초기하급수의 합공식

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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

$${}_2H_2(a,b;c,d;1)= \sum_{-\infty}^\infty\frac{(a)_n(b)_n}{(c)_n(d)_n}= \frac{\Gamma(d)\Gamma(e)\Gamma(1-a)\Gamma(1-b)\Gamma(c+d-a-b-1)}{\Gamma(c-a)\Gamma(c-b)\Gamma(d-a)\Gamma(d-b)}$$

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.