超幾何関数

超幾何関数（ちょうきかかんすう、英: hypergeometric function）は以下の超幾何級数で定義される特殊関数である。 F ( a , b ; c ; z ) := 2 F 1 [ a , b c ; z ] = ∑ n = 0 ∞ ( a ) n ( b ) n ( c ) n n ! z n {\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 は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。 超幾何関数は多くの初等関数や特殊関数を包含する。 対数関数、逆三角関数 log ⁡ ( 1 + z ) = z ∑ n = 0 ∞ ( − 1 ) n n + 1 z n = z ⋅ 2 F 1 [ 1 , 1 2 ; − z ] log ⁡ ( 1 + z 1 − z ) = 2 z ∑ n = 0 ∞ 1 2 n + 1 z 2 n = 2 z ⋅ 2 F 1 [ 1 2 , 1 3 2 ; z 2 ] sin − 1 ⁡ z = z ∑ n = 0 ∞ ( 2 n − 1 ) ! ! ( 2 n ) ! ! ( 2 n + 1 ) z 2 n = z ⋅ 2 F 1 [ 1 2 , 1 2 3 2 ; z 2 ] tan − 1 ⁡ z = z ∑ n = 0 ∞ ( − 1 ) n 2 n + 1 z 2 n = z ⋅ 2 F 1 [ 1 2 , 1 3 2 ; − z 2 ] {\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}}} 完全楕円積分 K ( k ) = π 2 ∑ n = 0 ∞ ( ( 2 n − 1 ) ! ! ( 2 n ) ! ! ) 2 k 2 n = π 2 ⋅ 2 F 1 [ 1 2 , 1 2 1 ; k 2 ] E ( k ) = π 2 ∑ n = 0 ∞ ( ( 2 n − 1 ) ! ! ( 2 n ) ! ! ) 2 k 2 n 1 − 2 n = π 2 ⋅ 2 F 1 [ 1 2 , − 1 2 1 ; k 2 ] {\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}}} ガウスの超幾何関数はオイラー積分で表される。 F ( a , b ; c ; z ) = Γ ( c ) Γ ( a ) Γ ( c − a ) ∫ 0 1 t a − 1 ( 1 − t ) c − a − 1 ( 1 − t z ) − b d t ( 0 < ℜ a < ℜ c , | z | < 1 ) {\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 F ( a , b ; c ; z ) = Γ ( c ) Γ ( a ) Γ ( c − a ) ⋅ Γ ( a ) Γ ( c − a ) Γ ( c ) ⋅ ∑ n = 0 ∞ ( a ) n ( b ) n ( c ) n n ! z n = Γ ( c ) Γ ( a ) Γ ( c − a ) ∑ n = 0 ∞ Γ ( a + n ) Γ ( c − a ) ( b ) n Γ ( c + n ) n ! z n = Γ ( c ) Γ ( a ) Γ ( c − a ) ∑ n = 0 ∞ B ( a + n , c − a ) ( b ) n n ! z n = Γ ( c ) Γ ( a ) Γ ( c − a ) ∑ n = 0 ∞ ( ∫ 0 1 t a + n − 1 ( 1 − t ) c − a − 1 d t ) ( b ) n n ! z n = Γ ( c ) Γ ( a ) Γ ( c − a ) ∫ 0 1 t a − 1 ( 1 − t ) c − a − 1 ( ∑ n = 0 ∞ ( b ) n n ! ( t z ) n ) d t = Γ ( c ) Γ ( a ) Γ ( c − a ) ∫ 0 1 t a − 1 ( 1 − t ) c − a − 1 ( 1 − t z ) − b d t {\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。 ガウスの超幾何関数のオイラー積分表示に z = 1 {\displaystyle z=1} を代入するとガウスの超幾何定理を得る。 F ( a , b ; c ; 1 ) = Γ ( c ) Γ ( a ) Γ ( c − a ) ∫ 0 ∞ t a − 1 ( 1 − t ) c − a − b − 1 d t = Γ ( c ) B ( a , c − a − b ) Γ ( a ) Γ ( c − a ) = Γ ( c ) Γ ( c − a − b ) Γ ( c − a ) Γ ( c − b ) ( ℜ a + ℜ b < ℜ c , c ∉ Z ∖ N ) {\displaystyle {\begin{aligned}F(a,b;c;1)&={\frac {\Gamma (c)}{\Gamma (a)\Gamma (c-a)}}\int _{0}^{\infty }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 a = − n {\displaystyle a=-n} を代入するとヴァンデルモンドの恒等式（英語版）を得る。 F ( − n , b ; c ; 1 ) = Γ ( c ) Γ ( c − b + n ) Γ ( c + n ) Γ ( c − b ) = ( c − b ) n ( c ) n {\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}}}}