"Basic hypergeometric series"의 두 판 사이의 차이
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imported>Pythagoras0 잔글 (찾아 바꾸기 – “<h5>” 문자열을 “==” 문자열로) |
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| 1번째 줄: | 1번째 줄: | ||
| − | ==theory | + | ==theory== |
* [http://pythagoras0.springnote.com/pages/4145675 오일러의 오각수정리(pentagonal number theorem)]<br><math>(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</math><br> | * [http://pythagoras0.springnote.com/pages/4145675 오일러의 오각수정리(pentagonal number theorem)]<br><math>(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</math><br> | ||
| 8번째 줄: | 8번째 줄: | ||
| − | ==q-Pochhammer | + | ==q-Pochhammer== |
* partition generating function | * partition generating function | ||
| 22번째 줄: | 22번째 줄: | ||
| − | ==q-hypergeometric series | + | ==q-hypergeometric series== |
<math>\sum_{n\geq 0}^{\infty}\frac{q^{n^2/2}}{(q)_n}\sim \exp(\frac{\pi^2}{12t}-\frac{t}{48})</math> | <math>\sum_{n\geq 0}^{\infty}\frac{q^{n^2/2}}{(q)_n}\sim \exp(\frac{\pi^2}{12t}-\frac{t}{48})</math> | ||
| 34번째 줄: | 34번째 줄: | ||
| − | ==KdV Hirota polynomials | + | ==KdV Hirota polynomials== |
* Series[1/QPochhammer[q, q^2] - 1/QPochhammer[q^2, q^4], {q, 0, 100}] | * Series[1/QPochhammer[q, q^2] - 1/QPochhammer[q^2, q^4], {q, 0, 100}] | ||
| 43번째 줄: | 43번째 줄: | ||
| − | ==related items | + | ==related items== |
* [[asymptotic analysis of basic hypergeometric series]] | * [[asymptotic analysis of basic hypergeometric series]] | ||
2012년 10월 28일 (일) 15:20 판
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]