Ramanujan's Lost Notebook (eBook)

Preface 7
Contents 8
Introduction 12
The Heine Transformation 16
Introduction 16
Heine's Method 17
Ramanujan's Proof of the q-Gauss Summation Theorem 21
Corollaries of (1.2.1) and (1.2.5) 25
Corollaries of (1.2.6) and (1.2.7) 33
Corollaries of (1.2.8), (1.2.9), and (1.2.10) 35
Corollaries of Section 1.2 and Auxiliary Results 38
The Sears--Thomae Transformation 55
Introduction 55
Direct Corollaries of (2.1.1) and (2.1.3) 55
Extended Corollaries of (2.1.1) and (2.1.3) 56
Bilateral Series 62
Introduction 62
Background 63
The 11 Identity 65
The 22 Identities 71
Identities Arising from the Quintuple Product Identity 77
Miscellaneous Bilateral Identities 83
Well-Poised Series 89
Introduction 89
Applications of (4.1.3) 90
Applications of Bailey's Formulas 97
Bailey's Lemma and Theta Expansions 105
Introduction 105
The Main Lemma 105
Corollaries of (5.2.3) 107
Corollaries of (5.2.4) and Related Results 115
Partial Theta Functions 121
Introduction 121
A General Identity 122
Consequences of Theorem 6.2.1 123
The function (a,q) 137
Euler's Identity and Its Extensions 141
The Warnaar Theory 149
Special Identities 156
Introduction 156
Generalized Modular Relations 157
Extending Abel's Lemma 165
Innocents Abroad 172
Theta Function Identities 180
Introduction 180
Cubic Identities 182
Septic Identities 187
Ramanujan's Cubic Class Invariant 202
Introduction 202
n and the Modular j-Invariant 206
n and the Class Invariant Gn 210
n and Modular Equations 211
n and Modular Equations in the Theory of Signature 3 215
n and Kronecker's Limit Formula 221
The Remaining Five Values 224
Some Modular Functions of Level 72 225
Computations of n Using the Shimura Reciprocity Law 228
Miscellaneous Results on Elliptic Functions and Theta Functions 232
A Quasi-theta Product 232
An Equivalent Formulation of (10.1.1) in Terms of Hyperbolic Series 233
Further Remarks on Ramanujan's Quasi-theta Product 238
A Generalization of the Dedekind Eta Function 241
Two Entries on Page 346 245
A Continued Fraction 247
Class Invariants 248
Formulas for the Power Series Coefficients of Certain Quotients of Eisenstein Series 250
Introduction 250
The Key Theorem 254
The Coefficients of 1/Q(q) 264
The Coefficients of Q(q)/R(q) 280
The Coefficients of (P(q)/3)/R(q) and (P(q)/3)2/R(q) 287
The Coefficients of (P(q)/23)/Q(q) 291
Eight Identities for Eisenstein Series and Theta Functions 294
The Coefficients of 1/B(q) 297
Formulas for the Coefficients of Further Eisenstein Series 305
The Coefficients of 1/B2(q) 307
A Calculation from p 319
Letters from Matlock House 320
Introduction 320
A Lower Bound 321
An Upper Bound 329
Eisenstein Series and Modular Equations 333
Introduction 333
Preliminary Results 334
Quintic Identities: First Method 337
Quintic Identities: Second Method 344
Septic Identities 351
Septic Differential Equations 359
Series Representable in Terms of Eisenstein Series 361
Introduction 361
The Series T2k(q) 362
The Series Un(q) 368
Eisenstein Series and Approximations to 371
Introduction 371
Eisenstein Series and the Modular j-Invariant 372
Eisenstein Series and Equations in : First Method 373
Eisenstein Series and Equations in : Second Method 376
Page 213 381
Ramanujan's Series for 1/ 381
Miscellaneous Results on Eisenstein Series 391
A generalization of Eisenstein Series 391
Representations of Eisenstein Series in Terms of Elliptic Function Parameters 392
Values of Certain Eisenstein Series 393
Some Elementary Identities 394
Location Guide 397
Provenance 403
References 406
Index 420