参考：具体数学
1. 超几何函数 HYPERGEOMETPIC FUNCTIONS
定义：
\[F\left(\begin{gathered}a_1,\cdots,a_m\\b_1,\cdots,b_n\end{gathered}\middle|z\right)=F(a_1,\cdots,a_m;b_1,\cdots,b_n;z)=\sum_{k\ge0}\frac{a_1^{\overline{k}}\cdots a_m^{\overline{k}}z^k}{b_1^{\overline{k}}\cdots b_n^{\overline{k}}k!} \]特殊情况：
\[F\left(\begin{gathered}1\\1\end{gathered}\middle|z\right)=\sum_{k\ge0}\frac{z^k}{k!}=e^z \]\[F\left(\begin{gathered}1,a\\1\end{gathered}\middle|z\right)=\sum_{k\ge0}\frac{a^{\overline{k}}z^k}{k!}=\sum_{k\ge0}\dbinom{a+k-1}{k}z^k=(1-z)^{-a} \]\[zF\left(\begin{gathered}1,1\\2\end{gathered}\middle|-z\right)=z\sum_{k\ge0}\frac{k!k!(-z)^k}{k!(k+1)!}=\sum_{k\ge0}\frac{(-1)^{k-1}}{k}z^k=\ln(1+z) \]级数到超几何函数：
\[t_k:=\frac{a_1^{\overline{k}}\cdots a_m^{\overline{k}}z^k}{b_1^{\overline{k}}\cdots b_n^{\overline{k}}k!} \]\[\frac{t_{k+1}}{t_k}=\frac{(k+a_1)\cdots(k+a_m)z}{(k+b_1)\cdots(k+b_n)(k+1)} \]故若无穷级数的相邻项比值为关于 \(k\) 的多项式之商，即可用超几何函数表示，如：
\[S=\sum_{k\ge0}\dbinom{r+n-k}{n-k} \]\[\frac{t_{k+1}}{t_k}=\frac{(r-n-k-1)!r!(n-k)!}{r!(n-k-1)!(r+n-k)!}=\frac{(k+1)(k-n)(1)}{(k-n-r)(k+1)} \]\[S=\dbinom{r+n}{n}F\left(\begin{gathered}1,-n\\-n-r\end{gathered}\middle|1\right)=\dbinom{r+n}{n}\frac{\Gamma(-r-1)\Gamma(-n-r)}{\Gamma(-n-r-1)\Gamma(-r)}=\dbinom{r+n+1}{n} \] 标签：ge0,frac,函数,gathered,笔记,overline,cdots,几何,sum From： https://www.cnblogs.com/JerryTcl/p/16808020.html