Introduction
1 The Rogers-Ramanujan Continued Fraction
and Its Modular Properties
1.1 Introduction
1.2 Two-Variable Generalizations of (1.1.10) and(1. 1.11) 13
1.3 Hybrids of(11.10)and(1.1.11)
1.4 Factorizations of(1.1.10) and(1. 1.11)
1.5 Modular equations
1.6 Theta-Function Identities of Degree 5
1.7 Refinements of the Previous Identities
1.8 Identities Involving the Parameter k=R(q)R(q2)
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
2.1 Introduction
2.2 Explicit Evaluations Using Eta-Function Identities
2.3 General Formulas for Evaluating R(e-2mVn) and S(e-TVn).66
2.4 Page 210 of Ramanujan's Lost Notebook
2.5 Some Theta-Function Identities
2.6 Ramanujans General Explicit Formulas for the
Rogers-Ramanujan Continued Fraction 79
3 A Fragment on the Rogers-Ramanujan and Cubic
Continued fractions
3.1 Introduction
3.23 The RogersTheory-RamanujofanujaContinuedsCubicFractionContinued Fraction,...86
3.4 Explicit Evaluations of G(a)
4 Rogers-Ramanujan Continued Fraction- Partitions,
Lambert series
4.1 Introduction.....,,,,.,...........
4.2 Connections with Partitions
4.3 Further Identities Involving the Power Series Coefficients of
C(q)and1/C(q)……
4.4 Generalized Lambert Series
4.5 Further g-Series Representations for C(a)
5 Finite Rogers-Ramanujan Continued Fractions......
125
5.1 Introduction.........
5.2 Finite Rogers-Ramanujan Continued Fractions......
126
53 A generalization of Entry5.2.1..………∵
5.4 Class invariant
5.5 A Finite Generalized Rogers-Ramanujan Continued Fraction 140
6 Other q-continued fractions
6.1 Introduction
6.2
The Main Theore
6.3
A Second General Continued Fraction
6.4 A Third General Continued Fraction...........
159
6.5 A Transformation Formula
6.6 Zeros................
,165
6.7 Two Entries on Page 200 of Ramanujan's Lost Notebook.. 169
6.8 An Elementary Continued Fraction
7 Asymptotic Formulas for Continued Fractions
7.1 Introduction
7.2 The Main Theorem
7.3 Two Asymptotic Formulas Found on Page 45 of
Ramanujans Lost Notebook
7.4 An Asymptotic Formula for R(a, q)
8 Ramanujan,s Continued Fraction for(q
8.1 Introduction
8.2 A Proof of Ramanujan's Formula(8.1.2)
3 The Special Case a= w of(8.1.2)
8.4 Two Continued Fractions Related to(q; q)oo/(q; oo... 213
8.5 An Asymptotic Expansion
9 The Rogers-Fine Identity
1 Introduction........
9.2
Series Transformations
9.3
The Series
nan(n 1)/2
n=09n(3n 1)/2
9. 4 The Series
9.5 The Series
n=o gun
2n
10 An Empirical Study of the Rogers-Ramanujan Identities. 241
10.1 Introduction.......,,,,,∴,.241
10.2 The First Argument
10.3 The Second Argument
10.4 The Third Argument
10.5 The Fourth Argument
11 Rogers-Ramanujan-Slater-Type Identities
........ 251
11.1 Introduction.
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
11.5 False Theta Functions
12 Partial fractions..,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,..261
12.1 Introduction.,,,,,,,,,,,,,,,,,,,,,,,....261
12.2 The Basic Partial Fractions
12.3 Applications of the Partial Fraction Decompositions
12.4 Partial Fractions Plus
12.5 Related Identities ......................................
279
12.6 Remarks on the Partial Fraction Method
13.2 Stieltjes-Wigert Polynomial............
285
13 Hadamard Products for Two q-Series
13.1 Introduction
13.3 The Hadamard Factorization..............
288
13. 4 Some Theta series
13.5 a Formal Power Series..,,,,,,,,,,,,,,,...,291
136 The Zeros of K。(2x)
13.7 Small Zeros of Koo(z)
13.8 A New Polynomial Sequence
13.9 The Zeros of pn(a)
13.10 A Theta Function Expansion
13.11 Ramanujan's Product for poo(a)
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