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DLMF: §35.3 Multivariate Gamma and Beta Functions ‣ Properties ‣ Chapter 35 Functions of Matrix Argument

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35 Functions of Matrix Argument Properties 35.2 Laplace Transform 35.4 Partitions and Zonal Polynomials

§35.3 Multivariate Gamma and Beta Functions

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Permalink:: http://dlmf.nist.gov/35.3
See also:: Annotations for Ch.35

Contents

§35.3(i) Definitions

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Defines:: $\Gamma_{\NVar{m}}\left(\NVar{a}\right)$ : multivariate gamma function
Keywords:: definition, multivariate gamma function
Notes:: See Wishart (1928), Ingham (1933), Gindikin (1964).
Permalink:: http://dlmf.nist.gov/35.3.i
See also:: Annotations for §35.3 and Ch.35

35.3.1		$\Gamma_{m}\left(a\right)=\int_{\boldsymbol{\Omega}}\operatorname{etr}\left(-% \mathbf{X}\right)\left\|\mathbf{X}\right\|^{a-\frac{1}{2}(m+1)}\,\mathrm{d}{% \mathbf{X}},$
$\Re\left(a\right)>\frac{1}{2}(m-1)$ .
ⓘ Symbols: $\operatorname{etr}\left(\NVar{\mathbf{A}}\right)$ : exponential of trace, $\int$ : integral, $\Gamma_{\NVar{m}}\left(\NVar{a}\right)$ : multivariate gamma function, $\Re$ : real part, $a$ : complex variable, $\mathbf{X}$ : real symmetric $m\times m$ matrix, $\left\|\NVar{\mathbf{X}}\right\|$ : determinant of $\NVar{\mathbf{X}}$ , $\,\mathrm{d}$ : differential of matrix, ${\boldsymbol{\Omega}}$ : space of positive-definite real symmetric $m\times m$ matrices and $m$ : positive integer Permalink: http://dlmf.nist.gov/35.3.E1 Encodings: TeX, pMML, png See also: Annotations for §35.3(i), §35.3 and Ch.35

35.3.2		$\Gamma_{m}\left(s_{1},\dots,s_{m}\right)=\int_{\boldsymbol{\Omega}}% \operatorname{etr}\left(-\mathbf{X}\right)\left\|\mathbf{X}\right\|^{s_{m}-\frac% {1}{2}(m+1)}\prod_{j=1}^{m-1}\|(\mathbf{X})_{j}\|^{s_{j}-s_{j+1}}\,\mathrm{d}{% \mathbf{X}},$
$s_{j}\in\mathbb{C}$ , $\Re\left(s_{j}\right)>\frac{1}{2}(j-1)$ , $j=1,\dots,m$ .
ⓘ Symbols: $\mathbb{C}$ : complex plane, $\in$ : element of, $\operatorname{etr}\left(\NVar{\mathbf{A}}\right)$ : exponential of trace, $\int$ : integral, $\Gamma_{\NVar{m}}\left(\NVar{a}\right)$ : multivariate gamma function, $\Re$ : real part, $\mathbf{X}$ : real symmetric $m\times m$ matrix, $\left\|\NVar{\mathbf{X}}\right\|$ : determinant of $\NVar{\mathbf{X}}$ , $\|(\NVar{\mathbf{X}})_{\NVar{j}}\|$ : $\NVar{j}$ th principal minor of $\NVar{\mathbf{X}}$ , $\,\mathrm{d}$ : differential of matrix, ${\boldsymbol{\Omega}}$ : space of positive-definite real symmetric $m\times m$ matrices, $j$ : nonnegative integer and $m$ : positive integer Permalink: http://dlmf.nist.gov/35.3.E2 Encodings: TeX, pMML, png See also: Annotations for §35.3(i), §35.3 and Ch.35

35.3.3		$\mathrm{B}_{m}\left(a,b\right)=\int\limits_{\boldsymbol{{0}}<\mathbf{X}<% \mathbf{I}}\left\|\mathbf{X}\right\|^{a-\frac{1}{2}(m+1)}\left\|\mathbf{I}-% \mathbf{X}\right\|^{b-\frac{1}{2}(m+1)}\,\mathrm{d}{\mathbf{X}},$
$\Re\left(a\right),\Re\left(b\right)>\frac{1}{2}(m-1)$ .
ⓘ Defines: $\mathrm{B}_{\NVar{m}}\left(\NVar{a},\NVar{b}\right)$ : multivariate beta function Symbols: $\int$ : integral, $\Re$ : real part, $a$ : complex variable, $\mathbf{I}$ : $m\times m$ identity matrix, $\mathbf{X}$ : real symmetric $m\times m$ matrix, $\left\|\NVar{\mathbf{X}}\right\|$ : determinant of $\NVar{\mathbf{X}}$ , $\,\mathrm{d}$ : differential of matrix, $b$ : complex variable, $<$ : less-than for matrices, $m$ : positive integer and $\boldsymbol{{0}}$ : $m\times m$ matrix of zeros Keywords: definition, multivariate beta function Permalink: http://dlmf.nist.gov/35.3.E3 Encodings: TeX, pMML, png See also: Annotations for §35.3(i), §35.3 and Ch.35

