8 Incomplete Gamma and Related Functions Related Functions 8.16 Generalizations 8.18 Asymptotic Expansions of $\mathop{I_{{x}}\/}\nolimits\!\left(a,b\right)$

§8.17 Incomplete Beta Functions

Keywords:: incomplete beta functions
Referenced by:: §8.17(i)
Permalink:: http://dlmf.nist.gov/8.17

§8.17(i) Definitions and Basic Properties
§8.17(ii) Hypergeometric Representations
§8.17(iii) Integral Representation
§8.17(iv) Recurrence Relations
§8.17(v) Continued Fraction
§8.17(vi) Sums
§8.17(vii) Addendum to 8.17(i) Definitions and Basic Properties

§8.17(i) Definitions and Basic Properties

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).
Keywords:: incomplete beta functions
Referenced by:: §8.17(vii), 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

Throughout §§8.17 and 8.18 we assume that , , 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 , , and , and also to complex values.

8.17.1 $\mathop{\mathrm{B}_{{x}}\/}\nolimits\!\left(a,b\right)=\int_{0}^{x}t^{{a-1}}(1% -t)^{{b-1}}dt,$

Defines:: $\mathop{\mathrm{B}_{{x}}\/}\nolimits\!\left(a,b\right)$ : incomplete beta function
Symbols:: : differential of , $\int$ : integral, : parameter, : parameter and : variable
A&S Ref:: 6.6.1
Permalink:: http://dlmf.nist.gov/8.17.E1
Encodings:: TeX, pMML, png

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:: $\mathop{I_{{x}}\/}\nolimits\!\left(a,b\right)$ : incomplete beta function
Symbols:: $\mathop{\mathrm{B}\/}\nolimits\!\left(a,b\right)$ : beta function, $\mathop{\mathrm{B}_{{x}}\/}\nolimits\!\left(a,b\right)$ : incomplete beta function, : parameter, : parameter and : variable
A&S Ref:: 6.6.2
Permalink:: http://dlmf.nist.gov/8.17.E2
Encodings:: TeX, pMML, png

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

8.17.3 $\mathop{\mathrm{B}\/}\nolimits\!\left(a,b\right)=\frac{\mathop{\Gamma\/}% \nolimits\!\left(a\right)\mathop{\Gamma\/}\nolimits\!\left(b\right)}{\mathop{% \Gamma\/}\nolimits\!\left(a+b\right)}.$

Symbols:: $\mathop{\mathrm{B}\/}\nolimits\!\left(a,b\right)$ : beta function, $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, : parameter and : parameter
Permalink:: http://dlmf.nist.gov/8.17.E3
Encodings:: TeX, pMML, png

8.17.4 $\mathop{I_{{x}}\/}\nolimits\!\left(a,b\right)=1-\mathop{I_{{1-x}}\/}\nolimits% \!\left(b,a\right).$

Symbols:: $\mathop{I_{{x}}\/}\nolimits\!\left(a,b\right)$ : incomplete beta function, : parameter, : parameter and : 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

8.17.5 $\mathop{I_{{x}}\/}\nolimits\!\left(m,n-m+1\right)=\sum_{{j=m}}^{n}\binomial{n}% {j}x^{j}(1-x)^{{n-j}},$

positive integers; $0\leq x<1$ .

Symbols:: $\mathop{I_{{x}}\/}\nolimits\!\left(a,b\right)$ : incomplete beta function, $\binom{m}{n}$ : binomial coefficient, : nonnegative integer and : 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

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:: $\mathop{I_{{x}}\/}\nolimits\!\left(a,b\right)$ : incomplete beta function, : parameter and : variable
Referenced by:: §8.17(i)
Permalink:: http://dlmf.nist.gov/8.17.E6
Encodings:: TeX, pMML, png

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

§8.17(ii) Hypergeometric Representations

Notes:: See Temme (1996b, §11.3).
Keywords:: hypergeometric function, incomplete beta functions
Referenced by:: §8.17(iii)
Permalink:: http://dlmf.nist.gov/8.17.ii

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

Symbols:: $\mathop{F\/}\nolimits\!\left(a,b;c;z\right)$ : hypergeometric function, $\mathop{\mathrm{B}_{{x}}\/}\nolimits\!\left(a,b\right)$ : incomplete beta function, : parameter, : parameter and : variable
A&S Ref:: 6.6.8
Permalink:: http://dlmf.nist.gov/8.17.E7
Encodings:: TeX, pMML, png

