"초기하급수의 합공식"의 두 판 사이의 차이

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==개요==
  
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==Chu-Vandermonde 공식==
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<math>\,_2F_1(-n,b;c;1)=\dfrac{(c-b)_{n}}{(c)_{n}}</math>
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아래 가우스 공식에서 <math>a=-n</math>인 경우에 얻어진다
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==가우스 공식==
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<math>\,_2F_1(a,b;c;1)=\dfrac{\Gamma(c)\,\Gamma(c-a-b)}{\Gamma(c-a)\Gamma(c-b)}</math>
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<math>\;_2F_1 \left(a,b;\frac{1}{2}+\frac{a}{2}+\frac{b}{2};\frac{1}{2}\right) =
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\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==
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<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==
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<math>\;_3F_2 (a,b,c;1+a-b,1+a-c;1)=
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\frac{\Gamma(1+a/2)\Gamma(1+a/2-b-c)\Gamma(1+a-b)\Gamma(1+a-c)}
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{\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==
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<math>\;_2F_1 \left(a,1-a;c;\frac{1}{2}\right)=
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\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==
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<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==
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[http://dx.doi.org/10.1016/0022-247X%2890%2990375-P http://dx.doi.org/10.1016/0022-247X(90)90375-P]
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<math>{}_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)} </math>
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http://en.wikipedia.org/wiki/Bilateral_hypergeometric_series#Dougall.27s_bilateral_sum
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==역사==
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* http://www.google.com/search?hl=en&tbs=tl:1&q=
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* [[수학사 연표]]
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==메모==
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*  [http://www.mathematik.uni-kassel.de/%7Ekoepf/hyper.html http://www.mathematik.uni-kassel.de/~koepf/hyper.html]
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==관련된 항목들==
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* [[로그 사인 적분 (log sine integrals)]]
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==사전 형태의 자료==
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* http://ko.wikipedia.org/wiki/
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* [http://en.wikipedia.org/wiki/Vandermonde%27s_identity http://en.wikipedia.org/wiki/Vandermonde's_identity]
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* http://en.wikipedia.org/wiki/
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* http://mathworld.wolfram.com/HypergeometricSummation.html
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==관련논문==
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* 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.
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* Wang, Chenying, and Xiaojing Chen. ‘A New Proof for Gasper’s Nonterminating Cubic <math>_7F_6</math>-Series Summation Identity’. arXiv:1410.5636 [math], 21 October 2014. http://arxiv.org/abs/1410.5636.
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* 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.
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==메타데이터==
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===위키데이터===
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* ID :  [https://www.wikidata.org/wiki/Q4907583 Q4907583]
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===Spacy 패턴 목록===
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* [{'LOWER': 'bilateral'}, {'LOWER': 'hypergeometric'}, {'LEMMA': 'series'}]

2021년 2월 17일 (수) 05:00 기준 최신판

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


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.

메타데이터

위키데이터

Spacy 패턴 목록

  • [{'LOWER': 'bilateral'}, {'LOWER': 'hypergeometric'}, {'LEMMA': 'series'}]
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