# §8.17 Incomplete Beta Functions

## §8.17(i) Definitions and Basic Properties

Throughout §§8.17 and 8.18 we assume that $a>0$, $b>0$, and $0\leq x\leq 1$. However, in the case of §8.17 it is straightforward to continue most results analytically to other real values of $a$, $b$, and $x$, and also to complex values.

 8.17.1 $\mathrm{B}_{x}\left(a,b\right)=\int_{0}^{x}t^{a-1}(1-t)^{b-1}\,\mathrm{d}t,$ ⓘ Defines: $\mathrm{B}_{\NVar{x}}\left(\NVar{a},\NVar{b}\right)$: incomplete beta function Symbols: $\,\mathrm{d}\NVar{x}$: differential of $x$, $\int$: integral, $a$: parameter, $b$: parameter and $x$: variable A&S Ref: 6.6.1 Permalink: http://dlmf.nist.gov/8.17.E1 Encodings: TeX, pMML, png See also: Annotations for §8.17(i), §8.17 and Ch.8
 8.17.2 $I_{x}\left(a,b\right)=\mathrm{B}_{x}\left(a,b\right)/\mathrm{B}\left(a,b\right),$ ⓘ Defines: $I_{\NVar{x}}\left(\NVar{a},\NVar{b}\right)$: incomplete beta function Symbols: $\mathrm{B}\left(\NVar{a},\NVar{b}\right)$: beta function, $\mathrm{B}_{\NVar{x}}\left(\NVar{a},\NVar{b}\right)$: incomplete beta function, $a$: parameter, $b$: parameter and $x$: variable A&S Ref: 6.6.2 Permalink: http://dlmf.nist.gov/8.17.E2 Encodings: TeX, pMML, png See also: Annotations for §8.17(i), §8.17 and Ch.8

where, as in §5.12, $\mathrm{B}\left(a,b\right)$ denotes the beta function:

 8.17.3 $\mathrm{B}\left(a,b\right)=\frac{\Gamma\left(a\right)\Gamma\left(b\right)}{% \Gamma\left(a+b\right)}.$ ⓘ Symbols: $\mathrm{B}\left(\NVar{a},\NVar{b}\right)$: beta function, $\Gamma\left(\NVar{z}\right)$: gamma function, $a$: parameter and $b$: parameter Permalink: http://dlmf.nist.gov/8.17.E3 Encodings: TeX, pMML, png See also: Annotations for §8.17(i), §8.17 and Ch.8
 8.17.4 $I_{x}\left(a,b\right)=1-I_{1-x}\left(b,a\right).$ ⓘ Symbols: $I_{\NVar{x}}\left(\NVar{a},\NVar{b}\right)$: incomplete beta function, $a$: parameter, $b$: parameter and $x$: variable A&S Ref: 6.6.3 (Symmetry relation) Referenced by: §8.17(v), §8.18(i) Permalink: http://dlmf.nist.gov/8.17.E4 Encodings: TeX, pMML, png See also: Annotations for §8.17(i), §8.17 and Ch.8
 8.17.5 $I_{x}\left(m,n-m+1\right)=\sum_{j=m}^{n}\genfrac{(}{)}{0.0pt}{}{n}{j}x^{j}(1-x% )^{n-j},$ $m,n$ positive integers; $0\leq x<1$. ⓘ Symbols: $I_{\NVar{x}}\left(\NVar{a},\NVar{b}\right)$: incomplete beta function, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$: binomial coefficient, $n$: nonnegative integer and $x$: variable A&S Ref: 6.6.4 (Relation to the Binomial Expansion) Referenced by: §8.17(i), §8.17(vii), Erratum (V1.0.5) for Subsection 8.17(i) Permalink: http://dlmf.nist.gov/8.17.E5 Encodings: TeX, pMML, png See also: Annotations for §8.17(i), §8.17 and Ch.8

Addendum: For a companion equation see (8.17.24).

 8.17.6 $I_{x}\left(a,a\right)=\tfrac{1}{2}I_{4x(1-x)}\left(a,\tfrac{1}{2}\right),$ $0\leq x\leq\frac{1}{2}$. ⓘ Symbols: $I_{\NVar{x}}\left(\NVar{a},\NVar{b}\right)$: incomplete beta function, $a$: parameter and $x$: variable Referenced by: §8.17(i) Permalink: http://dlmf.nist.gov/8.17.E6 Encodings: TeX, pMML, png See also: Annotations for §8.17(i), §8.17 and Ch.8

For a historical profile of $\mathrm{B}_{x}\left(a,b\right)$ see Dutka (1981).

