Basic hypergeometric series
imported>Pythagoras0님의 2017년 12월 7일 (목) 22:57 판 (→하위페이지)
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
- Series[1/QPochhammer[q, q], {q, 0, 100}]
- Dedekind eta
- Series[QPochhammer[q, q], {q, 0, 100}]
q-hypergeometric series
\(\sum_{n\geq 0}^{\infty}\frac{q^{n^2/2}}{(q)_n}\sim \exp(\frac{\pi^2}{12t}-\frac{t}{48})\)
- 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
- 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]
- Basic hypergeometric 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