초기하급수의 합공식

수학노트
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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.