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

 

• [http://www.springerlink.com/content/j22163577187156l/ Common extension of bilateral series for Andrews’ q-Bailey and q-Gauss sums

Wenchang Chu and Chenying Wang]

 

 

하위페이지

• 3 q-series
  
  • 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