一般化された超幾何関数
数学において、一般化された超幾何関数(いっぱんかされたちょうきかかんすう、英: generalized hypergeometric function)は、一般に
r F s [ a 1 , a 2 , ... , a r b 1 , b 2 , ... , b s ; z ] := ∑ n = 0 ∞ ( a 1 ) n ( a 2 ) n ⋯ ( a r ) n ( b 1 ) n ( b 2 ) n ⋯ ( b s ) n z n n ! {\displaystyle _{r}F_{s}\left[{\begin{matrix}a_{1},a_{2},\dotsc ,a_{r}\\b_{1},b_{2},\dotsc ,b_{s}\end{matrix}};z\right]:=\sum _{n=0}^{\infty }{\frac {(a_{1})_{n}(a_{2})_{n}\dotsb (a_{r})_{n}}{(b_{1})_{n}(b_{2})_{n}\dotsb (b_{s})_{n}}}{\frac {z^{n}}{n!}}}
の形式で表される級数である。ただし、
c n + 1 c n = ∏ j = 1 r ( 1 + a j n ) ⋅ ∏ j = 1 r − 1 ( 1 − b j n ) ⋅ ( 1 − 1 n ) + O ( n − 2 ) = 1 + ∑ j = 1 r a j n − ∑ j = 1 r − 1 b j n − 1 n + O ( n − 2 ) ( n ≫ a j , b k ) | c n + 1 c n | 2 = ( 1 + ∑ j = 1 r R a j n − ∑ j = 1 r − 1 R b j n − 1 n ) 2 + ( ∑ j − 1 r I a j n − ∑ j = 1 r − 1 I b j n ) 2 + O ( n − 2 ) = 1 + 2 n ( ∑ j = 1 r R a j − ∑ j = 1 r − 1 R b j − 1 ) + O ( n − 2 ) ( n ≫ a j , b k ) | c n c n + 1 | = 1 − 1 n ( ∑ j = 1 r R a j − ∑ j = 1 r − 1 R b j − 1 ) + O ( n − 2 ) ( n ≫ a j , b k ) {\displaystyle {\begin{aligned}{\frac {c_{n+1}}{c_{n}}}&=\prod _{j=1}^{r}\left(1+{\frac {a_{j}}{n}}\right)\cdot \prod _{j=1}^{r-1}\left(1-{\frac {b_{j}}{n}}\right)\cdot \left(1-{\frac {1}{n}}\right)+O\left(n^{-2}\right)\\&=1+\sum _{j=1}^{r}{\frac {a_{j}}{n}}-\sum _{j=1}^{r-1}{\frac {b_{j}}{n}}-{\frac {1}{n}}+O\left(n^{-2}\right)\qquad (n\gg a_{j},b_{k})\\\left|{\frac {c_{n+1}}{c_{n}}}\right|^{2}&=\left(1+\sum _{j=1}^{r}{\frac {\Re {a_{j}}}{n}}-\sum _{j=1}^{r-1}{\frac {\Re {b_{j}}}{n}}-{\frac {1}{n}}\right)^{2}+\left(\sum _{j-1}^{r}{\frac {\Im {a_{j}}}{n}}-\sum _{j=1}^{r-1}{\frac {\Im {b_{j}}}{n}}\right)^{2}+O\left(n^{-2}\right)\\&=1+{\frac {2}{n}}\left(\sum _{j=1}^{r}\Re {a_{j}}-\sum _{j=1}^{r-1}\Re {b_{j}}-1\right)+O\left(n^{-2}\right)\qquad (n\gg a_{j},b_{k})\\\left|{\frac {c_{n}}{c_{n+1}}}\right|&=1-{\frac {1}{n}}\left(\sum _{j=1}^{r}\Re {a_{j}}-\sum _{j=1}^{r-1}\Re {b_{j}}-1\right)+O\left(n^{-2}\right)\qquad (n\gg a_{j},b_{k})\\\end{aligned}}}