# multivariate beta function

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##### 1: 35.3 Multivariate Gamma and Beta Functions
###### §35.3 Multivariate Gamma and BetaFunctions
35.3.3 $\mathrm{B}_{m}\left(a,b\right)=\int\limits_{\boldsymbol{{0}}<\mathbf{X}<% \mathbf{I}}\left|\mathbf{X}\right|^{a-\frac{1}{2}(m+1)}\left|\mathbf{I}-% \mathbf{X}\right|^{b-\frac{1}{2}(m+1)}\mathrm{d}{\mathbf{X}},$ $\Re\left(a\right),\Re\left(b\right)>\frac{1}{2}(m-1)$.
35.3.7 $\mathrm{B}_{m}\left(a,b\right)=\frac{\Gamma_{m}\left(a\right)\Gamma_{m}\left(b% \right)}{\Gamma_{m}\left(a+b\right)}.$
35.3.8 $\mathrm{B}_{m}\left(a,b\right)=\int_{\boldsymbol{\Omega}}\left|\mathbf{X}% \right|^{a-\frac{1}{2}(m+1)}\left|\mathbf{I}+\mathbf{X}\right|^{-(a+b)}\mathrm% {d}{\mathbf{X}},$ $\Re\left(a\right),\Re\left(b\right)>\frac{1}{2}(m-1)$.
##### 2: 35.1 Special Notation
 $a,b$ complex variables. …
The main functions treated in this chapter are the multivariate gamma and beta functions, respectively $\Gamma_{m}\left(a\right)$ and $\mathrm{B}_{m}\left(a,b\right)$, and the special functions of matrix argument: Bessel (of the first kind) $A_{\nu}\left(\mathbf{T}\right)$ and (of the second kind) $B_{\nu}\left(\mathbf{T}\right)$; confluent hypergeometric (of the first kind) ${{}_{1}F_{1}}\left(a;b;\mathbf{T}\right)$ or $\displaystyle{{}_{1}F_{1}}\left({a\atop b};\mathbf{T}\right)$ and (of the second kind) $\Psi\left(a;b;\mathbf{T}\right)$; Gaussian hypergeometric ${{}_{2}F_{1}}\left(a_{1},a_{2};b;\mathbf{T}\right)$ or $\displaystyle{{}_{2}F_{1}}\left({a_{1},a_{2}\atop b};\mathbf{T}\right)$; generalized hypergeometric ${{}_{p}F_{q}}\left(a_{1},\dots,a_{p};b_{1},\dots,b_{q};\mathbf{T}\right)$ or $\displaystyle{{}_{p}F_{q}}\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};% \mathbf{T}\right)$. … Related notations for the Bessel functions are $\mathcal{J}_{\nu+\frac{1}{2}(m+1)}(\mathbf{T})=A_{\nu}\left(\mathbf{T}\right)/% A_{\nu}\left(\boldsymbol{{0}}\right)$ (Faraut and Korányi (1994, pp. 320–329)), $K_{m}(0,\dots,0,\nu\mathpunct{|}\mathbf{S},\mathbf{T})=\left|\mathbf{T}\right|% ^{\nu}B_{\nu}\left(\mathbf{S}\mathbf{T}\right)$ (Terras (1988, pp. 49–64)), and $\mathcal{K}_{\nu}(\mathbf{T})=\left|\mathbf{T}\right|^{\nu}B_{\nu}\left(% \mathbf{S}\mathbf{T}\right)$ (Faraut and Korányi (1994, pp. 357–358)).
##### 3: 35.6 Confluent Hypergeometric Functions of Matrix Argument
35.6.4 ${{}_{1}F_{1}}\left({a\atop b};\mathbf{T}\right)=\frac{1}{\mathrm{B}_{m}\left(a% ,b-a\right)}\int\limits_{\boldsymbol{{0}}<\mathbf{X}<\mathbf{I}}\mathrm{etr}% \left(\mathbf{T}\mathbf{X}\right)\left|\mathbf{X}\right|^{a-\frac{1}{2}(m+1)}% \left|\mathbf{I}-\mathbf{X}\right|^{b-a-\frac{1}{2}(m+1)}\mathrm{d}{\mathbf{X}},$ $\Re\left(a\right),\Re\left(b-a\right)>\frac{1}{2}(m-1)$.