§35.3(ii) Properties

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Keywords:: multivariate beta function, multivariate gamma function, properties
Notes:: See Wishart (1928), Ingham (1933), Gårding (1947), Herz (1955), Olkin (1959).
Permalink:: http://dlmf.nist.gov/35.3.ii
See also:: Annotations for §35.3 and Ch.35

35.3.4		$\Gamma_{m}\left(a\right)=\pi^{m(m-1)/4}\prod_{j=1}^{m}\Gamma\left(a-\tfrac{1}{% 2}(j-1)\right).$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\Gamma_{\NVar{m}}\left(\NVar{a}\right)$ : multivariate gamma function, $a$ : complex variable, $j$ : nonnegative integer and $m$ : positive integer Permalink: http://dlmf.nist.gov/35.3.E4 Encodings: TeX, pMML, png See also: Annotations for §35.3(ii), §35.3 and Ch.35

35.3.5		$\Gamma_{m}\left(s_{1},\dots,s_{m}\right)=\pi^{m(m-1)/4}\prod_{j=1}^{m}\Gamma% \left(s_{j}-\tfrac{1}{2}(j-1)\right).$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\Gamma_{\NVar{m}}\left(\NVar{a}\right)$ : multivariate gamma function, $j$ : nonnegative integer and $m$ : positive integer Permalink: http://dlmf.nist.gov/35.3.E5 Encodings: TeX, pMML, png See also: Annotations for §35.3(ii), §35.3 and Ch.35

35.3.6		$\Gamma_{m}\left(a,\dots,a\right)=\Gamma_{m}\left(a\right).$
ⓘ Symbols: $\Gamma_{\NVar{m}}\left(\NVar{a}\right)$ : multivariate gamma function, $a$ : complex variable and $m$ : positive integer Permalink: http://dlmf.nist.gov/35.3.E6 Encodings: TeX, pMML, png See also: Annotations for §35.3(ii), §35.3 and Ch.35

35.3.7		$\mathrm{B}_{m}\left(a,b\right)=\frac{\Gamma_{m}\left(a\right)\Gamma_{m}\left(b% \right)}{\Gamma_{m}\left(a+b\right)}.$
ⓘ Symbols: $\mathrm{B}_{\NVar{m}}\left(\NVar{a},\NVar{b}\right)$ : multivariate beta function, $\Gamma_{\NVar{m}}\left(\NVar{a}\right)$ : multivariate gamma function, $a$ : complex variable, $b$ : complex variable and $m$ : positive integer Permalink: http://dlmf.nist.gov/35.3.E7 Encodings: TeX, pMML, png See also: Annotations for §35.3(ii), §35.3 and Ch.35

35.3.8		$\mathrm{B}_{m}\left(a,b\right)=\int_{\boldsymbol{\Omega}}\left\|\mathbf{X}% \right\|^{a-\frac{1}{2}(m+1)}\left\|\mathbf{I}+\mathbf{X}\right\|^{-(a+b)}\,% \mathrm{d}{\mathbf{X}},$
$\Re\left(a\right),\Re\left(b\right)>\frac{1}{2}(m-1)$ .
ⓘ Symbols: $\int$ : integral, $\mathrm{B}_{\NVar{m}}\left(\NVar{a},\NVar{b}\right)$ : multivariate beta function, $\Re$ : real part, $a$ : complex variable, $\mathbf{I}$ : $m\times m$ identity matrix, $\mathbf{X}$ : real symmetric $m\times m$ matrix, $\left\|\NVar{\mathbf{X}}\right\|$ : determinant of $\NVar{\mathbf{X}}$ , $\,\mathrm{d}$ : differential of matrix, ${\boldsymbol{\Omega}}$ : space of positive-definite real symmetric $m\times m$ matrices, $b$ : complex variable and $m$ : positive integer Permalink: http://dlmf.nist.gov/35.3.E8 Encodings: TeX, pMML, png See also: Annotations for §35.3(ii), §35.3 and Ch.35

35.2 Laplace Transform 35.4 Partitions and Zonal Polynomials

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