Cover image for Ramanujan's lost notebook
Title:
Ramanujan's lost notebook
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New York, NY : Springer, 2005
ISBN:
9780387255293
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30000004604843 QA29.R3 A52 2005 Open Access Book Book
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Summary

Summary

In the library at Trinity College, Cambridge in 1976, George Andrews of Pennsylvania State University discovered a sheaf of pages in the handwriting of Srinivasa Ramanujan. Soon designated as "Ramanujan's Lost Notebook," it contains considerable material on mock theta functions and undoubtedly dates from the last year of Ramanujan's life. In this book, the notebook is presented with additional material and expert commentary.


Table of Contents

Prefacep. ix
Introductionp. 1
1 The Rogers-Ramanujan Continued Fraction and Its Modular Propertiesp. 9
1.1 Introductionp. 9
1.2 Two-Variable Generalizations of (1.1.10) and (1.1.11)p. 13
1.3 Hybrids of (1.1.10) and (1.1.11)p. 18
1.4 Factorizations of (1.1.10) and (1.1.11)p. 21
1.5 Modular Equationsp. 24
1.6 Theta-Function Identities of Degree 5p. 26
1.7 Refinements of the Previous Identitiesp. 28
1.8 Identities Involving the Parameter k = R(q)R[superscript 2](q[superscript 2])p. 33
1.9 Other Representations of Theta Functions Involving R(q)p. 39
1.10 Explicit Formulas Arising from (1.1.11)p. 44
2 Explicit Evaluations of the Rogers-Ramanujan Continued Fractionp. 57
2.1 Introductionp. 57
2.2 Explicit Evaluations Using Eta-Function Identitiesp. 59
2.3 General Formulas for Evaluating R [characters not reproducible] and S [characters not reproducible]p. 66
2.4 Page 210 of Ramanujan's Lost Notebookp. 71
2.5 Some Theta-Function Identitiesp. 75
2.6 Ramanujan's General Explicit Formulas for the Rogers-Ramanujan Continued Fractionp. 79
3 A Fragment on the Rogers-Ramanujan and Cubic Continued Fractionsp. 85
3.1 Introductionp. 85
3.2 The Rogers-Ramanujan Continued Fractionp. 86
3.3 The Theory of Ramanujan's Cubic Continued Fractionp. 94
3.4 Explicit Evaluations of G(q)p. 100
4 The Rogers-Ramanujan Continued Fraction and Its Partitions and Lambert Seriesp. 107
4.1 Introductionp. 107
4.2 Connections with Partitionsp. 108
4.3 Further Identities Involving the Power Series Coefficients of C(q) and 1/C(q)p. 114
4.4 Generalized Lambert Seriesp. 116
4.5 Further q-Series Representations for C(q)p. 121
5 Finite Rogers-Ramanujan Continued Fractionsp. 125
5.1 Introductionp. 125
5.2 Finite Rogers-Ramanujan Continued Fractionsp. 126
5.3 A generalization of Entry 5.2.1p. 133
5.4 Class Invariantsp. 137
5.5 A Finite Generalized Rogers-Ramanujan Continued Fractionp. 140
6 Other q-continued Fractionsp. 143
6.1 Introductionp. 143
6.2 The Main Theoremp. 144
6.3 A Second General Continued Fractionp. 158
6.4 A Third General Continued Fractionp. 159
6.5 A Transformation Formulap. 162
6.6 Zerosp. 165
6.7 Two Entries on Page 200 of Ramanujan's Lost Notebookp. 169
6.8 An Elementary Continued Fractionp. 172
7 Asymptotic Formulas for Continued Fractionsp. 179
7.1 Introductionp. 179
7.2 The Main Theoremp. 181
