# 超幾何関数

${\displaystyle F(a,b;c;z):={_{2}F_{1}}\left[{\begin{matrix}a,b\\c\end{matrix}};z\right]=\sum _{n=0}^{\infty }{\frac {(a)_{n}(b)_{n}}{(c)_{n}\;n!}}z^{n}}$

ただし、(x)nポッホハマー記号で表した昇冪 (x)0 = 1(x)n = x (x+1) (x+2)…(x+n−1) である。

## 概要

{\displaystyle {\begin{aligned}\log(1+z)&=z\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{n+1}}z^{n}=z\cdot {_{2}F_{1}}\left[{\begin{matrix}1,1\\2\end{matrix}};-z\right]\\\log \left({\frac {1+z}{1-z}}\right)&=2z\sum _{n=0}^{\infty }{\frac {1}{2n+1}}z^{2n}=2z\cdot {_{2}F_{1}}\left[{\begin{matrix}{\frac {1}{2}},1\\{\frac {3}{2}}\end{matrix}};z^{2}\right]\\\sin ^{-1}z&=z\sum _{n=0}^{\infty }{\frac {(2n-1)!!}{(2n)!!(2n+1)}}z^{2n}=z\cdot {_{2}F_{1}}\left[{\begin{matrix}{\frac {1}{2}},{\frac {1}{2}}\\{\frac {3}{2}}\end{matrix}};z^{2}\right]\\\tan ^{-1}z&=z\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{2n+1}}z^{2n}=z\cdot {_{2}F_{1}}\left[{\begin{matrix}{\frac {1}{2}},1\\{\frac {3}{2}}\end{matrix}};-z^{2}\right]\\\end{aligned}}}

{\displaystyle {\begin{aligned}K(k)&={\frac {\pi }{2}}\sum _{n=0}^{\infty }{\left({\frac {(2n-1)!!}{(2n)!!}}\right)^{2}k^{2n}}={\frac {\pi }{2}}\cdot {_{2}F_{1}}\left[{\begin{matrix}{\frac {1}{2}},{\frac {1}{2}}\\1\end{matrix}};k^{2}\right]\\E(k)&={\frac {\pi }{2}}\sum _{n=0}^{\infty }{\left({\frac {(2n-1)!!}{(2n)!!}}\right)^{2}{\frac {k^{2n}}{1-2n}}}={\frac {\pi }{2}}\cdot {_{2}F_{1}}\left[{\begin{matrix}{\frac {1}{2}},-{\frac {1}{2}}\\1\end{matrix}};k^{2}\right]\\\end{aligned}}}

## オイラー積分表示

ガウスの超幾何関数はオイラー積分で表される[1][2]

${\displaystyle F(a,b;c;z)={\frac {\Gamma (c)}{\Gamma (a)\Gamma (c-a)}}\int _{0}^{1}t^{a-1}(1-t)^{c-a-1}(1-tz)^{-b}dt\qquad (0<\Re {a}<\Re {c},|z|<1)}$

これは

{\displaystyle {\begin{aligned}F(a,b;c;z)&={\frac {\Gamma (c)}{\Gamma (a)\Gamma (c-a)}}\cdot {\frac {\Gamma (a)\Gamma (c-a)}{\Gamma (c)}}\cdot \sum _{n=0}^{\infty }{\frac {(a)_{n}(b)_{n}}{(c)_{n}\;n!}}z^{n}\\&={\frac {\Gamma (c)}{\Gamma (a)\Gamma (c-a)}}\sum _{n=0}^{\infty }{\frac {\Gamma (a+n)\Gamma (c-a)(b)_{n}}{\Gamma (c+n)\;n!}}z^{n}\\&={\frac {\Gamma (c)}{\Gamma (a)\Gamma (c-a)}}\sum _{n=0}^{\infty }\mathrm {B} (a+n,c-a){\frac {(b)_{n}}{n!}}z^{n}\\&={\frac {\Gamma (c)}{\Gamma (a)\Gamma (c-a)}}\sum _{n=0}^{\infty }\left(\int _{0}^{1}t^{a+n-1}(1-t)^{c-a-1}dt\right){\frac {(b)_{n}}{n!}}z^{n}\\&={\frac {\Gamma (c)}{\Gamma (a)\Gamma (c-a)}}\int _{0}^{1}t^{a-1}(1-t)^{c-a-1}\left(\sum _{n=0}^{\infty }{\frac {(b)_{n}}{n!}}(tz)^{n}\right)dt\\&={\frac {\Gamma (c)}{\Gamma (a)\Gamma (c-a)}}\int _{0}^{1}t^{a-1}(1-t)^{c-a-1}(1-tz)^{-b}dt\\\end{aligned}}}

として導かれる。

## 超幾何定理

ガウスの超幾何関数のオイラー積分表示に${\displaystyle z=1}$ を代入するとガウスの超幾何定理を得る[2][3]

{\displaystyle {\begin{aligned}F(a,b;c;1)&={\frac {\Gamma (c)}{\Gamma (a)\Gamma (c-a)}}\int _{0}^{1}t^{a-1}(1-t)^{c-a-b-1}dt\\&={\frac {\Gamma (c)\mathrm {B} (a,c-a-b)}{\Gamma (a)\Gamma (c-a)}}\\&={\frac {\Gamma (c)\Gamma (c-a-b)}{\Gamma (c-a)\Gamma (c-b)}}\qquad (\Re {a}+\Re {b}<\Re {c},c\not \in \mathbb {Z} \setminus \mathbb {N} )\\\end{aligned}}}

となる。更に${\displaystyle a=-n}$ を代入するとを得る[4]

${\displaystyle F(-n,b;c;1)={\frac {\Gamma (c)\Gamma (c-b+n)}{\Gamma (c+n)\Gamma (c-b)}}={\frac {(c-b)_{n}}{(c)_{n}}}}$

## 脚注

1. ^ 原岡喜重. (2002). 超幾何関数. 朝倉書店.
2. ^ a b 時弘哲治. (2006). 工学における特殊関数. 共立出版.
3. ^ Weisstein, Eric W. "Gauss's Hypergeometric Theorem". mathworld.wolfram.com (英語).
4. ^ Weisstein, Eric W. "Chu-Vandermonde Identity". mathworld.wolfram.com (英語).