# §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 $\mathop{\mathrm{B}_{x}\/}\nolimits\!\left(a,b\right)=\int_{0}^{x}t^{a-1}(1-t)^% {b-1}\mathrm{d}t,$ Defines: $\mathop{\mathrm{B}_{\NVar{x}}\/}\nolimits\!\left(\NVar{a},\NVar{b}\right)$: 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)
 8.17.2 $\mathop{I_{x}\/}\nolimits\!\left(a,b\right)=\mathop{\mathrm{B}_{x}\/}\nolimits% \!\left(a,b\right)/\mathop{\mathrm{B}\/}\nolimits\!\left(a,b\right),$ Defines: $\mathop{I_{\NVar{x}}\/}\nolimits\!\left(\NVar{a},\NVar{b}\right)$: incomplete beta function Symbols: $\mathop{\mathrm{B}\/}\nolimits\!\left(\NVar{a},\NVar{b}\right)$: beta function, $\mathop{\mathrm{B}_{\NVar{x}}\/}\nolimits\!\left(\NVar{a},\NVar{b}\right)$: 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)

where, as in §5.12, $\mathop{\mathrm{B}\/}\nolimits\!\left(a,b\right)$ denotes the Beta function:

 8.17.3 $\mathop{\mathrm{B}\/}\nolimits\!\left(a,b\right)=\frac{\mathop{\Gamma\/}% \nolimits\!\left(a\right)\mathop{\Gamma\/}\nolimits\!\left(b\right)}{\mathop{% \Gamma\/}\nolimits\!\left(a+b\right)}.$
 8.17.4 $\mathop{I_{x}\/}\nolimits\!\left(a,b\right)=1-\mathop{I_{1-x}\/}\nolimits\!% \left(b,a\right).$ Symbols: $\mathop{I_{\NVar{x}}\/}\nolimits\!\left(\NVar{a},\NVar{b}\right)$: 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)
 8.17.5 $\mathop{I_{x}\/}\nolimits\!\left(m,n-m+1\right)=\sum_{j=m}^{n}\genfrac{(}{)}{0% .0pt}{}{n}{j}x^{j}(1-x)^{n-j},$ $m,n$ positive integers; $0\leq x<1$. Symbols: $\mathop{I_{\NVar{x}}\/}\nolimits\!\left(\NVar{a},\NVar{b}\right)$: incomplete beta function, $\binom{\NVar{m}}{\NVar{n}}$: 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), Subsection 8.17(i) Permalink: http://dlmf.nist.gov/8.17.E5 Encodings: TeX, pMML, png See also: Annotations for 8.17(i)

Addendum: For a companion equation see (8.17.24).

 8.17.6 $\mathop{I_{x}\/}\nolimits\!\left(a,a\right)=\tfrac{1}{2}\mathop{I_{4x(1-x)}\/}% \nolimits\!\left(a,\tfrac{1}{2}\right),$ $0\leq x\leq\frac{1}{2}$. Symbols: $\mathop{I_{\NVar{x}}\/}\nolimits\!\left(\NVar{a},\NVar{b}\right)$: 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)

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

## §8.17(ii) Hypergeometric Representations

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

For the hypergeometric function $\mathop{F\/}\nolimits\!\left(a,b;c;z\right)$ see §15.2(i).

