拉马努金遗失笔记（第四卷）

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)