8 Incomplete Gamma and Related Functions Related Functions 8.16 Generalizations 8.18 Asymptotic Expansions of

I_{x}\left(a,b\right)

§8.17 Incomplete Beta Functions

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Keywords:: incomplete beta functions
Referenced by:: §8.17(i)
Permalink:: http://dlmf.nist.gov/8.17
See also:: Annotations for Ch.8

§8.17(i) Definitions and Basic Properties

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Keywords:: basic properties, historical profile, incomplete beta functions
Notes:: For (8.17.5) combine (8.17.8) and (15.8.1). For (8.17.6) combine (8.17.8), (15.8.18), and (15.8.1).
Referenced by:: §8.17(vii), Erratum (V1.0.5) for Subsection 8.17(i)
Permalink:: http://dlmf.nist.gov/8.17.i
Errata (effective with 1.0.5):: The conditions for the validity of (8.17.5) were added. Previously, no conditions were stated.
Reported 2011-03-23 by Stephen Bourn
See also:: Annotations for §8.17 and Ch.8

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

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Keywords:: hypergeometric function, incomplete beta functions, relation to hypergeometric function, relations to other functions
Notes:: See Temme (1996b, §11.3).
Referenced by:: §8.17(iii)
Permalink:: http://dlmf.nist.gov/8.17.ii
See also:: Annotations for §8.17 and Ch.8

8.17.7	$\displaystyle\mathrm{B}_{x}\left(a,b\right)$	$\displaystyle=\frac{x^{a}}{a}F\left(a,1-b;a+1;x\right),$
ⓘ Symbols: $F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Gauss’ hypergeometric function, $\mathrm{B}_{\NVar{x}}\left(\NVar{a},\NVar{b}\right)$ : incomplete beta function, $a$ : parameter, $b$ : parameter and $x$ : variable A&S Ref: 6.6.8 Permalink: http://dlmf.nist.gov/8.17.E7 Encodings: TeX, pMML, png See also: Annotations for §8.17(ii), §8.17 and Ch.8
8.17.8	$\displaystyle\mathrm{B}_{x}\left(a,b\right)$	$\displaystyle=\frac{x^{a}(1-x)^{b}}{a}F\left(a+b,1;a+1;x\right),$
ⓘ Symbols: $F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Gauss’ hypergeometric function, $\mathrm{B}_{\NVar{x}}\left(\NVar{a},\NVar{b}\right)$ : incomplete beta function, $a$ : parameter, $b$ : parameter and $x$ : variable Referenced by: §8.17(i) Permalink: http://dlmf.nist.gov/8.17.E8 Encodings: TeX, pMML, png See also: Annotations for §8.17(ii), §8.17 and Ch.8
8.17.9	$\displaystyle\mathrm{B}_{x}\left(a,b\right)$	$\displaystyle=\frac{x^{a}(1-x)^{b-1}}{a}F\left({1,1-b\atop a+1};\frac{x}{x-1}% \right).$
ⓘ Symbols: $F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Gauss’ hypergeometric function, $\mathrm{B}_{\NVar{x}}\left(\NVar{a},\NVar{b}\right)$ : incomplete beta function, $a$ : parameter, $b$ : parameter and $x$ : variable Referenced by: §8.18(i) Permalink: http://dlmf.nist.gov/8.17.E9 Encodings: TeX, pMML, png See also: Annotations for §8.17(ii), §8.17 and Ch.8

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

§8.17(iii) Integral Representation

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Keywords:: incomplete beta functions, integral representation
Notes:: See Temme (1996b, p. 290).
Permalink:: http://dlmf.nist.gov/8.17.iii
See also:: Annotations for §8.17 and Ch.8

With $a>0$ , $b>0$ , and $0<x<1$ ,

8.17.10		$I_{x}\left(a,b\right)=\frac{x^{a}(1-x)^{b}}{2\pi i}\int_{c-i\infty}^{c+i\infty% }s^{-a}(1-s)^{-b}\frac{\,\mathrm{d}s}{s-x},$
ⓘ Symbols: $I_{\NVar{x}}\left(\NVar{a},\NVar{b}\right)$ : incomplete beta function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{i}$ : imaginary unit, $\int$ : integral, $a$ : parameter, $b$ : parameter, $x$ : variable and $c$ : parameter Permalink: http://dlmf.nist.gov/8.17.E10 Encodings: TeX, pMML, png See also: Annotations for §8.17(iii), §8.17 and Ch.8

where $x<c<1$ 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

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Keywords:: incomplete beta functions, recurrence relations
Notes:: See Temme (1996b, pp. 289–290).
Permalink:: http://dlmf.nist.gov/8.17.iv
See also:: Annotations for §8.17 and Ch.8

With

8.17.11	$\displaystyle x^{\prime}$	$\displaystyle=1-x,$
8.17.11	$\displaystyle c$	$\displaystyle=a+b-1,$
ⓘ 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	$\displaystyle aI_{x}\left(a+1,b\right)$	$\displaystyle=(a+cx)I_{x}\left(a,b\right)-cxI_{x}\left(a-1,b\right),$
ⓘ Symbols: $I_{\NVar{x}}\left(\NVar{a},\NVar{b}\right)$ : incomplete beta function, $a$ : parameter, $b$ : parameter, $x$ : variable and $c$ Permalink: http://dlmf.nist.gov/8.17.E16 Encodings: TeX, pMML, png See also: Annotations for §8.17(iv), §8.17 and Ch.8
8.17.17	$\displaystyle bI_{x}\left(a,b+1\right)$	$\displaystyle=(b+cx^{\prime})I_{x}\left(a,b\right)-cx^{\prime}I_{x}\left(a,b-1% \right),$
ⓘ Symbols: $I_{\NVar{x}}\left(\NVar{a},\NVar{b}\right)$ : incomplete beta function, $a$ : parameter, $b$ : parameter, $x$ : variable, $c$ and $x^{\prime}$ Permalink: http://dlmf.nist.gov/8.17.E17 Encodings: TeX, pMML, png See also: Annotations for §8.17(iv), §8.17 and Ch.8

