目錄
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)
1.3 Hybrids of (1.1.10) 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)R2(q2)
1.9 Other Representations of Theta Functions Involving R(q)
1.10 Explicit Formulas Arising from (1.1.11 )
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 and S
2.4 Page 210 of Ramanujan's Lost Notebook
2.5 Some Theta-Function Identities
2.6 Ramanujan's General Explicit Formulas for the Rogers-Ramanujan Continued Fraction
3 A Fragment on the Rogers-Ramanujan and Cubic Continued Fractions
3.1 Introduction
3.2 The Rogers-Ramanujan Continued Fraction
3.3 The Theory of Ramanujan's Cubic Continued Fraction
3.4 Explicit Evaluations of G(q)
4 Rogers-Ramanujan Continued Fraction - Partitions,Lambert Series
4.1 Introduction
4.2 Connections with Partitions
4.3 Fhrther Identities Involving the Power Series Coefficients of C(q) and 1/C(q)
4.4 Generalized Lambert Series
4.5 Further q-Series Representations for C(q)
5 Finite Rogers-Ramanujan Continued Fractions
5.1 Introduction
5.2 Finite Rogers-Ramanujan Continued Fractions
5.4 Class Invariants
5.5 A Finite Generalized Rogers-Ramanujan Continued Fraction
6 Other q-continued Fractions
6.1 Introduction
6.2 The Main Theorem
6.3 A Second General Continued Fraction
6.4 A Third General Continued Fraction
6.5 A Transformation Formula
6.6 Zeros
6.7 Two Entries on Page 200 of Ramanujan's Lost Notebook
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 Ramanujan's Lost Notebook
7.4 An Asymptotic Formula for R(a, q)
8 Ramanujan's Continued Fraction for (q2; q3)/(q; q3)
8.1 Introduction
8.2 A Proof of Ramanujan's Formula (8.1 :2)
8.3 The Special Case a = w of (8.1.2 )
8.4 Two Continued Fractions Related to (q2; q3)/(q; q3)
8.5 An Asymptotic Expansion
9 The Rogers-Fine Identity
10 An Empirical Study of the Rogers-Ramanujan Identities .
10.1 Introduction
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
11.1 Introduction
11.2 Identities Associated with Modulus 5
11.3 Identities Associated with the Moduli 3, 6, and 12
11.4 Identities Associated with the Modulus 7
11.5 False Theta Functions
12 Partial Fractions
12.1 Introduction
12.2 The Basic Partial Fractions
12.3 Applications of the Partial Fraction Decompositions
12.4 Partial Fractions Plus
12.5 Related Identities
12.6 Remarks on the Partial Fraction Method
13 Hadamard Products for Two q-Series
13.1 Introduction
13.2 Stieltjes-Wigert Polynomials
13.3 The Hadamard Factorization
13.4 Some Theta Series
13.5 A Formal Power Series
13.6 The Zeros of K~(zx)
13.7 Small Zeros of K~(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 p~(a)
14 Integrals of Theta Functions
14.1 Introduction
14.2 Preliminary Results
14.3 The Identities on Page 207
14.4 Integral Representations of the Rogers-Ramanujan Continued Fraction
15 Incomplete Elliptic Integrals
15.1 Introduction
15.2 Preliminary Results
15.3 Two Simpler Integrals
15.4 Elliptic Integrals of Order 5 (I)
15.5 Elliptic Integrals of Order 5 (II)
15.6 Elliptic Integrals of Order 5 (III)
15.7 Elliptic Integrals of Order 15
15.8 Elliptic Integrals of Order 14
15.9 An Elliptic Integral of Order 35
15.10 Constructions of New Incomplete Elliptic Integral Identities
16 Infinite Integrals of q-Products
16.1 Introduction
16.2 Proofs
17 Modular Equations in Ramanujan's Lost Notebook
17.1 Introduction
17.2 Eta-Function Identities
17.3 Summary of Modular Equations of Six Kinds
17.4 A Fragment on Page 349
18 Fragments on Lambert Series
18.1 Introduction
18.2 Entries from the Two Fragments
Location Guide
Provenance
References
Index
附錄I 拉馬努金的中國知音：數學家劉治國的「西天取經」之旅
附錄II 劉治國教授訪談
編輯手記
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