35.6.6 $\mathrm{B}_{m}\left(b_{1},b_{2}\right)\left|\mathbf{T}\right|^{b_{1}+b_{2}-% \frac{1}{2}(m+1)}{{}_{1}F_{1}}\left({a_{1}+a_{2}\atop b_{1}+b_{2}};\mathbf{T}% \right)=\int_{\boldsymbol{{0}}<\mathbf{X}<\mathbf{T}}\left|\mathbf{X}\right|^{% b_{1}-\frac{1}{2}(m+1)}{{}_{1}F_{1}}\left({a_{1}\atop b_{1}};\mathbf{X}\right)% {\left|\mathbf{T}-\mathbf{X}\right|}^{b_{2}-\frac{1}{2}(m+1)}{{}_{1}F_{1}}% \left({a_{2}\atop b_{2}};\mathbf{T}-\mathbf{X}\right)\mathrm{d}{\mathbf{X}},$ $\Re\left(b_{1}\right),\Re\left(b_{2}\right)>\frac{1}{2}(m-1)$.
##### 4: 35.4 Partitions and Zonal Polynomials
35.4.9 $\int\limits_{\boldsymbol{{0}}<\mathbf{X}<\mathbf{I}}\left|\mathbf{X}\right|^{a% -\frac{1}{2}(m+1)}\*\left|\mathbf{I}-\mathbf{X}\right|^{b-\frac{1}{2}(m+1)}Z_{% \kappa}\left(\mathbf{T}\mathbf{X}\right)\mathrm{d}{\mathbf{X}}=\frac{{\left[a% \right]_{\kappa}}}{{\left[a+b\right]_{\kappa}}}\mathrm{B}_{m}\left(a,b\right)Z% _{\kappa}\left(\mathbf{T}\right).$
##### 5: 19.16 Definitions
19.16.9 $R_{-a}\left(\mathbf{b};\mathbf{z}\right)=\frac{1}{\mathrm{B}\left(a,a^{\prime}% \right)}\int_{0}^{\infty}t^{a^{\prime}-1}\prod^{n}_{j=1}(t+z_{j})^{-b_{j}}% \mathrm{d}t=\frac{1}{\mathrm{B}\left(a,a^{\prime}\right)}\int_{0}^{\infty}t^{a% -1}\prod^{n}_{j=1}(1+tz_{j})^{-b_{j}}\mathrm{d}t,$ $b_{1}+\cdots+b_{n}>a>0$, $b_{j}\in\mathbb{R}$, $z_{j}\in\mathbb{C}\setminus(-\infty,0]$,
19.16.12 $R_{-a}\left(b_{1},\dots,b_{4};c-1,c-k^{2},c,c-\alpha^{2}\right)=\frac{2({\sin}% ^{2}\phi)^{1-a^{\prime}}}{\mathrm{B}\left(a,a^{\prime}\right)}\int_{0}^{\phi}(% \sin\theta)^{2a-1}{({\sin}^{2}\phi-{\sin}^{2}\theta)}^{a^{\prime}-1}\*(\cos% \theta)^{1-2b_{1}}{(1-k^{2}{\sin}^{2}\theta)}^{-b_{2}}{(1-\alpha^{2}{\sin}^{2}% \theta)}^{-b_{4}}\mathrm{d}\theta,$
19.16.19 $R_{-a}\left(b_{1},\dots,b_{n};0,z_{2},\dots,z_{n}\right)=\frac{\mathrm{B}\left% (a,a^{\prime}-b_{1}\right)}{\mathrm{B}\left(a,a^{\prime}\right)}R_{-a}\left(b_% {2},\dots,b_{n};z_{2},\dots,z_{n}\right),$ $a+a^{\prime}>0$, $a^{\prime}>b_{1}$.
19.16.24 $R_{-a}\left(\mathbf{b};\mathbf{z}\right)=\frac{z_{1}^{a^{\prime}-b_{1}}}{% \mathrm{B}\left(b_{1},a^{\prime}-b_{1}\right)}\int_{0}^{\infty}t^{b_{1}-1}(t+z% _{1})^{-a^{\prime}}\*R_{-a}\left(\mathbf{b};0,t+z_{2},\dots,t+z_{n}\right)% \mathrm{d}t,$ $a^{\prime}>b_{1}$, $a+a^{\prime}>b_{1}>0$.
##### 6: 19.23 Integral Representations
19.23.8 $R_{-a}\left(\mathbf{b};\mathbf{z}\right)=\frac{2}{\mathrm{B}\left(b_{1},b_{2}% \right)}\int_{0}^{\pi/2}{(z_{1}{\cos}^{2}\theta+z_{2}{\sin}^{2}\theta)}^{-a}\*% (\cos\theta)^{2b_{1}-1}(\sin\theta)^{2b_{2}-1}\mathrm{d}\theta,$ $b_{1},b_{2}>0$; $\Re z_{1},\Re z_{2}>0$.
