目录 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 Integ
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