# multivariate gamma function

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##### 1: 35.3 Multivariate Gamma and Beta Functions
###### §35.3(ii) Properties
35.3.7 $\mathrm{B}_{m}\left(a,b\right)=\frac{\Gamma_{m}\left(a\right)\Gamma_{m}\left(b% \right)}{\Gamma_{m}\left(a+b\right)}.$
##### 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 $\displaystyle{{}_{p}F_{q}}\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};% \mathbf{T}\right)$. An alternative notation for the multivariate gamma function is $\Pi_{m}(a)=\Gamma_{m}\left(a+\tfrac{1}{2}(m+1)\right)$ (Herz (1955, p. 480)). 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
35.6.2 $\Psi\left(a;b;\mathbf{T}\right)=\frac{1}{\Gamma_{m}\left(a\right)}\int_{% \boldsymbol{\Omega}}\mathrm{etr}\left(-\mathbf{T}\mathbf{X}\right)\left|% \mathbf{X}\right|^{a-\frac{1}{2}(m+1)}\*{\left|\mathbf{I}+\mathbf{X}\right|}^{% b-a-\frac{1}{2}(m+1)}\mathrm{d}{\mathbf{X}},$ $\Re\left(a\right)>\frac{1}{2}(m-1)$, $\mathbf{T}\in{\boldsymbol{\Omega}}$.
35.6.3 $L^{(\gamma)}_{\nu}\left(\mathbf{T}\right)=\frac{\Gamma_{m}\left(\gamma+\nu+% \frac{1}{2}(m+1)\right)}{\Gamma_{m}\left(\gamma+\frac{1}{2}(m+1)\right)}\*{{}_% {1}F_{1}}\left({-\nu\atop\gamma+\frac{1}{2}(m+1)};\mathbf{T}\right),$ $\Re\left(\gamma\right),\Re\left(\gamma+\nu\right)>-1$.
35.6.5 $\int_{\boldsymbol{\Omega}}\mathrm{etr}\left(-\mathbf{T}\mathbf{X}\right)\left|% \mathbf{X}\right|^{b-\frac{1}{2}(m+1)}{{}_{1}F_{1}}\left({a\atop b};\mathbf{S}% \mathbf{X}\right)\mathrm{d}{\mathbf{X}}=\Gamma_{m}\left(b\right)\left|\mathbf{% I}-\mathbf{S}\mathbf{T}^{-1}\right|^{-a}\left|\mathbf{T}\right|^{-b},$ $\mathbf{T}>\mathbf{S}$, $\Re\left(b\right)>\frac{1}{2}(m-1)$.
35.6.8 $\int_{\boldsymbol{\Omega}}\left|\mathbf{T}\right|^{c-\frac{1}{2}(m+1)}\Psi% \left(a;b;\mathbf{T}\right)\mathrm{d}{\mathbf{T}}=\frac{\Gamma_{m}\left(c% \right)\Gamma_{m}\left(a-c\right)\Gamma_{m}\left(c-b+\frac{1}{2}(m+1)\right)}{% \Gamma_{m}\left(a\right)\Gamma_{m}\left(a-b+\frac{1}{2}(m+1)\right)},$ $\Re\left(a\right)>\Re\left(c\right)+\frac{1}{2}(m-1)>m-1$, $\Re\left(c-b\right)>-1$.
35.6.10 $\lim_{a\to\infty}\Gamma_{m}\left(a\right)\Psi\left(a+\nu;\nu+\tfrac{1}{2}(m+1)% ;a^{-1}\mathbf{T}\right)=B_{\nu}\left(\mathbf{T}\right).$
##### 4: 35.8 Generalized Hypergeometric Functions of Matrix Argument
35.8.4 $A_{\nu}\left(\mathbf{T}\right)=\dfrac{1}{\Gamma_{m}\left(\nu+\frac{1}{2}(m+1)% \right)}{{}_{0}F_{1}}\left({-\atop\nu+\frac{1}{2}(m+1)};-\mathbf{T}\right),$ $\mathbf{T}\in\boldsymbol{\mathcal{S}}$.
