Ramanujan's Lost Notebook

by Andrews, George E.; Berndt, Bruce C.

ISBN13: 9780387255293

ISBN10: 038725529X

Format: Hardcover

Pub. Date: 2005-05-06

Publisher(s): Springer Verlag

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Summary

This volume is the first of approximately four volumes devoted to providing statements, proofs, and discussions of all the claims made by Srinivasa Ramanujan in his lost notebook and all his other manuscripts and letters published with the lost notebook. In addition to the lost notebook, this publication contains copies of unpublished manuscripts in the Oxford library, in particular, his famous unpublished manuscript on the partition and tau-functions; fragments of both published and unpublished papers; miscellaneous sheets; and Ramanujan's letters to G. H. Hardy, written from nursing homes during Ramanujan's final two years in England. This volume contains accounts of 442 entries (counting multiplicities) made by Ramanujan in the aforementioned publication. The present authors have organized these claims into eighteen chapters, containing anywhere from two entries in Chapter 13 to sixty-one entries in Chapter 17.

Preface

Introduction

(8)

The Rogers--Ramanujan Continued Fraction and Its Modular Properties

(48)

Introduction

(4)

Two-Variable Generalizations of (1.1.10) and (1.1.11)

(5)

Hybrids of (1.1.10) and (1.1.11)

(3)

Factorizations of (1.1.10) and (1.1.11)

(3)

Modular Equations

(2)

Theta-Function Identities of Degree 5

(2)

Refinements of the Previous Identities

(5)

Identities Involving the Parameter k = R(q)R2(q2)

(6)

Other Representations of Theta Functions Involving R(q)

(5)

Explicit Formulas Arising from (1.1.11)

(13)

Explicit Evaluations of the Rogers--Ramanujan Continued Fraction

(28)

Introduction

(2)

Explicit Evaluations Using Eta-Function Identities

(7)

General Formulas for Evaluating R(e-2π√n) and S(e-π√n)

(5)

Page 210 of Ramanujan's Lost Notebook

(4)

Some Theta-Function Identities

(4)

Ramanujan's General Explicit Formulas for the Rogers--Ramanujan Continued Fraction

(6)

A Fragment on the Rogers--Ramanujan and Cubic Continued Fractions

(22)

Introduction

(1)

The Rogers-Ramanujan Continued Fraction

(8)

The Theory of Ramanujan's Cubic Continued Fraction

(6)

Explicit Evaluations of G(q)

100

(7)

The Rogers-Ramanujan Continued Fraction and Its Partitions and Lambert Series

107

(18)

Introduction

107

(1)

Connections with Partitions

108

(6)

Further Identities Involving the Power Series Coefficients of C(q) and 1/C(q)

114

(2)

Generalized Lambert Series

116

(5)

Further q-Series Representations for C(q)

121

(4)

Finite Rogers-Ramanujan Continued Fractions

125

(18)

Introduction

125

(1)

Finite Rogers--Ramanujan Continued Fractions

126

(7)

A generalization of Entry 5.2.1

133

(4)

Class Invariants

137

(3)

A Finite Generalized Rogers--Ramanujan Continued Fraction

140

(3)

Other q-continued Fractions

143

(36)

Introduction

143

(1)

The Main Theorem

144

(14)

A Second General Continued Fraction

158

(1)

A Third General Continued Fraction

159

(3)

A Transformation Formula

162

(3)

Zeros

165

(4)

Two Entries on Page 200 of Ramanujan's Lost Notebook

169

(3)

An Elementary Continued Fraction

172

(7)

Asymptotic Formulas for Continued Fractions

179

(18)

Introduction

179

(2)

The Main Theorem

181

(6)

Two Asymptotic Formulas Found on Page 45 of Ramanujan's Lost Notebook

187

(6)

An Asymptotic Formula for R(a, q)

193

(4)

Ramanujan's Continued Fraction for (q2; q3)∞/(q; q3)∞

197

(26)

Introduction

197

(2)

A Proof of Ramanujan's Formula (8.1.2)

199

(11)

The Special Case a = ω of (8.1.2)

210

(3)

Two Continued Fractions Related to (q2;q3)∞/(q; q3)∞

213

(1)

An Asymptotic Expansion

214

(9)

The Rogers-Fine Identity

223

(18)

Introduction

223

(1)

Series Transformations

223

(4)

The Series Σ∞n=0(-1)n qn(n+1)/2

227

(5)

The Series Σ∞n=0v qn(3n+1)/2(1-q2n+1)

232

(5)

The Series Σ∞n=0 q3n2+2n (1 - q2n+1)

237

(4)

An Empirical Study of the Rogers--Ramanujan Identities

241

(10)

Introduction

241

(1)

The First Argument

241

(6)

The Second Argument

247

(1)

The Third Argument

247

(1)

The Fourth Argument

248

(3)

Rogers--Ramanujan--Slater--Type Identities

251

(10)

Introduction

251

(1)

Identities Associated with Modulus 5

252

(1)

Identities Associated with the Moduli 3, 6, and 12

253

(3)

Identities Associated with the Modulus 7

256

(1)

False Theta Functions

256

(5)

Partial Fractions

261

(24)

Introduction

261

(1)

The Basic Partial Fractions

262

(3)

Applications of the Partial Fraction Decompositions

265

(7)

Partial Fractions Plus

272

(7)

Related Identities

279

(5)

Remarks on the Partial Fraction Method

284

(1)

Hadamard Products for Two q-Series

285

(24)

Introduction

285

(1)

Stieltjes-Wigert Polynomials

286

(2)

The Hadamard Factorization

288

(1)

Some Theta Series

289

(2)

A Formal Power Series

291

(4)

The Zeros of K∞ (zx)

295

(2)

Small Zeros of K∞ (z)

297

(1)

A New Polynomial Sequence

297

(5)

The Zeros of Pn (a)

302

(2)

A Theta Function Expansion

304

(1)

Ramanujan's Product for p∞(a)

305

(4)

Integrals of Theta Functions

309

(18)

Introduction

309

(1)

Preliminary Results

310

(4)

The Identities on Page 207

314

(9)

Integral Representations of the Rogers-Ramanujan Continued Fraction

323

(4)

Incomplete Elliptic Integrals

327

(40)

Introduction

327

(1)

Preliminary Results

328

(2)

Two Simpler Integrals

330

(3)

Elliptic Integrals of Order 5 (I)

333

(6)

Elliptic Integrals of Order 5 (II)

339

(3)

Elliptic Integrals of Order 5 (III)

342

(7)

Elliptic Integrals of Order 15

349

(7)

Elliptic Integrals of Order 14

356

(5)

An Elliptic Integral of Order 35

361

(4)

Constructions of New Incomplete Elliptic Integral Identities

365

(2)

Infinite Integrals of q-Products

367

(6)

Introduction

367

(1)

Proofs

368

(5)

Modular Equations in Ramanujan's Lost Notebook

373

(22)

Introduction

373

(2)

Eta-Function Identities

375

(9)

Summary of Modular Equations of Six Kinds

384

(8)

A Fragment on Page 349

392

(3)

Fragments on Lambert Series

395

(14)

Introduction

395

(1)

Entries from the Two Fragments

396

(13)

Location Guide

409

(6)

Provenance

415

(4)

References

419

(14)

Index

433

Ramanujan's Lost Notebook

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