"초기하급수의 합공식"의 두 판 사이의 차이
Pythagoras0 (토론 | 기여) |
Pythagoras0 (토론 | 기여) |
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==개요== | ==개요== | ||
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==Chu-Vandermonde 공식== | ==Chu-Vandermonde 공식== | ||
| 7번째 줄: | 7번째 줄: | ||
<math>\,_2F_1(-n,b;c;1)=\dfrac{(c-b)_{n}}{(c)_{n}}</math> | <math>\,_2F_1(-n,b;c;1)=\dfrac{(c-b)_{n}}{(c)_{n}}</math> | ||
| − | 아래 가우스 | + | 아래 가우스 공식에서 <math>a=-n</math>인 경우에 얻어진다 |
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==가우스 공식== | ==가우스 공식== | ||
| 20번째 줄: | 20번째 줄: | ||
\frac{\Gamma(\frac{1}{2})\Gamma(\frac{1}{2}+\frac{a}{2}+\frac{b}{2})}{\Gamma(\frac{1}{2}+\frac{a}{2})\Gamma(\frac{1}{2}+\frac{b}{2})}</math> | \frac{\Gamma(\frac{1}{2})\Gamma(\frac{1}{2}+\frac{a}{2}+\frac{b}{2})}{\Gamma(\frac{1}{2}+\frac{a}{2})\Gamma(\frac{1}{2}+\frac{b}{2})}</math> | ||
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| − | == | + | == 쿰머 공식== |
| − | + | <math>\,_2F_1(a,b;1+a-b;-1)=\dfrac{\Gamma(1+a-b)\,\Gamma(1+\frac{1}{2}a)}{\Gamma(1+a)\Gamma(1+\frac{1}{2}a-b)}</math> | |
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==딕슨 공식== | ==딕슨 공식== | ||
| 38번째 줄: | 38번째 줄: | ||
{\Gamma(1+a)\Gamma(1+a-b-c)\Gamma(1+a/2-b)\Gamma(1+a/2-c)}</math> | {\Gamma(1+a)\Gamma(1+a-b-c)\Gamma(1+a/2-b)\Gamma(1+a/2-c)}</math> | ||
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| − | == | + | == Bailey 공식== |
<math>\;_2F_1 \left(a,1-a;c;\frac{1}{2}\right)= | <math>\;_2F_1 \left(a,1-a;c;\frac{1}{2}\right)= | ||
\frac{\Gamma(\frac{c}{2})\Gamma(\frac{1}{2}+\frac{c}{2})}{\Gamma(\frac{c}{2}+\frac{a}{2})\Gamma(\frac{1}{2}+\frac{c}{2}-\frac{a}{2})}</math> | \frac{\Gamma(\frac{c}{2})\Gamma(\frac{1}{2}+\frac{c}{2})}{\Gamma(\frac{c}{2}+\frac{a}{2})\Gamma(\frac{1}{2}+\frac{c}{2}-\frac{a}{2})}</math> | ||
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==Pfaff 공식== | ==Pfaff 공식== | ||
| − | + | <math>\,_3F_2(a,b,-n;c,1+a+b-c-n;1)=\dfrac{(c-a)_{n}(c-b)_{n}}{(c)_{n}(c-a-b)_{n}}</math> | |
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==Dougall 공식== | ==Dougall 공식== | ||
| 69번째 줄: | 69번째 줄: | ||
http://en.wikipedia.org/wiki/Bilateral_hypergeometric_series#Dougall.27s_bilateral_sum | http://en.wikipedia.org/wiki/Bilateral_hypergeometric_series#Dougall.27s_bilateral_sum | ||
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==역사== | ==역사== | ||
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* http://www.google.com/search?hl=en&tbs=tl:1&q= | * http://www.google.com/search?hl=en&tbs=tl:1&q= | ||
* [[수학사 연표]] | * [[수학사 연표]] | ||
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==메모== | ==메모== | ||