35.8.5 ${{}_{3}F_{2}}\left({a_{1},a_{2},a_{3}\atop b_{1},b_{2}};\mathbf{I}\right)=% \frac{\Gamma_{m}\left(b_{2}\right)\Gamma_{m}\left(c\right)}{\Gamma_{m}\left(b_% {2}-a_{3}\right)\Gamma_{m}\left(c+a_{3}\right)}\*{{}_{3}F_{2}}\left({b_{1}-a_{% 1},b_{1}-a_{2},a_{3}\atop b_{1},c+a_{3}};\mathbf{I}\right),$ $\Re\left(b_{2}\right),\Re\left(c\right)>\frac{1}{2}(m-1)$.
35.8.6 ${{}_{3}F_{2}}\left({a_{1},a_{2},a_{3}\atop b_{1},b_{2}};\mathbf{I}\right)=% \frac{\Gamma_{m}\left(b_{1}-a_{1}\right)\Gamma_{m}\left(b_{1}-a_{2}\right)}{% \Gamma_{m}\left(b_{1}\right)\Gamma_{m}\left(b_{1}-a_{1}-a_{2}\right)}\*\frac{% \Gamma_{m}\left(b_{1}-a_{3}\right)\Gamma_{m}\left(b_{1}-a_{1}-a_{2}-a_{3}% \right)}{\Gamma_{m}\left(b_{1}-a_{1}-a_{3}\right)\Gamma_{m}\left(b_{1}-a_{2}-a% _{3}\right)}.$
35.8.7 ${{}_{3}F_{2}}\left({a_{1},a_{2},a_{3}\atop b_{1},b_{2}};\mathbf{I}\right)=% \frac{\Gamma_{m}\left(b_{1}\right)\Gamma_{m}\left(b_{2}\right)\Gamma\left(c% \right)}{\Gamma_{m}\left(a_{1}\right)\Gamma_{m}\left(c+a_{2}\right)\Gamma\left% (c+a_{3}\right)}\*{{}_{3}F_{2}}\left({b_{1}-a_{1},b_{2}-a_{2},c\atop c+a_{2},c% +a_{3}};\mathbf{I}\right),$ $\Re\left(b_{1}\right)$, $\Re\left(b_{2}\right)$, $\Re\left(c\right)>\frac{1}{2}(m-1)$.
35.8.12 ${\int_{\boldsymbol{\Omega}}\mathrm{etr}\left(-\mathbf{T}\mathbf{X}\right)\left% |\mathbf{X}\right|^{\gamma-\frac{1}{2}(m+1)}\*{{}_{p}F_{q}}\left({a_{1},\dots,% a_{p}\atop b_{1},\dots,b_{q}};-\mathbf{X}\right)\mathrm{d}{\mathbf{X}}}=\Gamma% _{m}\left(\gamma\right)\left|\mathbf{T}\right|^{-\gamma}{{}_{p+1}F_{q}}\left({% a_{1},\dots,a_{p},\gamma\atop b_{1},\dots,b_{q}};-\mathbf{T}^{-1}\right),$ $\Re\left(\gamma\right)>\frac{1}{2}(m-1)$.
##### 5: 35.4 Partitions and Zonal Polynomials
35.4.8 $\int_{\boldsymbol{\Omega}}\mathrm{etr}\left(-\mathbf{T}\mathbf{X}\right)\,% \left|\mathbf{X}\right|^{a-\frac{1}{2}(m+1)}Z_{\kappa}\left(\mathbf{X}\right)% \mathrm{d}{\mathbf{X}}=\Gamma_{m}\left(a+\kappa\right)\,\left|\mathbf{T}\right% |^{-a}Z_{\kappa}\left(\mathbf{T}^{-1}\right),$
##### 6: 35.7 Gaussian Hypergeometric Function of Matrix Argument
35.7.2 $P^{(\gamma,\delta)}_{\nu}\left(\mathbf{T}\right)=\frac{\Gamma_{m}\left(\gamma+% \nu+\frac{1}{2}(m+1)\right)}{\Gamma_{m}\left(\gamma+\frac{1}{2}(m+1)\right)}\*% {{}_{2}F_{1}}\left({-\nu,\gamma+\delta+\nu+\frac{1}{2}(m+1)\atop\gamma+\frac{1% }{2}(m+1)};\mathbf{T}\right),$ $\boldsymbol{{0}}<\mathbf{T}<\mathbf{I}$; $\gamma,\delta,\nu\in\mathbb{C}$; $\Re\left(\gamma\right)>-1$.
