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Ramanujan's Lost Notebook : Part II :  Part II - George E. Andrews

Ramanujan's Lost Notebook : Part II

Part II

Hardcover Published: December 2008
ISBN: 9780387777658
Number Of Pages: 420

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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lost notebook, which is broadly interpreted to include all material published in The Lost Notebook and Other Unpublished Papers in 1988. The primary topics addressed in the authors' second volume on the lost notebook are q-series, Eisenstein series, and theta functions. Most of the entries on q-series are located in the heart of the original lost notebook, while the entries on Eisenstein series are either scattered in the lost notebook or are found in letters that Ramanujan wrote to G.H. Hardy from nursing homes.

Industry Reviews

From the reviews: "This volume contains 16 chapters comprising 314 entries. The material is arranged thematically with the main topics being some of Ramanujan's favorites q series theta functions ... . the authors treatment is extremely thorough. Each chapter contains an introduction with appropriate background. References to all other known proofs of the entries are provided. ... Fans of Ramanujan's mathematics are sure to be delighted by this book. ... Many entries are just begging for further study and will undoubtedly be inspiring research for decades to come." (Jeremy Lovejoy, Mathematical Reviews, Issue 2010 f)

Prefacep. vii
Introductionp. 1
The Heine Transformationp. 5
Introductionp. 5
Heine's Methodp. 6
Ramanujan's Proof of the q-Gauss Summation Theoremp. 10
Corollaries of (1.2.1) and (1.2.5)p. 14
Corollaries of (1.2.6) and (1.2.7)p. 22
Corollaries of (1.2.8), (1.2.9), and (1.2.10)p. 24
Corollaries of Section 1.2 and Auxiliary Resultsp. 27
The Sears-Thomae Transformationp. 45
Introductionp. 45
Direct Corollaries of (2.1.1) and (2.1.3)p. 45
Extended Corollaries of (2.1.1) and (2.1.3)p. 46
Bilateral Seriesp. 53
Introductionp. 53
Backgroundp. 54
The 11 Identityp. 56
The 22 Identitiesp. 62
Identities Arising from the Quintuple Product Identityp. 68
Miscellaneous Bilateral Identitiesp. 74
Well-Poised Seriesp. 81
Introductionp. 81
Applications of (4.1.3)p. 82
Applications of Bailey's Formulasp. 89
Bailey's Lemma and Theta Expansionsp. 97
Introductionp. 97
The Main Lemmap. 97
Corollaries of (5.2.3)p. 99
Corollaries of (5.2.4) and Related Resultsp. 107
Partial Theta Functionsp. 113
Introductionp. 113
A General Identityp. 114
Consequences of Theorem 6.2.1p. 115
The function (a, q)p. 129
Euler's Identity and Its Extensionsp. 133
The Warnaar Theoryp. 141
Special Identitiesp. 149
Introductionp. 149
Generalized Modular Relationsp. 150
Extending Abel's Lemmap. 158
Innocents Abroadp. 165
Theta Function Identitiesp. 173
Introductionp. 173
Cubic Identitiesp. 175
Septic Identitiesp. 180
Ramanujan's Cubic Class Invariantp. 195
Introductionp. 195
n and the Modular j-Invariantp. 199
n and the Class Invariant Gnp. 203
n and Modular Equationsp. 204
n and Modular Equations in the Theory of Signature 3p. 208
n and Kronecker's Limit Formulap. 214
The Remaining Five Valuesp. 217
Some Modular Functions of Level 72p. 218
Computations of n Using the Shimura Reciprocity Lawp. 221
Miscellaneous Results on Elliptic Functions and Theta Functionsp. 225
A Quasi-theta Productp. 225
An Equivalent Formulation of (10.1.1) in Terms of Hyperbolic Seriesp. 226
Further Remarks on Ramanujan's Quasi-theta Productp. 231
A Generalization of the Dedekind Eta Functionp. 234
Two Entries on Page 346p. 238
A Continued Fractionp. 240
Class Invariantsp. 241
Formulas for the Power Series Coefficients of Certain Quotients of Eisenstein Seriesp. 243
Introductionp. 243
The Key Theoremp. 247
The Coefficients of 1/Q(q)p. 257
The Coefficients of Q(q)/R(q)p. 273
The Coefficients of ( P(q)/3)R(q) and ( P(q)/3)2/R(q)p. 280
The Coefficients of ( P(q)/2 &sqrt; 3)/Q(q)p. 284
Eight Identities for Eisenstein Series and Theta Functionsp. 287
The Coefficients of 1/B(q)p. 290
Formulas for the Coefficients of Further Eisenstein Seriesp. 298
The Coefficients of 1/B2(q)p. 300
A Calculation from [176]p. 312
Letters from Matlock Housep. 313
Introductionp. 313
A Lower Boundp. 314
An Upper Boundp. 322
Eisenstein Series and Modular Equationsp. 327
Introductionp. 327
Preliminary Resultsp. 328
Quintic Identities: First Methodp. 331
Quintic Identities: Second Methodp. 338
Septic Identitiesp. 345
Septic Differential Equationsp. 353
Series Representable in Terms of Eisenstein Seriesp. 355
Introductionp. 355
The Series T2k(q)p. 356
The Series Un(q)p. 362
Eisenstein Series and Approximations to p. 365
Introductionp. 365
Eisenstein Series and the Modular j-Invariantp. 366
Eisenstein Series and Equations in : First Methodp. 367
Eisenstein Series and Equations in : Second Methodp. 370
Page 213p. 375
Ramanujan's Series for 1/p. 375
Miscellaneous Results on Eisenstein Seriesp. 385
A generalization of Eisenstein Seriesp. 385
Representations of Eisenstein Series in Terms of Elliptic Function Parametersp. 386
Values of Certain Eisenstein Seriesp. 387
Some Elementary Identitiesp. 388
Location Guidep. 391
Provenancep. 397
Referencesp. 401
Indexp. 415
Table of Contents provided by Ingram. All Rights Reserved.

ISBN: 9780387777658
ISBN-10: 0387777652
Audience: Professional
Format: Hardcover
Language: English
Number Of Pages: 420
Published: December 2008
Publisher: Springer-Verlag New York Inc.
Country of Publication: US
Dimensions (cm): 23.5 x 15.5  x 2.54
Weight (kg): 0.73

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