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Incomplete beta-function

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The function defined by the formula

$$I_x(a,b)=\frac1{B(a,b)}\int\limits_0^xt^{a-1}(1-t)^{b-1}dt,$$

$$0\leq x\leq1,\quad a>0,\quad b>0,$$

where

$$B(a,b)=\int\limits_0^1t^{a-1}(1-t)^{b-1}dt=\frac{\Gamma(a)\Gamma(b)}{\Gamma(a+b)}$$

is the beta-function. If $a$ is an integer, then

$$1-I_x(a,b)=\frac{(1-x)^b}{B(a,b)}\sum_{i=0}^{a-1}(-1)^i\begin{pmatrix}a-1\\i\end{pmatrix}\frac{(1-x)^i}{b+i}=$$

$$=(1-x)^{a+b-1}\sum_{i=0}^{a-1}\begin{pmatrix}a+b-1\\i\end{pmatrix}\left(\frac x{1-x}\right)^i.$$

Series representation:

$$I_x(a,b)=\frac{x^a(1-x)^b}{aB(a,b)}\left\lbrace1+\sum_{n=0}^\infty\frac{B(a+1,n+1)}{B(a+b,n+1)}x^{n+1}\right\rbrace,$$

$$0<x<1.$$

Continued fraction representation:

$$I_x(a,b)=\frac{x^a(1-x)^b}{aB(a,b)}\left\lbrace\frac{1|}{|1}+\frac{d_1|}{|1}+\frac{d_2|}{|1}+\dots\right\rbrace,$$

where

$$d_{2m+1}=-\frac{(a+m)(a+b+m)x}{(a+2m)(a+2m+1)},$$

$$d_{2m}=\frac{m(b-m)x}{(a+2m-1)(a+2m)}.$$

Asymptotic representation for large $a$ and $b$:

$$I_x(a,b)=\Phi\left\lbrace3\frac{(bx)^{1/3}\left(1-\frac1{9b}\right)-[a(1-x)]^{1/3}\left(1-\frac1{9a}\right)}{\sqrt{\frac{[a(1-x)]^{2/3}}a+\frac{(bx)^{2/3}}b}}\right\rbrace+$$

$${}+O\left(\frac1{\min{(a,b)}}\right),$$

where

$$\Phi(z)=\frac1{\sqrt{2\pi}}\int\limits_{-\infty}^ze^{-t^2/2}dt.$$

Asymptotic representation for large $b$ and bounded $a$:

$$I_x(a,b)=I\left(\frac{x(2b+a-1)}{2-x},a\right)+O(b^{-2}),$$

where

$$I(z,a)=\frac1{\Gamma(a)}\int\limits_0^ze^{-t}t^{a-1}dt.$$

Connection with the hypergeometric function:

$$I_x(a,b)=\frac{x^a}{aB(x,a)}F(a,1-b;a+1;x).$$

Recurrence relations:

$$I_x(a,b)=1-I_{1-x}(b,a),$$

$$I_x(a,b)=xI_x(a-1,b)+(1-x)I_x(a,b-1),$$

$$I_x(a,a)=\frac12I_{1-y}\left(a,\frac12\right),\quad y=4\left(x-\frac12\right)^2,\quad0<x\leq\frac12.$$

References

[1] M. Abramowitz, I.A. Stegun, "Handbook of mathematical functions" , Dover, reprint (1973)
[2] K. Pearson, "Tables of the incomplete beta-function" , Cambridge Univ. Press (1932)
How to Cite This Entry:
Incomplete beta-function. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Incomplete_beta-function&oldid=40746
This article was adapted from an original article by V.I. Pagurova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article