35.7.7 ${{{}_{2}F_{1}}\left({a,b\atop c};\mathbf{I}\right)=\frac{\Gamma_{m}\left(c% \right)\Gamma_{m}\left(c-a-b\right)}{\Gamma_{m}\left(c-a\right)\Gamma_{m}\left% (c-b\right)}},$ $\Re\left(c\right),\Re\left(c-a-b\right)>\frac{1}{2}(m-1)$.
35.7.8 ${{{}_{2}F_{1}}\left({a,b\atop c};\mathbf{T}\right)=\frac{\Gamma_{m}\left(c% \right)\Gamma_{m}\left(c-a-b\right)}{\Gamma_{m}\left(c-a\right)\Gamma_{m}\left% (c-b\right)}}\*{{}_{2}F_{1}}\left({a,b\atop a+b-c+\frac{1}{2}(m+1)};\mathbf{I}% -\mathbf{T}\right),$ $\boldsymbol{{0}}<\mathbf{T}<\mathbf{I}$; ${\frac{1}{2}}(j+1)-a\in\mathbb{N}$ for some $j=1,\ldots,m$; ${\frac{1}{2}}(j+1)-c\notin\mathbb{N}$ and $c-a-b-{\frac{1}{2}}(m-j)\notin\mathbb{N}$ for all $j=1,\ldots,m$.
##### 7: 19.31 Probability Distributions
19.31.2 $\int_{{\mathbb{R}}^{n}}(\mathbf{x}^{\mathrm{T}}\mathbf{A}\mathbf{x})^{\mu}\exp% \left(-\mathbf{x}^{\mathrm{T}}\mathbf{B}\mathbf{x}\right)\mathrm{d}x_{1}\cdots% \mathrm{d}x_{n}=\frac{\pi^{n/2}\Gamma\left(\mu+\tfrac{1}{2}n\right)}{\sqrt{% \det\mathbf{B}}\Gamma\left(\tfrac{1}{2}n\right)}R_{\mu}\left(\tfrac{1}{2},% \dots,\tfrac{1}{2};\lambda_{1},\dots,\lambda_{n}\right),$ $\mu>-\tfrac{1}{2}n$.
##### 8: 35.5 Bessel Functions of Matrix Argument
35.5.1 $A_{\nu}\left(\boldsymbol{{0}}\right)=\frac{1}{\Gamma_{m}\left(\nu+\frac{1}{2}(% m+1)\right)},$ $\nu\in\mathbb{C}$.
##### 9: 19.23 Integral Representations
19.23.9 $R_{-a}\left(\mathbf{b};\mathbf{z}\right)=\frac{4\Gamma\left(b_{1}+b_{2}+b_{3}% \right)}{\Gamma\left(b_{1}\right)\Gamma\left(b_{2}\right)\Gamma\left(b_{3}% \right)}\int_{0}^{\pi/2}\!\!\!\!\int_{0}^{\pi/2}\left(\sum_{j=1}^{3}z_{j}l_{j}% ^{2}\right)^{-a}\*\prod_{j=1}^{3}l_{j}^{2b_{j}-1}\sin\theta\mathrm{d}\theta% \mathrm{d}\phi,$ $b_{j}>0$, $\Re z_{j}>0$.
##### 10: 19.28 Integrals of Elliptic Integrals
19.28.4 $\int_{0}^{1}t^{\sigma-1}(1-t)^{c-1}R_{-a}\left(b_{1},b_{2};t,1\right)\mathrm{d% }t=\frac{\Gamma\left(c\right)\Gamma\left(\sigma\right)\Gamma\left(\sigma+b_{2}% -a\right)}{\Gamma\left(\sigma+c-a\right)\Gamma\left(\sigma+b_{2}\right)},$ $c=b_{1}+b_{2}>0$, $\Re\sigma>\max(0,a-b_{2})$.