Basic hypergeometric series

theory

• 오일러의 오각수정리(pentagonal number theorem)
  \((1-x)(1-x^2)(1-x^3) \cdots = 1 - x - x^2 + x^5 + x^7 - x^{12} - x^{15} + x^{22} + x^{26} + \cdots\)
  
• 오일러공식
  \(\prod_{n=0}^{\infty}(1+zq^n)=\sum_{n\geq 0}\frac{q^{n(n-1)/2}}{(1-q)(1-q^2)\cdots(1-q^n)} z^n\)
  

 

 

q-Pochhammer

• partition generating function

1. Series[1/QPochhammer[q, q], {q, 0, 100}]

• Dedekind eta

1. Series[QPochhammer[q, q], {q, 0, 100}]

 

 

q-hypergeometric series

\(\sum_{n\geq 0}^{\infty}\frac{q^{n^2/2}}{(q)_n}\sim \exp(\frac{\pi^2}{12t}-\frac{t}{48})\)

 

1. f[q_] := QHypergeometricPFQ[{}, {}, q, -q^(1/2)]
   g[q_] := Exp[-(Pi^2/(12 Log[q])) + Log[q]/48]
   Table[N[f[1 - 1/10^(i)]/g[1 - 1/10^(i)], 50], {i, 1, 5}] // TableForm

 

 

KdV Hirota polynomials

• Series[1/QPochhammer[q, q^2] - 1/QPochhammer[q^2, q^4], {q, 0, 100}]
• KdV equation

 

 

related items

• asymptotic analysis of basic hypergeometric series
• hypergeometric functions and representation theory
• Bailey pair and lemma
• Bailey lattice
• sources of Bailey pairs
• determinantal identities and Airy kernel
• elliptic hypergeometric series
• finitized q-series identity
• integer partitions
• q-analogue of summation formulas
• Slater list
• Slater 31
• Slater 32
• Slater 34
• Slater 36
• Slater 37
• Slater 47
• Slater 83
• Slater 86
• Slater 92
• Slater 98
• useful techniques in q-series

 

==memo

• Common extension of bilateral series for Andrews’ q-Bailey and q-Gauss sums Wenchang Chu and Chenying Wang

 

computational resource

• https://drive.google.com/file/d/1ko4taip_awmsywmG0oV3zbW7TnRtKyhb/view