不完全ベータ関数
I 0 ( a , b ) = 0 {\displaystyle I_{0}(a,b)=0\,} I 1 ( a , b ) = 1 {\displaystyle I_{1}(a,b)=1\,} I x ( a , 1 ) = x a {\displaystyle I_{x}(a,1)=x^{a}\,} I x ( 1 , b ) = 1 − ( 1 − x ) b {\displaystyle I_{x}(1,b)=1-(1-x)^{b}\,} I x ( a , b ) = 1 − I 1 − x ( b , a ) {\displaystyle I_{x}(a,b)=1-I_{1-x}(b,a)\,} I x ( a + 1 , b ) = I x ( a , b ) − x a ( 1 − x ) b a B ( a , b ) {\displaystyle I_{x}(a+1,b)=I_{x}(a,b)-{\frac {x^{a}(1-x)^{b}}{a\mathrm {B} (a,b)}}\,} I x ( a , b + 1 ) = I x ( a , b ) + x a ( 1 − x ) b b B ( a , b ) {\displaystyle I_{x}(a,b+1)=I_{x}(a,b)+{\frac {x^{a}(1-x)^{b}}{b\mathrm {B} (a,b)}}\,} B ( x ; a , b ) = ( − 1 ) a B ( x x − 1 ; a , 1 − a − b ) {\displaystyle \mathrm {B} (x;a,b)=(-1)^{a}\mathrm {B} \left({\frac {x}{x-1}};a,1-a-b\right)}