7.3 Two Asymptotic Formulas Found on Page 45 of Ramanujan's Lost Notebookp. 187
7.4 An Asymptotic Formula for R(a,q)p. 193
8 Ramanujan's Continued Fraction for (q[superscript 2];q[superscript 3])[infinity]/(q;q[superscript 3])[infinity]p. 197
8.1 Introductionp. 197
8.2 A Proof of Ramanujan's Formula (8.1.2)p. 199
8.3 The Special Case a = w of (8.1.2)p. 210
8.4 Two Continued Fractions Related to (q[superscript 2];q[superscript 3])[infinity]/(q;q[superscript 3])[infinity]p. 213
8.5 An Asymptotic Expansionp. 214
9 The Rogers-Fine Identityp. 223
9.1 Introductionp. 223
9.2 Series Transformationsp. 223
9.3 The Series [characters not reproducible]p. 227
9.4 The Series [characters not reproducible]p. 232
9.5 The Series [characters not reproducible]p. 237
10 An Empirical Study of the Rogers-Ramanujan Identitiesp. 241
10.1 Introductionp. 241
10.2 The First Argumentp. 241
10.3 The Second Argumentp. 247
10.4 The Third Argumentp. 247
10.5 The Fourth Argumentp. 248
11 Rogers-Ramanujan-Slater-Type Identitiesp. 251
11.1 Introductionp. 251
11.2 Identities Associated with Modulus 5p. 252
11.3 Identities Associated with the Moduli 3, 6, and 12p. 253
11.4 Identities Associated with the Modulus 7p. 256
11.5 False Theta Functionsp. 256
12 Partial Fractionsp. 261
12.1 Introductionp. 261
12.2 The Basic Partial Fractionsp. 262
12.3 Applications of the Partial Fraction Decompositionsp. 265
12.4 Partial Fractions Plusp. 272
12.5 Related Identitiesp. 279
12.6 Remarks on the Partial Fraction Methodp. 284
13 Hadamard Products for Two q-Seriesp. 285
13.1 Introductionp. 285
13.2 Stieltjes-Wigert Polynomialsp. 286
13.3 The Hadamard Factorizationp. 288
13.4 Some Theta Seriesp. 289
13.5 A Formal Power Seriesp. 291
13.6 The Zeros of K[subscript infinity](zx)p. 295
13.7 Small Zeros of K[subscript infinity](z)p. 297
13.8 A New Polynomial Sequencep. 297
13.9 The Zeros of p[subscript n](a)p. 302
13.10 A Theta Function Expansionp. 304
13.11 Ramanujan's Product for p[subscript infinity](a)p. 305
14 Integrals of Theta Functionsp. 309
14.1 Introductionp. 309
14.2 Preliminary Resultsp. 310
14.3 The Identities on Page 207p. 314
14.4 Integral Representations of the Rogers-Ramanujan Continued Fractionp. 323
15 Incomplete Elliptic Integralsp. 327
15.1 Introductionp. 327
15.2 Preliminary Resultsp. 328
15.3 Two Simpler Integralsp. 330
15.4 Elliptic Integrals of Order 5 (I)p. 333
15.5 Elliptic Integrals of Order 5 (II)p. 339
15.6 Elliptic Integrals of Order 5 (III)p. 342
15.7 Elliptic Integrals of Order 15p. 349
15.8 Elliptic Integrals of Order 14p. 356
15.9 An Elliptic Integral of Order 35p. 361
15.10 Constructions of New Incomplete Elliptic Integral Identitiesp. 365
16 Infinite Integrals of q-Productsp. 367
16.1 Introductionp. 367
16.2 Proofsp. 368
17 Modular Equations in Ramanujan's Lost Notebookp. 373
17.1 Introductionp. 373
17.2 Eta-Function Identitiesp. 375
17.3 Summary of Modular Equations of Six Kindsp. 384
17.4 A Fragment on Page 349p. 392
18 Fragments on Lambert Seriesp. 395
18.1 Introductionp. 395
18.2 Entries from the Two Fragmentsp. 396
Location Guidep. 409
Provenancep. 415
Referencesp. 419
Indexp. 433