# Formelsammlung Mathematik: Unendliche Reihen: Hypergeometrische Reihen

Zurück zu Unendliche Reihen

##### 1.1
${\displaystyle \sum _{k=0}^{\infty }{\frac {\Gamma (a+k)\,\Gamma (b+k)}{k!\,\Gamma (c+k)}}={\frac {\Gamma (a)\,\Gamma (b)\,\Gamma (c-a-b)}{\Gamma (c-a)\,\Gamma (c-b)}}\qquad {\text{Re}}(c-a-b)>0}$

##### 1.2
${\displaystyle \sum _{k=0}^{\infty }{\frac {1}{k!\,(\alpha +k)!\,(\beta -k)!\,(\gamma -k)!}}={\frac {(\alpha +\beta +\gamma )!}{\beta !\,\gamma !\,(\alpha +\beta )!\,(\alpha +\gamma )!}}\qquad {\text{Re}}(\alpha +\beta +\gamma )>-1}$

##### 2
${\displaystyle \sum _{n=0}^{\infty }(-1)^{n}\,{\frac {\Gamma (a+n)\,\Gamma (b+n)}{n!\;\Gamma (1+a-b+n)}}={\frac {\Gamma \left({\frac {a}{2}}\right)\,\Gamma (b)}{2\,\Gamma \left(1+{\frac {a}{2}}-b\right)}}}$

##### 3.1
${\displaystyle \sum _{n=-\infty }^{\infty }{\frac {\Gamma (a+n)\,\Gamma (b+n)}{\Gamma (c+n)\,\Gamma (d+n)}}={\frac {\pi ^{2}}{\sin a\pi \,\sin b\pi }}\,{\frac {\Gamma (c+d-a-b-1)}{\Gamma (c-a)\,\Gamma (d-a)\,\Gamma (c-b)\,\Gamma (d-b)}}\qquad {\begin{matrix}{\text{Re}}(c+d-a-b)>1\;,\;a,b\in \mathbb {C} \setminus \mathbb {Z} \\\\c-a\;,\;d-a\;,\;c-b\;,\;d-b\notin \mathbb {Z} ^{\leq 0}\end{matrix}}}$

##### 3.2
${\displaystyle \sum _{n=-\infty }^{\infty }{\frac {1}{\Gamma (a+n)\,\Gamma (b+n)\,\Gamma (c-n)\,\Gamma (d-n)}}={\frac {\Gamma (a+b+c+d-3)}{\Gamma (a+c-1)\,\Gamma (a+d-1)\,\Gamma (b+c-1)\,\Gamma (b+d-1)}}\qquad {\text{Re}}(a+b+c+d)>3}$

##### 4
${\displaystyle \sum _{n=0}^{\infty }{\frac {\Gamma (a+n)\,\Gamma (b+n)\,\Gamma (c+n)}{n!\;\Gamma (1+a-b+n)\,\Gamma (1+a-c+n)}}={\frac {\Gamma \left({\frac {a}{2}}\right)\Gamma (b)\,\Gamma (c)\,\Gamma \left(1+{\frac {a}{2}}-b-c\right)}{2\;\Gamma \left(1+{\frac {a}{2}}-b\right)\,\Gamma \left(1+{\frac {a}{2}}-c\right)\,\Gamma \left(1+a-b-c\right)}}}$

##### 5
${\displaystyle \sum _{n=1}^{\infty }{\frac {1}{n}}\left({\frac {{\sqrt {\pi }}\,\,\Gamma (n)}{2\,\Gamma \left(n+{\frac {1}{2}}\right)}}\right)^{2}=2\pi G-{\frac {7}{2}}\zeta (3)}$.

##### 6
${\displaystyle \sum _{k=0}^{\infty }{\frac {\Gamma (p+2k)}{\Gamma (p+k+1)}}\,{\frac {x^{k}}{k!}}={\frac {1}{p}}\left({\frac {2}{1+{\sqrt {1-4x}}}}\right)^{p}\qquad |x|<{\frac {1}{4}}\quad ,\quad p\in \mathbb {C} \setminus \mathbb {Z} ^{\leq 0}}$

##### 7
${\displaystyle \sum _{k=0}^{\infty }{2k+p \choose k}\,x^{k}={\frac {2^{p}}{{\sqrt {1-4x}}\cdot \left(1+{\sqrt {1-4x}}\,\right)^{p}}}\qquad |x|<{\frac {1}{4}}\quad ,\quad p\in \mathbb {C} }$