19.23.10 $R_{-a}\left(\mathbf{b};\mathbf{z}\right)=\frac{1}{\mathrm{B}\left(a,a^{\prime}% \right)}\int_{0}^{1}u^{a-1}(1-u)^{a^{\prime}-1}\*\prod_{j=1}^{n}(1-u+uz_{j})^{% -b_{j}}\mathrm{d}u,$ $a,a^{\prime}>0$; $a+a^{\prime}=\sum_{j=1}^{n}b_{j}$; $z_{j}\in\mathbb{C}\setminus(-\infty,0]$.
##### 7: 35.7 Gaussian Hypergeometric Function of Matrix Argument
35.7.5 ${{}_{2}F_{1}}\left({a,b\atop c};\mathbf{T}\right)=\frac{1}{\mathrm{B}_{m}\left% (a,c-a\right)}\int\limits_{\boldsymbol{{0}}<\mathbf{X}<\mathbf{I}}\left|% \mathbf{X}\right|^{a-\frac{1}{2}(m+1)}\*{\left|\mathbf{I}-\mathbf{X}\right|}^{% c-a-\frac{1}{2}(m+1)}{\left|\mathbf{I}-\mathbf{T}\mathbf{X}\right|}^{-b}% \mathrm{d}{\mathbf{X}},$ $\Re\left(a\right),\Re\left(c-a\right)>\frac{1}{2}(m-1)$, $\boldsymbol{{0}}<\mathbf{T}<\mathbf{I}$.
##### 8: 35.8 Generalized Hypergeometric Functions of Matrix Argument
35.8.13 $\int\limits_{\boldsymbol{{0}}<\mathbf{X}<\mathbf{I}}\left|\mathbf{X}\right|^{a% _{1}-\frac{1}{2}(m+1)}{\left|\mathbf{I}-\mathbf{X}\right|}^{b_{1}-a_{1}-\frac{% 1}{2}(m+1)}\*{{}_{p}F_{q}}\left({a_{2},\dots,a_{p+1}\atop b_{2},\dots,b_{q+1}}% ;\mathbf{T}\mathbf{X}\right)\mathrm{d}{\mathbf{X}}=\frac{1}{\mathrm{B}_{m}% \left(b_{1}-a_{1},a_{1}\right)}{{}_{p+1}F_{q+1}}\left({a_{1},\dots,a_{p+1}% \atop b_{1},\dots,b_{q+1}};\mathbf{T}\right),$ $\Re\left(b_{1}-a_{1}\right),\Re\left(a_{1}\right)>\frac{1}{2}(m-1)$.
##### 9: 19.28 Integrals of Elliptic Integrals
Also, $\mathrm{B}$ again denotes the beta function5.12). …
19.28.2 $\int_{0}^{1}t^{\sigma-1}R_{G}\left(0,t,1\right)\mathrm{d}t=\frac{\sigma}{4% \sigma+2}\left(\mathrm{B}\left(\sigma,\tfrac{1}{2}\right)\right)^{2},$
19.28.3 $\int_{0}^{1}t^{\sigma-1}(1-t)R_{D}\left(0,t,1\right)\mathrm{d}t=\frac{3}{4% \sigma+2}\left(\mathrm{B}\left(\sigma,\tfrac{1}{2}\right)\right)^{2}.$
19.28.4 $\int_{0}^{1}t^{\sigma-1}(1-t)^{c-1}R_{-a}\left(b_{1},b_{2};t,1\right)\mathrm{d% }t=\frac{\Gamma\left(c\right)\Gamma\left(\sigma\right)\Gamma\left(\sigma+b_{2}% -a\right)}{\Gamma\left(\sigma+c-a\right)\Gamma\left(\sigma+b_{2}\right)},$ $c=b_{1}+b_{2}>0$, $\Re\sigma>\max(0,a-b_{2})$.
##### 10: Bibliography P
• E. Pairman (1919) Tables of Digamma and Trigamma Functions. In Tracts for Computers, No. 1, K. Pearson (Ed.),
• P. I. Pastro (1985) Orthogonal polynomials and some $q$-beta integrals of Ramanujan. J. Math. Anal. Appl. 112 (2), pp. 517–540.
• K. Pearson (Ed.) (1968) Tables of the Incomplete Beta-function. 2nd edition, Published for the Biometrika Trustees at the Cambridge University Press, Cambridge.
• M. D. Perlman and I. Olkin (1980) Unbiasedness of invariant tests for MANOVA and other multivariate problems. Ann. Statist. 8 (6), pp. 1326–1341.
• H. N. Phien (1990) A note on the computation of the incomplete beta function. Adv. Eng. Software 12 (1), pp. 39–44.