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Contents
Introduction.,,,,,,,,,,,,,,,,,,,,,,,,,,,,.,.,.1
1 The heine Transformation
Introduction
1.2 Heine,'s Method
1.3 Ramanujan's Proof of the g-Gauss Summation Theorem... 10
4 Corollaries of (1.2. 1) and(1.2.5)
1.5 Corollaries of(1.2.)and(1.2.7)
1.6 Corollaries of(1.28),(1.2.9),and(1.2.10)
1.7 Corollaries of Section 1.2 and Auxiliary Results
2 The Sears-Thomae Transformation
2.1 Introduction
2.2 Direct Corollaries of (2.1.1)and(2.1.3)
2.3 Extended Corollaries of (2.1. 1)and(2.1.3)
3 Bilateral series
3.1 Introduction
3.2 Background...…
33 The 1/1 Identity..……
3.4 The 20/2 Identities
3.5 Identities Arising from the Quintuple Product Identity ... 68
3.6 Miscellaneous bilateral Identities
4 Well-Poised series
4.1 Introduction
4.2 Applications of(4.1. 3)
4.3 Applications of Bailey' s Formulas........
5 Bailey's Lemma and Theta Expansions
5.1 Introduction...…
5.2 The Main Lemma
5.3 Corollaries of (5.2.3
5.4 Corollaries of (5.2.4)and Related Results.........107
6 Partial Theta Functions
6.1 Introduction....……………
6.2 A General Identity..………
6.3 Consequences of Theorem6.2.1.…………115
6.4 The function v(a, q)
6.5 Euler's Identity and Its Extensions
6.6 The Warnaar Theory
7 Special Identities..…………………149
7.1 Introduction
7.2 Generalized Modular Relations.……
7.3 Extending Abel's Lemma........
7.4 Innocents Abroad
8 Theta Function Identities…………173
8.1 Introduction
8.2 Cubic Identities
8.3 Septic Identities
Ramanujan's Cubic Class Invariant 1 Introduction...................... 195
9.2 An and the Modular j- Invariant..……………199
9.3 An and the Class Invariant G,,,……203
9.4 An and Modular Equations
9.5 An and Modular Equations in the Theory of Signature 3. .. 208
6 An and Kronecker's Limit Formula 9.7 The Remaining Five Values..……………217
9.9 Computations of入 Using the Shimura Reciprocity Lap…….218
9.8 Some Modular Functions of Level 72
10 Miscellaneous Results on Elliptic Functions and Theta
Functions....,,.,,..,,,,.,,,
10.1 A Quasi-theta Product……225
10.2 An Equivalent Formulation of(10. 1.1)in Terms of Hyperbolic Series…
10.3 Further Remarks on Ramanujans Quasi-theta Product...... 231
10.4 A Generalization of the Dedekind Eta Function....... 234
10.5 Two Entries on Page346,…,…238
10.6 A Continued Fraction..………
10.7 Class Invariants
11 Formulas for the Power Series Coefficients of Certain
Quotients of Eisenstein Series......
11.1 Introduction
112 The Key Theorem.………………247
11.3 The Coefficients of 1/Q(q)
11.4 The Coefficients of Q(o/R(q)
11.5 The Coefficients of(T P(a)/3)/R(g) and(P(a)/3)/R(q).. 280
1.6 The Coefficients of(πP(q)/2√3)/Q(q)…………………284
11.7 Eight Identities for Eisenstein Series and Theta Functions.. 287
118 The Coefficients of1/B(q)………290
11.9 Formulas for the Coefficients of Further Eisenstein Series
1.10 The Coefficients of 1/B2(q)………
11.11 A Calculation from [176
12 Letters from Matlock House...........:.... 313
12.1 Introduction
12.2 A Lower Bound
12.3 An Upper Bound
13 Eisenstein Series and Modular Equations 327
13.1 Introduction ........
13.2 Preliminary Results
13.3 Quintic Identities: First Method
13.4 Quintic Identities: Second Method..………338
13.5 Septic Identities
13.6 Septic Differential Equations……………353
14 Series Representable in Terms of Eisenstein Series 355
14.1 Introduction 355
14.2 The Series T2k(g)
14.3 The Series Un(q).…………
15 Eisenstein Series and Approximations to丌.………365
15.1 Introduction...............
15.2 Eisenstein Series and the Modular j-Invariant
15.3 Eisenstein Series and Equations in T: First Method ..... 367
15.4 Eisenstein Series and Equations in T: Second Method.... 370 15.5Page213. 15.6 Ramanujan,'s Series for 1/T 16 Miscellaneous Results on Eisenstein Series......... 385
16.1 A generalization of Eisenstein Series.………
16.2 Representations of Eisenstein Series in Terms of Elliptic
Function parameters
16.3 Values of Certain Eisenstein Series
16.4 Some Elementary Identities
Index...,,.,,,,,.,,.,.
附录I拉马努金的中国知音:数学家刘治国的“西天取经”之旅
附录II刘治国教授访谈
编辑手记
