ベータ関数
複素数 x {\displaystyle x} に対して, 以下が成り立つ.
B ( 1 , x ) = 1 x . {\displaystyle \mathrm {B} (1,\,x)={\frac {1}{x}}.}
B ( x , 1 − x ) = π sin  ( π x ) ( x ∉ Z ) . {\displaystyle \mathrm {B} (x,\,1-x)={\frac {\pi }{\sin(\pi x)}}\qquad (x\notin \mathbb {Z} ).}
B ( 1 2 , x ) = 2 2 x − 1 { Γ ( x ) } 2 Γ ( 2 x ) . {\displaystyle \mathrm {B} \left({\frac {1}{2}},\,x\right)={\frac {2^{2x-1}\{\Gamma (x)\}^{2}}{\Gamma (2x)}}.}
特に, B ( 1 2 , 1 2 ) = π . {\displaystyle \mathrm {B} \left({\frac {1}{2}},\,{\frac {1}{2}}\right)=\pi .}
非負の整数 l {\displaystyle l} , m {\displaystyle m} に対して, 以下が成り立つ.
B ( l + 1 , m + 1 ) = l ! m ! ( l + m + 1 ) ! = 1 ( l + m + 1 ) ( l + m m ) . {\displaystyle \mathrm {B} (l+1,\,m+1)={\frac {l!\,m!