Ramanujan's Lost Notebook: Part I

437 Pages·2005·2.33 MB·English

by  George E. Andrews

Ramanujan’s Lost Notebook
Part I
George E. Andrews Bruce C. Berndt
Ramanujan’s Lost Notebook
Part I
GeorgeE.Andrews BruceBerndt
DepartmentofMathematics DepartmentofMathematics
EberlyCollegeofScience UniversityofIllinois,Urbana-Champaign
PennsylvaniaStateUniversity Urbana,IL61801
StateCollege,PA16802 USA
USA [email protected]
[email protected]
MathematicsSubjectClassification:14H42,33D15,33C75,40A15
LibraryofCongressControlNumber:2005923547
ISBN-10:0-387-25529-X Printedonacid-freepaper.
ISBN-13:978-0387-25529-3
©2005SpringerScience+BusinessMedia,Inc.
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Readers will learn in the introduction to this volume that mathemati-
cians owe a huge debt to R.A. Rankin and J.M. Whittaker for their efforts
in preserving Ramanujan’s “Lost Notebook.” If it were not for them, Ra-
manujan’slostnotebooklikelywouldhavebeenpermanentlylost.Rankinwas
borninGarlieston,Scotland,inOctober1915anddiedinGlasgowinJanuary
2001. For several years he was professor of Mathematics at the University of
Glasgow. An account of his life and work has been given by B.C. Berndt,
W. Kohnen, and K. Ono in [79]. Whittaker was born in March 1905 in Cam-
bridge and died in Sheffield in January 1984. At his retirement, he was vice-
chancellor of Sheffield University. A description of Whittaker’s life and work
has been written by W.K. Hayman [150].
Through long lapse of time,
This knowledge was lost.
But now, as you are devoted to truth,
I will reveal the supreme secret.
Bhagavad Gita, IV.2 & IV.3
Preface
This volume is the first of approximately four volumes devoted to the exami-
nation of all claims made by Srinivasa Ramanujan in The Lost Notebook and
OtherUnpublishedPapers.ThispublicationcontainsRamanujan’sfamouslost
notebook; copies of unpublished manuscripts in the Oxford library, in partic-
ular, his famous unpublished manuscript on the partition function and the
tau-function; fragments of both published and unpublished papers; miscella-
neous 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.
Contents
Preface ........................................................ ix
Introduction................................................... 1
1 The Rogers–Ramanujan Continued Fraction
and Its Modular Properties................................ 9
1.1 Introduction ........................................... 9
1.2 Two-Variable Generalizations of (1.1.10) and (1.1.11) ....... 13
1.3 Hybrids of (1.1.10) and (1.1.11) .......................... 18
1.4 Factorizations of (1.1.10) and (1.1.11)..................... 21
1.5 Modular Equations ..................................... 24
1.6 Theta-Function Identities of Degree 5 ..................... 26
1.7 Refinements of the Previous Identities..................... 28
1.8 Identities Involving the Parameter k =R(q)R2(q2).......... 33
1.9 Other Representations of Theta Functions Involving R(q) ... 39
1.10 Explicit Formulas Arising from (1.1.11) ................... 44
2 Explicit Evaluations of the Rogers–Ramanujan Continued
Fraction ................................................... 57
2.1 Introduction ........................................... 57
2.2 Explicit Evaluations Using Eta-Function Identities.......... 59
√ √
2.3 General Formulas for Evaluating R(e−2π n) and S(e−π n) .. 66
2.4 Page 210 of Ramanujan’s Lost Notebook .................. 71
2.5 Some Theta-Function Identities .......................... 75
2.6 Ramanujan’s General Explicit Formulas for the
Rogers–Ramanujan Continued Fraction ................... 79
3 A Fragment on the Rogers–Ramanujan and Cubic
Continued Fractions ....................................... 85
3.1 Introduction ........................................... 85
xii Contents
3.2 The Rogers–Ramanujan Continued Fraction ............... 86
3.3 The Theory of Ramanujan’s Cubic Continued Fraction...... 94
3.4 Explicit Evaluations of G(q) .............................100
4 Rogers–Ramanujan Continued Fraction – Partitions,
Lambert Series ............................................107
4.1 Introduction ...........................................107
4.2 Connections with Partitions .............................108
4.3 Further Identities Involving the Power Series Coefficients of