## §8.17(iii) Integral Representation

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

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

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)
 8.17.12 $\displaystyle\mathop{I_{x}\/}\nolimits\!\left(a,b\right)$ $\displaystyle=x\mathop{I_{x}\/}\nolimits\!\left(a-1,b\right)+x^{\prime}\mathop% {I_{x}\/}\nolimits\!\left(a,b-1\right),$ Symbols: $\mathop{I_{\NVar{x}}\/}\nolimits\!\left(\NVar{a},\NVar{b}\right)$: 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) 8.17.13 $\displaystyle(a+b)\mathop{I_{x}\/}\nolimits\!\left(a,b\right)$ $\displaystyle=a\mathop{I_{x}\/}\nolimits\!\left(a+1,b\right)+b\mathop{I_{x}\/}% \nolimits\!\left(a,b+1\right),$ Symbols: $\mathop{I_{\NVar{x}}\/}\nolimits\!\left(\NVar{a},\NVar{b}\right)$: 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)
 8.17.14 $(a+bx)\mathop{I_{x}\/}\nolimits\!\left(a,b\right)=xb\mathop{I_{x}\/}\nolimits% \!\left(a-1,b+1\right)+a\mathop{I_{x}\/}\nolimits\!\left(a+1,b\right),$
 8.17.15 $(b+ax^{\prime})\mathop{I_{x}\/}\nolimits\!\left(a,b\right)=ax^{\prime}\mathop{% I_{x}\/}\nolimits\!\left(a+1,b-1\right)+b\mathop{I_{x}\/}\nolimits\!\left(a,b+% 1\right),$ Symbols: $\mathop{I_{\NVar{x}}\/}\nolimits\!\left(\NVar{a},\NVar{b}\right)$: 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)
 8.17.16 $\displaystyle a\mathop{I_{x}\/}\nolimits\!\left(a+1,b\right)$ $\displaystyle=(a+cx)\mathop{I_{x}\/}\nolimits\!\left(a,b\right)-cx\mathop{I_{x% }\/}\nolimits\!\left(a-1,b\right),$ 8.17.17 $\displaystyle b\mathop{I_{x}\/}\nolimits\!\left(a,b+1\right)$ $\displaystyle=(b+cx^{\prime})\mathop{I_{x}\/}\nolimits\!\left(a,b\right)-cx^{% \prime}\mathop{I_{x}\/}\nolimits\!\left(a,b-1\right),$
 8.17.18 $\mathop{I_{x}\/}\nolimits\!\left(a,b\right)=\mathop{I_{x}\/}\nolimits\!\left(a% +1,b-1\right)+\frac{x^{a}(x^{\prime})^{b-1}}{a\mathop{\mathrm{B}\/}\nolimits\!% \left(a,b\right)},$
 8.17.19 $\mathop{I_{x}\/}\nolimits\!\left(a,b\right)=\mathop{I_{x}\/}\nolimits\!\left(a% -1,b+1\right)-\frac{x^{a-1}(x^{\prime})^{b}}{b\mathop{\mathrm{B}\/}\nolimits\!% \left(a,b\right)},$
 8.17.20 $\displaystyle\mathop{I_{x}\/}\nolimits\!\left(a,b\right)$ $\displaystyle=\mathop{I_{x}\/}\nolimits\!\left(a+1,b\right)+\frac{x^{a}(x^{% \prime})^{b}}{a\mathop{\mathrm{B}\/}\nolimits\!\left(a,b\right)},$ 8.17.21 $\displaystyle\mathop{I_{x}\/}\nolimits\!\left(a,b\right)$ $\displaystyle=\mathop{I_{x}\/}\nolimits\!\left(a,b+1\right)-\frac{x^{a}(x^{% \prime})^{b}}{b\mathop{\mathrm{B}\/}\nolimits\!\left(a,b\right)}.$

## §8.17(v) Continued Fraction

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

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)

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

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 $\mathop{I_{1-x}\/}\nolimits\!\left(b,a\right)$ and using (8.17.4).
 8.17.24 $\mathop{I_{x}\/}\nolimits\!\left(m,n\right)=(1-x)^{n}\sum_{j=m}^{\infty}% \genfrac{(}{)}{0.0pt}{}{n+j-1}{j}x^{j},$ $m,n$ positive integers; $0\leq x<1$. Symbols: $\mathop{I_{\NVar{x}}\/}\nolimits\!\left(\NVar{a},\NVar{b}\right)$: incomplete beta function, $\binom{\NVar{m}}{\NVar{n}}$: 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), Other Changes 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 $\infty$. Reported 2011-03-23 by Stephen Bourn See also: Annotations for 8.17(vii)