## §8.17(ii) Hypergeometric Representations

 8.17.7 $\displaystyle\mathrm{B}_{x}\left(a,b\right)$ $\displaystyle=\frac{x^{a}}{a}F\left(a,1-b;a+1;x\right),$ 8.17.8 $\displaystyle\mathrm{B}_{x}\left(a,b\right)$ $\displaystyle=\frac{x^{a}(1-x)^{b}}{a}F\left(a+b,1;a+1;x\right),$ 8.17.9 $\displaystyle\mathrm{B}_{x}\left(a,b\right)$ $\displaystyle=\frac{x^{a}(1-x)^{b-1}}{a}F\left({1,1-b\atop a+1};\frac{x}{x-1}% \right).$

For the hypergeometric function $F\left(a,b;c;z\right)$ see §15.2(i).

## §8.17(iii) Integral Representation

With $a>0$, $b>0$, and $0,

 8.17.10 $I_{x}\left(a,b\right)=\frac{x^{a}(1-x)^{b}}{2\pi i}\int_{c-i\infty}^{c+i\infty% }s^{-a}(1-s)^{-b}\frac{\,\mathrm{d}s}{s-x},$

where $x and the branches of $s^{-a}$ and $(1-s)^{-b}$ are continuous on the path and assume their principal values when $s=c$.

Further integral representations can be obtained by combining the results given in §8.17(ii) with §15.6.

## §8.17(iv) Recurrence Relations

With

 8.17.11 $\displaystyle x^{\prime}$ $\displaystyle=1-x,$ $\displaystyle c$ $\displaystyle=a+b-1,$ ⓘ Defines: $c$ (locally) and $x^{\prime}$ (locally) Symbols: $a$: parameter, $b$: parameter and $x$: variable Permalink: http://dlmf.nist.gov/8.17.E11 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §8.17(iv), §8.17 and Ch.8
 8.17.12 $\displaystyle I_{x}\left(a,b\right)$ $\displaystyle=xI_{x}\left(a-1,b\right)+x^{\prime}I_{x}\left(a,b-1\right),$ ⓘ Symbols: $I_{\NVar{x}}\left(\NVar{a},\NVar{b}\right)$: incomplete beta function, $a$: parameter, $b$: parameter, $x$: variable and $x^{\prime}$ A&S Ref: 6.6.5 Permalink: http://dlmf.nist.gov/8.17.E12 Encodings: TeX, pMML, png See also: Annotations for §8.17(iv), §8.17 and Ch.8 8.17.13 $\displaystyle(a+b)I_{x}\left(a,b\right)$ $\displaystyle=aI_{x}\left(a+1,b\right)+bI_{x}\left(a,b+1\right),$ ⓘ Symbols: $I_{\NVar{x}}\left(\NVar{a},\NVar{b}\right)$: incomplete beta function, $a$: parameter, $b$: parameter and $x$: variable A&S Ref: 6.6.7 Permalink: http://dlmf.nist.gov/8.17.E13 Encodings: TeX, pMML, png See also: Annotations for §8.17(iv), §8.17 and Ch.8
 8.17.14 $(a+bx)I_{x}\left(a,b\right)=xbI_{x}\left(a-1,b+1\right)+aI_{x}\left(a+1,b% \right),$ ⓘ Symbols: $I_{\NVar{x}}\left(\NVar{a},\NVar{b}\right)$: incomplete beta function, $a$: parameter, $b$: parameter and $x$: variable Permalink: http://dlmf.nist.gov/8.17.E14 Encodings: TeX, pMML, png See also: Annotations for §8.17(iv), §8.17 and Ch.8
 8.17.15 $(b+ax^{\prime})I_{x}\left(a,b\right)=ax^{\prime}I_{x}\left(a+1,b-1\right)+bI_{% x}\left(a,b+1\right),$ ⓘ Symbols: $I_{\NVar{x}}\left(\NVar{a},\NVar{b}\right)$: incomplete beta function, $a$: parameter, $b$: parameter, $x$: variable and $x^{\prime}$ A&S Ref: 6.6.6 (An error has been corrected.) Permalink: http://dlmf.nist.gov/8.17.E15 Encodings: TeX, pMML, png See also: Annotations for §8.17(iv), §8.17 and Ch.8
 8.17.16 $\displaystyle aI_{x}\left(a+1,b\right)$ $\displaystyle=(a+cx)I_{x}\left(a,b\right)-cxI_{x}\left(a-1,b\right),$ 8.17.17 $\displaystyle bI_{x}\left(a,b+1\right)$ $\displaystyle=(b+cx^{\prime})I_{x}\left(a,b\right)-cx^{\prime}I_{x}\left(a,b-1% \right),$
 8.17.18 $I_{x}\left(a,b\right)=I_{x}\left(a+1,b-1\right)+\frac{x^{a}(x^{\prime})^{b-1}}% {a\mathrm{B}\left(a,b\right)},$
 8.17.19 $I_{x}\left(a,b\right)=I_{x}\left(a-1,b+1\right)-\frac{x^{a-1}(x^{\prime})^{b}}% {b\mathrm{B}\left(a,b\right)},$
 8.17.20 $\displaystyle I_{x}\left(a,b\right)$ $\displaystyle=I_{x}\left(a+1,b\right)+\frac{x^{a}(x^{\prime})^{b}}{a\mathrm{B}% \left(a,b\right)},$ 8.17.21 $\displaystyle I_{x}\left(a,b\right)$ $\displaystyle=I_{x}\left(a,b+1\right)-\frac{x^{a}(x^{\prime})^{b}}{b\mathrm{B}% \left(a,b\right)}.$