C(q) and 1/C(q) .......................................114
4.4 Generalized Lambert Series..............................116
4.5 Further q-Series Representations for C(q)..................121
5 Finite Rogers–Ramanujan Continued Fractions ............125
5.1 Introduction ...........................................125
5.2 Finite Rogers–Ramanujan Continued Fractions.............126
5.3 A generalization of Entry 5.2.1...........................133
5.4 Class Invariants ........................................137
5.5 A Finite Generalized Rogers–Ramanujan Continued Fraction 140
6 Other q-continued Fractions ...............................143
6.1 Introduction ...........................................143
6.2 The Main Theorem.....................................144
6.3 A Second General Continued Fraction.....................158
6.4 A Third General Continued Fraction......................159
6.5 A Transformation Formula ..............................162
6.6 Zeros .................................................165
6.7 Two Entries on Page 200 of Ramanujan’s Lost Notebook ....169
6.8 An Elementary Continued Fraction .......................172
7 Asymptotic Formulas for Continued Fractions .............179
7.1 Introduction ...........................................179
7.2 The Main Theorem.....................................181
7.3 Two Asymptotic Formulas Found on Page 45 of
Ramanujan’s Lost Notebook .............................187
7.4 An Asymptotic Formula for R(a,q) .......................193
8 Ramanujan’s Continued Fraction for (q2;q3)∞/(q;q3)∞....197
8.1 Introduction ...........................................197
8.2 A Proof of Ramanujan’s Formula (8.1.2) ..................199
8.3 The Special Case a=ω of (8.1.2).........................210
8.4 Two Continued Fractions Related to (q2;q3)∞/(q;q3)∞ .....213
8.5 An Asymptotic Expansion...............................214
Contents xiii
9 The Rogers–Fine Identity..................................223
9.1 Introduction ...........................................223
9.2 Series Tran(cid:1)sformations ..................................223
99..43 TThhee SSeerriieess (cid:1)∞n∞=0(q−n(13)nn+q1n)(/n2+(11)−/2q.2.n.+.1.).........................................223227
(cid:1)n=0
9.5 The Series ∞ q3n2+2n(1−q2n+1) ......................237
n=0
10 An Empirical Study of the Rogers–Ramanujan Identities ..241
10.1 Introduction ...........................................241
10.2 The First Argument ....................................241
10.3 The Second Argument ..................................247
10.4 The Third Argument ...................................247
10.5 The Fourth Argument ..................................248
11 Rogers–Ramanujan–Slater–Type Identities ................251
11.1 Introduction ...........................................251
11.2 Identities Associated with Modulus 5 .....................252
11.3 Identities Associated with the Moduli 3, 6, and 12..........253
11.4 Identities Associated with the Modulus 7..................256
11.5 False Theta Functions ..................................256
12 Partial Fractions...........................................261
12.1 Introduction ...........................................261
12.2 The Basic Partial Fractions..............................262
12.3 Applications of the Partial Fraction Decompositions ........265
12.4 Partial Fractions Plus...................................272
12.5 Related Identities ......................................279
12.6 Remarks on the Partial Fraction Method ..................284
13 Hadamard Products for Two q-Series......................285
13.1 Introduction ...........................................285
13.2 Stieltjes–Wigert Polynomials.............................286
13.3 The Hadamard Factorization ............................288
13.4 Some Theta Series......................................289
13.5 A Formal Power Series ..................................291
13.6 The Zeros of K∞(zx) ...................................295
13.7 Small Zeros of K∞(z)...................................297
13.8 A New Polynomial Sequence.............................297
13.9 The Zeros of pn(a) .....................................302
13.10 A Theta Function Expansion ............................304
13.11 Ramanujan’s Product for p∞(a)..........................305