一般化された超幾何関数
第 n {\displaystyle n} 項を c n {\displaystyle c_{n}} とする:
r F r − 1 [ a 1 , a 2 , ... , a r b 1 , b 2 , ... , b r − 1 ; z ] = ∑ n = 0 ∞ c n z n c n = ( a 1 ) n ( a 2 ) n ⋯ ( a r − 1 ) n ( a r ) n ( b 1 ) n ( b 2 ) n ⋯ ( b r − 1 ) n n ! {\displaystyle {\begin{aligned}{}_{r}F_{r-1}\left[{\begin{matrix}a_{1},a_{2},\dotsc ,a_{r}\\b_{1},b_{2},\dotsc ,b_{r-1}\end{matrix}};z\right]=\sum _{n=0}^{\infty }c_{n}z^{n}\\c_{n}={\frac {(a_{1})_{n}(a_{2})_{n}\dotsb (a_{r-1})_{n}(a_{r})_{n}}{(b_{1})_{n}(b_{2})_{n}\dotsb (b_{r-1})_{n}\;n!}}\end{aligned}}}
公比は
であり、