Formelsammlung Mathematik: Unendliche Reihen: Hypergeometrische Reihen

Zurück zu Unendliche Reihen

1.1

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

1.2

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

2

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

3.1

$\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}\frac{\mathrm{\Gamma }(a+n)\mathrm{\Gamma }(b+n)}{\mathrm{\Gamma }(c+n)\mathrm{\Gamma }(d+n)}=\frac{{\pi }^2}{\sin a\pi \sin b\pi }\frac{\mathrm{\Gamma }(c+d-a-b-1)}{\mathrm{\Gamma }(c-a)\mathrm{\Gamma }(d-a)\mathrm{\Gamma }(c-b)\mathrm{\Gamma }(d-b)}\begin{array}{c}\text{Re}(c+d-a-b)>1,a,b\in ℂ\setminus \mathbb{Z}\\ \\ c-a,d-a,c-b,d-b\notin {\mathbb{Z}}^{\le 0}\end{array}$

3.2

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

4

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

5

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

6

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

7

$\sum _{k=0}^{\mathrm{\infty }}(\genfrac{}{}{0ex}{}{2k+p}{k})x^k=\frac{2^p}{\sqrt{1-4x}\cdot {(1+\sqrt{1-4x})}^p}|x|<\frac{1}{4},p\in ℂ$