"Basic hypergeometric series"의 두 판 사이의 차이
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Pythagoras0 (토론 | 기여) (→메타데이터: 새 문단) |
Pythagoras0 (토론 | 기여) |
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| 4번째 줄: | 4번째 줄: | ||
* 오일러공식<math>\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</math> | * 오일러공식<math>\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</math> | ||
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==q-Pochhammer== | ==q-Pochhammer== | ||
| 18번째 줄: | 18번째 줄: | ||
# Series[QPochhammer[q, q], {q, 0, 100}] | # Series[QPochhammer[q, q], {q, 0, 100}] | ||
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==q-hypergeometric series== | ==q-hypergeometric series== | ||
| 26번째 줄: | 26번째 줄: | ||
<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> | ||
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# 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 | # 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 | ||
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==KdV Hirota polynomials== | ==KdV Hirota polynomials== | ||
| 39번째 줄: | 39번째 줄: | ||
* [[KdV equation]] | * [[KdV equation]] | ||
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==related items== | ==related items== | ||
| 66번째 줄: | 66번째 줄: | ||
* [[Slater 98]] | * [[Slater 98]] | ||
* [[useful techniques in q-series]] | * [[useful techniques in q-series]] | ||
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==memo== | ==memo== | ||
* [http://www.springerlink.com/content/j22163577187156l/ Common extension of bilateral series for Andrews’ q-Bailey and q-Gauss sums Wenchang Chu and Chenying Wang] | * [http://www.springerlink.com/content/j22163577187156l/ Common extension of bilateral series for Andrews’ q-Bailey and q-Gauss sums Wenchang Chu and Chenying Wang] | ||
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==computational resource== | ==computational resource== | ||
* https://drive.google.com/file/d/1ko4taip_awmsywmG0oV3zbW7TnRtKyhb/view | * https://drive.google.com/file/d/1ko4taip_awmsywmG0oV3zbW7TnRtKyhb/view | ||
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2020년 12월 28일 (월) 05:22 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
- 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
computational resource
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