广义超几何级数

广义超几何级数的定义

广义超几何级数的定义是

$$_{p}F_{q}\left(\alpha_{1},\alpha_{2},\cdots,\alpha_{p};\gamma_{1},\gamma_{2},\cdots,\gamma_{q};z\right)=\sum_{n=0}^{\infty}\frac{(\alpha_{1})_{n}(\alpha_{2})_{n}\cdots(\alpha_{p})_{n}}{n!(\gamma_{1})_{n}(\gamma_{2})_{n}\cdots(\gamma_{q})_{n}}z^{n},$$

其中$(\alpha)_{n}=\alpha(\alpha+1)(\alpha+2)\cdots(\alpha+n-1)$为$\alpha$的升$n$阶乘.

关于公式的记忆, 首先对于一般级数

$$f(z)=\sum_{n=0}^{\infty}c_{n}(z),$$

如果已知$c_{0}=\color{red}{\alpha}$,

$$\frac{c_{n+1}(z)}{c_{n}(z)}=\frac{({\color{blue}{ \alpha_{1} }}+n)({\color{blue}{ \alpha_{2} }}+n)\cdots({\color{blue}{ \alpha_{p} }}+n)}{({\color{green}{ \gamma_{1} }}+n)\cdots({\color{green}{ \gamma_{q} }}+n)}\frac{z}{(n+1)},$$

则$f(z)$有表示

$$f(z)={\color{red}{\alpha}}\cdot{}_{p}F_{q}\left({\color{blue}{ \alpha_{1},\alpha_{2},\cdots,\alpha_{p} }};{\color{green}{ \gamma_{1},\gamma_{2},\cdots,\gamma_{q} }};z\right).$$