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

Symbols:: $\mathop{F\/}\nolimits\!\left(a,b;c;z\right)$ : hypergeometric function, $\mathop{\mathrm{B}_{{x}}\/}\nolimits\!\left(a,b\right)$ : incomplete beta function, : parameter, : parameter and : variable
Referenced by:: §8.17(i)
Permalink:: http://dlmf.nist.gov/8.17.E8
Encodings:: TeX, pMML, png

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

Symbols:: $\mathop{F\/}\nolimits\!\left(a,b;c;z\right)$ : hypergeometric function, $\mathop{\mathrm{B}_{{x}}\/}\nolimits\!\left(a,b\right)$ : incomplete beta function, : parameter, : parameter and : variable
Referenced by:: §8.18(i)
Permalink:: http://dlmf.nist.gov/8.17.E9
Encodings:: TeX, pMML, png

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

§8.17(iii) Integral Representation

Notes:: See Temme (1996b, p. 290).
Keywords:: incomplete beta functions
Permalink:: http://dlmf.nist.gov/8.17.iii

With , , and ,

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

Symbols:: $\mathop{I_{{x}}\/}\nolimits\!\left(a,b\right)$ : incomplete beta function, : differential of , $\int$ : integral, : parameter, : parameter, : variable and : parameter
Permalink:: http://dlmf.nist.gov/8.17.E10
Encodings:: TeX, pMML, png

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

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

§8.17(iv) Recurrence Relations

Notes:: See Temme (1996b, pp. 289–290).
Keywords:: incomplete beta functions
Permalink:: http://dlmf.nist.gov/8.17.iv

With

8.17.11

$x^{{\prime}}=1-x,$

Symbols:: : parameter, : parameter, : variable, and $x^{{\prime}}$
Permalink:: http://dlmf.nist.gov/8.17.E11
Encodings:: TeX, TeX, pMML, pMML, png, png

8.17.12 $\mathop{I_{{x}}\/}\nolimits\!\left(a,b\right)=x\mathop{I_{{x}}\/}\nolimits\!% \left(a-1,b\right)+x^{{\prime}}\mathop{I_{{x}}\/}\nolimits\!\left(a,b-1\right),$

Symbols:: $\mathop{I_{{x}}\/}\nolimits\!\left(a,b\right)$ : incomplete beta function, : parameter, : parameter, : variable and $x^{{\prime}}$
A&S Ref:: 6.6.5
Permalink:: http://dlmf.nist.gov/8.17.E12
Encodings:: TeX, pMML, png

8.17.13 $(a+b)\mathop{I_{{x}}\/}\nolimits\!\left(a,b\right)=a\mathop{I_{{x}}\/}% \nolimits\!\left(a+1,b\right)+b\mathop{I_{{x}}\/}\nolimits\!\left(a,b+1\right),$

Symbols:: $\mathop{I_{{x}}\/}\nolimits\!\left(a,b\right)$ : incomplete beta function, : parameter, : parameter and : variable
A&S Ref:: 6.6.7
Permalink:: http://dlmf.nist.gov/8.17.E13
Encodings:: TeX, pMML, png

8.17.14 $(a+bx)\mathop{I_{{x}}\/}\nolimits\!\left(a,b\right)=xb\mathop{I_{{x}}\/}% \nolimits\!\left(a-1,b+1\right)+a\mathop{I_{{x}}\/}\nolimits\!\left(a+1,b% \right),$

Symbols:: $\mathop{I_{{x}}\/}\nolimits\!\left(a,b\right)$ : incomplete beta function, : parameter, : parameter and : variable
Permalink:: http://dlmf.nist.gov/8.17.E14
Encodings:: TeX, pMML, png

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:: $\mathop{I_{{x}}\/}\nolimits\!\left(a,b\right)$ : incomplete beta function, : parameter, : parameter, : 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

8.17.16 $a\mathop{I_{{x}}\/}\nolimits\!\left(a+1,b\right)=(a+cx)\mathop{I_{{x}}\/}% \nolimits\!\left(a,b\right)-cx\mathop{I_{{x}}\/}\nolimits\!\left(a-1,b\right),$

Symbols:: $\mathop{I_{{x}}\/}\nolimits\!\left(a,b\right)$ : incomplete beta function, : parameter, : parameter, : variable and
Permalink:: http://dlmf.nist.gov/8.17.E16
Encodings:: TeX, pMML, png

8.17.17 $b\mathop{I_{{x}}\/}\nolimits\!\left(a,b+1\right)=(b+cx^{{\prime}})\mathop{I_{{% x}}\/}\nolimits\!\left(a,b\right)-cx^{{\prime}}\mathop{I_{{x}}\/}\nolimits\!% \left(a,b-1\right),$

