Ramanujan's lost notebook,

front-matter.pdf......Page 1
Preface......Page 6
Contents......Page 7
Introduction......Page 11
Introduction......Page 15
Heine's Method......Page 16
Ramanujan's Proof of the q-Gauss Summation Theorem......Page 20
Corollaries of (1.2.1) and (1.2.5)......Page 24
Corollaries of (1.2.6) and (1.2.7)......Page 32
Corollaries of (1.2.8), (1.2.9), and (1.2.10)......Page 34
Corollaries of Section 1.2 and Auxiliary Results......Page 37
Direct Corollaries of (2.1.1) and (2.1.3)......Page 54
Extended Corollaries of (2.1.1) and (2.1.3)......Page 55
Introduction......Page 61
Background......Page 62
The 11 Identity......Page 64
The 22 Identities......Page 70
Identities Arising from the Quintuple Product Identity......Page 76
Miscellaneous Bilateral Identities......Page 82
Introduction......Page 88
Applications of (4.1.3)......Page 89
Applications of Bailey's Formulas......Page 96
The Main Lemma......Page 104
Corollaries of (5.2.3)......Page 106
Corollaries of (5.2.4) and Related Results......Page 114
Introduction......Page 120
A General Identity......Page 121
Consequences of Theorem 6.2.1......Page 122
The function (a,q)......Page 136
Euler's Identity and Its Extensions......Page 140
The Warnaar Theory......Page 148
Introduction......Page 155
Generalized Modular Relations......Page 156
Extending Abel's Lemma......Page 164
Innocents Abroad......Page 171
Introduction......Page 179
Cubic Identities......Page 181
Septic Identities......Page 186
Introduction......Page 201
n and the Modular j-Invariant......Page 205
n and the Class Invariant Gn......Page 209
n and Modular Equations......Page 210
n and Modular Equations in the Theory of Signature 3......Page 214
n and Kronecker's Limit Formula......Page 220
The Remaining Five Values......Page 223
Some Modular Functions of Level 72......Page 224
Computations of n Using the Shimura Reciprocity Law......Page 227
A Quasi-theta Product......Page 231
An Equivalent Formulation of (10.1.1) in Terms of Hyperbolic Series......Page 232
Further Remarks on Ramanujan's Quasi-theta Product......Page 237
A Generalization of the Dedekind Eta Function......Page 240
Two Entries on Page 346......Page 244
A Continued Fraction......Page 246
Class Invariants......Page 247
Introduction......Page 249
The Key Theorem......Page 253
The Coefficients of 1/Q(q)......Page 263
The Coefficients of Q(q)/R(q)......Page 279
The Coefficients of (P(q)/3)/R(q) and (P(q)/3)2/R(q)......Page 286
The Coefficients of (P(q)/23)/Q(q)......Page 290
Eight Identities for Eisenstein Series and Theta Functions......Page 293
The Coefficients of 1/B(q)......Page 296
Formulas for the Coefficients of Further Eisenstein Series......Page 304
The Coefficients of 1/B2(q)......Page 306
A Calculation from p......Page 318
Introduction......Page 319
A Lower Bound......Page 320
An Upper Bound......Page 328
Introduction......Page 332
Preliminary Results......Page 333
Quintic Identities: First Method......Page 336
Quintic Identities: Second Method......Page 343
Septic Identities......Page 350
Septic Differential Equations......Page 358
Introduction......Page 360
The Series T2k(q)......Page 361
The Series Un(q)......Page 367
Introduction......Page 370
Eisenstein Series and the Modular j-Invariant......Page 371
Eisenstein Series and Equations in : First Method......Page 372
Eisenstein Series and Equations in : Second Method......Page 375
Ramanujan's Series for 1/......Page 380
A generalization of Eisenstein Series......Page 390
Representations of Eisenstein Series in Terms of Elliptic Function Parameters......Page 391
Values of Certain Eisenstein Series......Page 392
Some Elementary Identities......Page 393
Location Guide......Page 396
Provenance......Page 402
References......Page 405
Index......Page 419