| − | * | + | * [http://www.mathematik.uni-kassel.de/%7Ekoepf/hyper.html http://www.mathematik.uni-kassel.de/~koepf/hyper.html] |
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==관련된 항목들== | ==관련된 항목들== | ||
| 95번째 줄: | 95번째 줄: | ||
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| − | ==사전 | + | ==사전 형태의 자료== |
* http://ko.wikipedia.org/wiki/ | * http://ko.wikipedia.org/wiki/ | ||
2020년 12월 28일 (월) 03:58 판
개요
Chu-Vandermonde 공식
\(\,_2F_1(-n,b;c;1)=\dfrac{(c-b)_{n}}{(c)_{n}}\)
아래 가우스 공식에서 \(a=-n\)인 경우에 얻어진다
가우스 공식
\(\,_2F_1(a,b;c;1)=\dfrac{\Gamma(c)\,\Gamma(c-a-b)}{\Gamma(c-a)\Gamma(c-b)}\)
\(\;_2F_1 \left(a,b;\frac{1}{2}+\frac{a}{2}+\frac{b}{2};\frac{1}{2}\right) = \frac{\Gamma(\frac{1}{2})\Gamma(\frac{1}{2}+\frac{a}{2}+\frac{b}{2})}{\Gamma(\frac{1}{2}+\frac{a}{2})\Gamma(\frac{1}{2}+\frac{b}{2})}\)
쿰머 공식
\(\,_2F_1(a,b;1+a-b;-1)=\dfrac{\Gamma(1+a-b)\,\Gamma(1+\frac{1}{2}a)}{\Gamma(1+a)\Gamma(1+\frac{1}{2}a-b)}\)
딕슨 공식
\(\;_3F_2 (a,b,c;1+a-b,1+a-c;1)= \frac{\Gamma(1+a/2)\Gamma(1+a/2-b-c)\Gamma(1+a-b)\Gamma(1+a-c)} {\Gamma(1+a)\Gamma(1+a-b-c)\Gamma(1+a/2-b)\Gamma(1+a/2-c)}\)
Bailey 공식
\(\;_2F_1 \left(a,1-a;c;\frac{1}{2}\right)= \frac{\Gamma(\frac{c}{2})\Gamma(\frac{1}{2}+\frac{c}{2})}{\Gamma(\frac{c}{2}+\frac{a}{2})\Gamma(\frac{1}{2}+\frac{c}{2}-\frac{a}{2})}\)
Pfaff 공식
\(\,_3F_2(a,b,-n;c,1+a+b-c-n;1)=\dfrac{(c-a)_{n}(c-b)_{n}}{(c)_{n}(c-a-b)_{n}}\)
Dougall 공식
http://dx.doi.org/10.1016/0022-247X(90)90375-P
\({}_2H_2(a,b;c,d;1)= \sum_{-\infty}^\infty\frac{(a)_n(b)_n}{(c)_n(d)_n}= \frac{\Gamma(d)\Gamma(e)\Gamma(1-a)\Gamma(1-b)\Gamma(c+d-a-b-1)}{\Gamma(c-a)\Gamma(c-b)\Gamma(d-a)\Gamma(d-b)} \)
http://en.wikipedia.org/wiki/Bilateral_hypergeometric_series#Dougall.27s_bilateral_sum
역사
메모
관련된 항목들
사전 형태의 자료
- http://ko.wikipedia.org/wiki/
- http://en.wikipedia.org/wiki/Vandermonde's_identity
- http://en.wikipedia.org/wiki/
- http://mathworld.wolfram.com/HypergeometricSummation.html
관련논문
- Dunkl, Charles F., and George Gasper. “The Sums of a Double Hypergeometric Series and of the First m+1 Terms of 3F2(a,b,c;(a+b+1)/2,2c;1) When c = -M Is a Negative Integer.” arXiv:1412.4022 [math], December 12, 2014. http://arxiv.org/abs/1412.4022.
- Wang, Chenying, and Xiaojing Chen. ‘A New Proof for Gasper’s Nonterminating Cubic \(_7F_6\)-Series Summation Identity’. arXiv:1410.5636 [math], 21 October 2014. http://arxiv.org/abs/1410.5636.
- Vyas, Yashoverdhan, and Kalpana Fatawat. “Extensions of the Classical Theorems for Very Well-Poised Hypergeometric Functions.” arXiv:1410.3241 [math], October 13, 2014. http://arxiv.org/abs/1410.3241.