Symbols:: $\mathop{I_{{x}}\/}\nolimits\!\left(a,b\right)$ : incomplete beta function, : parameter, : parameter, : variable, and $x^{{\prime}}$
Permalink:: http://dlmf.nist.gov/8.17.E17
Encodings:: TeX, pMML, png

8.17.18 $\mathop{I_{{x}}\/}\nolimits\!\left(a,b\right)=\mathop{I_{{x}}\/}\nolimits\!% \left(a+1,b-1\right)+\frac{x^{a}(x^{{\prime}})^{{b-1}}}{a\mathop{\mathrm{B}\/}% \nolimits\!\left(a,b\right)},$

Symbols:: $\mathop{\mathrm{B}\/}\nolimits\!\left(a,b\right)$ : beta function, $\mathop{I_{{x}}\/}\nolimits\!\left(a,b\right)$ : incomplete beta function, : parameter, : parameter, : variable and $x^{{\prime}}$
Permalink:: http://dlmf.nist.gov/8.17.E18
Encodings:: TeX, pMML, png

8.17.19 $\mathop{I_{{x}}\/}\nolimits\!\left(a,b\right)=\mathop{I_{{x}}\/}\nolimits\!% \left(a-1,b+1\right)-\frac{x^{{a-1}}(x^{{\prime}})^{b}}{b\mathop{\mathrm{B}\/}% \nolimits\!\left(a,b\right)},$

Symbols:: $\mathop{\mathrm{B}\/}\nolimits\!\left(a,b\right)$ : beta function, $\mathop{I_{{x}}\/}\nolimits\!\left(a,b\right)$ : incomplete beta function, : parameter, : parameter, : variable and $x^{{\prime}}$
Permalink:: http://dlmf.nist.gov/8.17.E19
Encodings:: TeX, pMML, png

8.17.20 $\mathop{I_{{x}}\/}\nolimits\!\left(a,b\right)=\mathop{I_{{x}}\/}\nolimits\!% \left(a+1,b\right)+\frac{x^{a}(x^{{\prime}})^{b}}{a\mathop{\mathrm{B}\/}% \nolimits\!\left(a,b\right)},$

Symbols:: $\mathop{\mathrm{B}\/}\nolimits\!\left(a,b\right)$ : beta function, $\mathop{I_{{x}}\/}\nolimits\!\left(a,b\right)$ : incomplete beta function, : parameter, : parameter, : variable and $x^{{\prime}}$
Permalink:: http://dlmf.nist.gov/8.17.E20
Encodings:: TeX, pMML, png

8.17.21 $\mathop{I_{{x}}\/}\nolimits\!\left(a,b\right)=\mathop{I_{{x}}\/}\nolimits\!% \left(a,b+1\right)-\frac{x^{a}(x^{{\prime}})^{b}}{b\mathop{\mathrm{B}\/}% \nolimits\!\left(a,b\right)}.$

Symbols:: $\mathop{\mathrm{B}\/}\nolimits\!\left(a,b\right)$ : beta function, $\mathop{I_{{x}}\/}\nolimits\!\left(a,b\right)$ : incomplete beta function, : parameter, : parameter, : variable and $x^{{\prime}}$
Permalink:: http://dlmf.nist.gov/8.17.E21
Encodings:: TeX, pMML, png

§8.17(v) Continued Fraction

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

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

Symbols:: $\mathop{\mathrm{B}\/}\nolimits\!\left(a,b\right)$ : beta function, $\mathop{I_{{x}}\/}\nolimits\!\left(a,b\right)$ : incomplete beta function, : parameter, $d_{m}$ : coefficients, : parameter and : variable
Referenced by:: §8.17(v)
Permalink:: http://dlmf.nist.gov/8.17.E22
Encodings:: TeX, pMML, png

where

8.17.23

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

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

Symbols:: : parameter, $d_{m}$ : coefficients, : parameter and : variable
Permalink:: http://dlmf.nist.gov/8.17.E23
Encodings:: TeX, TeX, pMML, pMML, png, png

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

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

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

§8.17(vi) Sums

Keywords:: incomplete beta functions
Permalink:: http://dlmf.nist.gov/8.17.vi

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

Notes:: The material in this subsection was added in Version 1.0.5. It will be incorporated in the next print edition.
Keywords:: incomplete beta functions
Permalink:: http://dlmf.nist.gov/8.17.vii

8.17.24 $\mathop{I_{{x}}\/}\nolimits\!\left(m,n\right)=(1-x)^{n}\sum_{{j=m}}^{{\infty}}% \binomial{n+j-1}{j}x^{j},$