lost Notebook and Other Unpublished Papers in 1988.  See more details below

Overview

This volume is the second of approximately four volumes that the authors plan to write on Ramanujan's lost notebook, which is broadly interpreted to include all material published in The Lost Notebook and Other Unpublished Papers in 1988.

Product Details

ISBN-13:
9781441926661
Publisher:
Springer New York
Publication date:
10/29/2010
Edition description:
Softcover reprint of hardcover 1st ed. 2009
Pages:
420
Product dimensions:
6.14(w) x 9.21(h) x 0.88(d)

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) 46

3 Bilateral Series 53

3.1 Introduction 53

3.2 Background 54

3.3 The 1ψ1 Identity 56

3.4 The 2ψ2 Identities 62

3.5 Identities Arising from the Quintuple Product Identity 68

3.6 Miscellaneous Bilateral Identities 74

4 Well-Poised Series 81

4.1 Introduction 81

4.2 Applications of (4.1.3) 82

4.3 Applications of Bailey's Formulas 89

5 Bailey's Lemma and Theta Expansions 97

5.1 Introduction 97

5.2 The Main Lemma 97

5.3 Corollaries of (5.2.3) 99

5.4 Corollaries of (5.2.4) and Related Results 107

6 Partial Theta Functions 113

6.1 Introduction 113

6.2 A General Identity 114

6.3 Consequences of Theorem 6.2.1 115

6.4 The function ψ (a, q) 129

6.5 Euler's Identity and Its Extensions 133

6.6 The Warnaar Theory 141

7 Special Identities 149

7.1 Introduction 149

7.2 Generalized Modular Relations 150

7.3 Extending Abel's Lemma 158

8 Theta Function Identities 173

8.1 Introduction 173

8.2 Cubic Identities 175

8.3 Septic Identities 180

9 Ramanujan's Cubic Class Invariant 195

9.1 Introduction 195

9.2 λn and the Modular j-Invariant 199

9.3 λn andthe Class Invariant Gn 203

9.4 λn and Modular Equations 204

9.5 λn and Modular Equations in the Theory of Signature 3 208

9.6 λn and Kronecker's Limit Formula 214

9.7 The Remaining Five Values 217

9.8 Some Modular Functions of Level 72 218

9.9 Computations of λn Using the Shimura Reciprocity Law 221

10 Miscellaneous Results on Elliptic Functions and Theta Functions 225

10.1 A Quasi-theta Product 225

10.2 An Equivalent Formulation of (10.1.1) in Terms of Hyperbolic Series 226

10.3 Further Remarks on Ramanujan's Quasi-theta Product 231

10.4 A Generalization of the Dedekind Eta Function 234

10.5 Two Entries on Page 346 238

10.6 A Continued Fraction 240

10.7 Class Invariants 241

11 Formulas for the Power Series Coefficients of Certain Quotients of Eisenstein Series 243

11.1 Introduction 243

11.2 The Key Theorem 247

11.3 The Coefficients of 1/Q(q) 257

11.4 The Coefficients of Q(q)/R(q) 273

11.5 The Coefficients of <$>(\pi P(q)/3)R(q)<$> and <$>(\pi P(q)/3)^2/R(q)<$> 280

11.6 The Coefficients of <$>(\pi P(q)/2 \sqrt {3})/Q(q)<$> 284

11.7 Eight Identities for Eisenstein Series and Theta Functions 287

11.8 The Coefficients of 1/B(q) 290

11.9 Formulas for the Coefficients of Further Eisenstein Series 298

11.10 The Coefficients of 1/B2(q) 300

11.11 A Calculation from [176] 312

12 Letters from Matlock House 313

12.1 Introduction 313

12.2 A Lower Bound 314

12.3 An Upper Bound 322

13 Eisenstein Series and Modular Equations 327

13.1 Introduction 327

13.2 Preliminary Results 328

13.3 Quintic Identities: First Method 331

13.4 Quintic Identities: Second Method 338

13.5 Septic Identities 345

13.6 Septic Differential Equations 353

14 Series Representable in Terms of Eisenstein Series 355

14.1 Introduction 355

14.2 The Series T2k(q) 356

14.3 The Series Un(q) 362

15 Eisenstein Series and Approximations to π 365

15.1 Introduction 365

15.2 Eisenstein Series and the Modular j-Invariant 366

15.3 Eisenstein Series and Equations in π: First Method 367

15.4 Eisenstein Series and Equations in π: Second Method 370

15.5 Page 213 375

15.6 Ramanujan's Series for 1/π 375

16 Miscellaneous Results on Eisenstein Series 385

16.1 A generalization of Eisenstein Series 385

16.2 Representations of Eisenstein Series in Terms of Elliptic Function Parameters 386

16.3 Values of Certain Eisenstein Series 387

16.4 Some Elementary Identities 388

Location Guide 391

Provenance 397

References 401

Index 415