## §8.17(v) Continued Fraction

 8.17.22 $I_{x}\left(a,b\right)=\frac{x^{a}(1-x)^{b}}{a\mathrm{B}\left(a,b\right)}\left(% \cfrac{1}{1+\cfrac{d_{1}}{1+\cfrac{d_{2}}{1+\cfrac{d_{3}}{1+}}}}\cdots\right),$

where

 8.17.23 $\displaystyle d_{2m}$ $\displaystyle=\frac{m(b-m)x}{(a+2m-1)(a+2m)},$ $\displaystyle d_{2m+1}$ $\displaystyle=-\frac{(a+m)(a+b+m)x}{(a+2m)(a+2m+1)}.$ ⓘ Symbols: $a$: parameter, $d_{m}$: coefficients, $b$: parameter and $x$: variable Permalink: http://dlmf.nist.gov/8.17.E23 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §8.17(v), §8.17 and Ch.8

The $4m$ and $4m+1$ convergents are less than $I_{x}\left(a,b\right)$, and the $4m+2$ and $4m+3$ convergents are greater than $I_{x}\left(a,b\right)$.

See also Cuyt et al. (2008, pp. 385–389).

The expansion (8.17.22) converges rapidly for $x<(a+1)/(a+b+2)$. For $x>(a+1)/(a+b+2)$ or $1-x<(b+1)/(a+b+2)$, more rapid convergence is obtained by computing $I_{1-x}\left(b,a\right)$ and using (8.17.4).

## §8.17(vi) Sums

For sums of infinite series whose terms involve the incomplete beta function see Hansen (1975, §62).

## §8.17(vii) Addendum to 8.17(i) Definitions and Basic Properties

 8.17.24 $I_{x}\left(m,n\right)=(1-x)^{n}\sum_{j=m}^{\infty}\genfrac{(}{)}{0.0pt}{}{n+j-% 1}{j}x^{j},$ $m,n$ positive integers; $0\leq x<1$. ⓘ Symbols: $I_{\NVar{x}}\left(\NVar{a},\NVar{b}\right)$: incomplete beta function, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$: binomial coefficient, $n$: nonnegative integer and $x$: variable A&S Ref: 26.5.26 (The upper limit of summation has been corrected.) Referenced by: §8.17(i), Erratum (V1.0.5) for Chapters 8, 20, 36 Permalink: http://dlmf.nist.gov/8.17.E24 Encodings: TeX, pMML, png Addition (effective with 1.0.5): This equation is the same as Equation (26.5.26) of Abramowitz and Stegun (1964), except that the upper limit of the summation has been corrected to $\infty$. Suggested 2011-03-23 by Stephen Bourn See also: Annotations for §8.17(vii), §8.17 and Ch.8

Compare (8.17.5).