8.17.18		$I_{x}\left(a,b\right)=I_{x}\left(a+1,b-1\right)+\frac{x^{a}(x^{\prime})^{b-1}}% {a\mathrm{B}\left(a,b\right)},$
ⓘ Symbols: $\mathrm{B}\left(\NVar{a},\NVar{b}\right)$ : beta function, $I_{\NVar{x}}\left(\NVar{a},\NVar{b}\right)$ : incomplete beta function, $a$ : parameter, $b$ : parameter, $x$ : variable and $x^{\prime}$ Permalink: http://dlmf.nist.gov/8.17.E18 Encodings: TeX, pMML, png See also: Annotations for §8.17(iv), §8.17 and Ch.8

8.17.19		$I_{x}\left(a,b\right)=I_{x}\left(a-1,b+1\right)-\frac{x^{a-1}(x^{\prime})^{b}}% {b\mathrm{B}\left(a,b\right)},$
ⓘ Symbols: $\mathrm{B}\left(\NVar{a},\NVar{b}\right)$ : beta function, $I_{\NVar{x}}\left(\NVar{a},\NVar{b}\right)$ : incomplete beta function, $a$ : parameter, $b$ : parameter, $x$ : variable and $x^{\prime}$ Permalink: http://dlmf.nist.gov/8.17.E19 Encodings: TeX, pMML, png See also: Annotations for §8.17(iv), §8.17 and Ch.8

8.17.20	$\displaystyle I_{x}\left(a,b\right)$	$\displaystyle=I_{x}\left(a+1,b\right)+\frac{x^{a}(x^{\prime})^{b}}{a\mathrm{B}% \left(a,b\right)},$
ⓘ Symbols: $\mathrm{B}\left(\NVar{a},\NVar{b}\right)$ : beta function, $I_{\NVar{x}}\left(\NVar{a},\NVar{b}\right)$ : incomplete beta function, $a$ : parameter, $b$ : parameter, $x$ : variable and $x^{\prime}$ Permalink: http://dlmf.nist.gov/8.17.E20 Encodings: TeX, pMML, png See also: Annotations for §8.17(iv), §8.17 and Ch.8
8.17.21	$\displaystyle I_{x}\left(a,b\right)$	$\displaystyle=I_{x}\left(a,b+1\right)-\frac{x^{a}(x^{\prime})^{b}}{b\mathrm{B}% \left(a,b\right)}.$
ⓘ Symbols: $\mathrm{B}\left(\NVar{a},\NVar{b}\right)$ : beta function, $I_{\NVar{x}}\left(\NVar{a},\NVar{b}\right)$ : incomplete beta function, $a$ : parameter, $b$ : parameter, $x$ : variable and $x^{\prime}$ Permalink: http://dlmf.nist.gov/8.17.E21 Encodings: TeX, pMML, png See also: Annotations for §8.17(iv), §8.17 and Ch.8

§8.17(v) Continued Fraction

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Keywords:: continued fraction, incomplete beta functions
Notes:: See Temme (1996b, p. 289). For the last paragraph, see Zhang and Jin (1996, p. 65).
Referenced by:: §3.10(ii)
Permalink:: http://dlmf.nist.gov/8.17.v
See also:: Annotations for §8.17 and Ch.8

8.17.22		$I_{x}\left(a,b\right)=\frac{x^{a}(1-x)^{b}}{a\mathrm{B}\left(a,b\right)}\left(% \cfrac{1}{1+\cfrac{d_{1}}{1+\cfrac{d_{2}}{1+\cfrac{d_{3}}{1+}}}}\cdots\right),$
ⓘ Symbols: $\mathrm{B}\left(\NVar{a},\NVar{b}\right)$ : beta function, $I_{\NVar{x}}\left(\NVar{a},\NVar{b}\right)$ : incomplete beta function, $a$ : parameter, $d_{m}$ : coefficients, $b$ : parameter and $x$ : variable Referenced by: §8.17(v) Permalink: http://dlmf.nist.gov/8.17.E22 Encodings: TeX, pMML, png See also: Annotations for §8.17(v), §8.17 and Ch.8

where

8.17.23	$\displaystyle d_{2m}$	$\displaystyle=\frac{m(b-m)x}{(a+2m-1)(a+2m)},$
8.17.23	$\displaystyle d_{2m+1}$	$\displaystyle=-\frac{(a+m)(a+b+m)x}{(a+2m)(a+2m+1)}.$
ⓘ 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

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Keywords:: incomplete beta functions, sums
Permalink:: http://dlmf.nist.gov/8.17.vi
See also:: Annotations for §8.17 and Ch.8

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

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Keywords:: basic properties, incomplete beta functions
Notes:: The material in this subsection was added in Version 1.0.5. It will be incorporated in the next print edition.
Permalink:: http://dlmf.nist.gov/8.17.vii
See also:: Annotations for §8.17 and Ch.8

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).

8.16 Generalizations 8.18 Asymptotic Expansions of

I_{x}\left(a,b\right)

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