§35.3 Multivariate Gamma and Beta Functions

Permalink:: http://dlmf.nist.gov/35.3

§35.3(i) Definitions
§35.3(ii) Properties

§35.3(i) Definitions

Notes:: See Wishart (1928), Ingham (1933), Gindikin (1964).
Defines:: $\mathop{\Gamma _{{m}}\/}\nolimits\!\left(a\right)$ : multivariate gamma function
Keywords:: multivariate gamma function
Permalink:: http://dlmf.nist.gov/35.3.i

35.3.1 $\mathop{\Gamma _{{m}}\/}\nolimits\!\left(a\right)=\int _{{\boldsymbol{\Omega}}}\mathop{\mathrm{etr}\/}\nolimits\!\left(-\mathbf{X}\right)|\mathbf{X}|^{{a-\frac{1}{2}(m+1)}}d\mathbf{X},$ $\realpart{(a)}>\frac{1}{2}(m-1)$ .

Symbols:: $dx$ : differential of $x$ , $\mathop{\mathrm{etr}\/}\nolimits\!\left(\mathbf{X}\right)$ : exponential of trace, $\int$ : integral, $\mathop{\Gamma _{{m}}\/}\nolimits\!\left(a\right)$ : multivariate gamma function, $\realpart{}$ : real part, $a$ : complex variable, ${\boldsymbol{\Omega}}$ : space of matrices and $m$ : positive integer
Permalink:: http://dlmf.nist.gov/35.3.E1
Encodings:: TeX, pMML, png

35.3.2 $\mathop{\Gamma _{{m}}\/}\nolimits\!\left(s_{1},\dots,s_{m}\right)=\int _{{\boldsymbol{\Omega}}}\mathop{\mathrm{etr}\/}\nolimits\!\left(-\mathbf{X}\right)|\mathbf{X}|^{{s_{m}-\frac{1}{2}(m+1)}}\prod _{{j=1}}^{{m-1}}|(\mathbf{X})_{j}|^{{s_{j}-s_{{j+1}}}}d\mathbf{X},$ $s_{j}\in\Complex$ , $\realpart{(s_{j})}>\frac{1}{2}(j-1)$ , $j=1,\dots,m$ .

Symbols:: $\Complex$ : complex plane (excluding infinity), $dx$ : differential of $x$ , $\in$ : element of, $\mathop{\mathrm{etr}\/}\nolimits\!\left(\mathbf{X}\right)$ : exponential of trace, $\int$ : integral, $\mathop{\Gamma _{{m}}\/}\nolimits\!\left(a\right)$ : multivariate gamma function, $\realpart{}$ : real part, ${\boldsymbol{\Omega}}$ : space of matrices, $j$ : nonnegative integer and $m$ : positive integer
Permalink:: http://dlmf.nist.gov/35.3.E2
Encodings:: TeX, pMML, png

35.3.3 $\mathop{\mathrm{B}_{{m}}\/}\nolimits\!\left(a,b\right)=\int\limits _{{\boldsymbol{{0}}<\mathbf{X}<\mathbf{I}}}|\mathbf{X}|^{{a-\frac{1}{2}(m+1)}}|\mathbf{I}-\mathbf{X}|^{{b-\frac{1}{2}(m+1)}}d\mathbf{X},$ $\realpart{(a)},\realpart{(b)}>\frac{1}{2}(m-1)$ .

Defines:: $\mathop{\mathrm{B}_{{m}}\/}\nolimits\!\left(a,b\right)$ : multivariate beta function
Symbols:: $dx$ : differential of $x$ , $\int$ : integral, $\realpart{}$ : real part, $a$ : complex variable, $b$ : complex variable and $m$ : positive integer
Keywords:: multivariate beta function
Permalink:: http://dlmf.nist.gov/35.3.E3
Encodings:: TeX, pMML, png

§35.3(ii) Properties

Notes:: See Wishart (1928), Ingham (1933), Gårding (1947), Herz (1955), Olkin (1959).
Keywords:: multivariate beta function, multivariate gamma function
Permalink:: http://dlmf.nist.gov/35.3.ii

35.3.4 $\mathop{\Gamma _{{m}}\/}\nolimits\!\left(a\right)=\pi^{{m(m-1)/4}}\prod _{{j=1}}^{m}\mathop{\Gamma\/}\nolimits\!\left(a-\tfrac{1}{2}(j-1)\right).$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\Gamma _{{m}}\/}\nolimits\!\left(a\right)$ : 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

35.3.5 $\mathop{\Gamma _{{m}}\/}\nolimits\!\left(s_{1},\dots,s_{m}\right)=\pi^{{m(m-1)/4}}\prod _{{j=1}}^{m}\mathop{\Gamma\/}\nolimits\!\left(s_{j}-\tfrac{1}{2}(j-1)\right).$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\Gamma _{{m}}\/}\nolimits\!\left(a\right)$ : multivariate gamma function, $j$ : nonnegative integer and $m$ : positive integer
Permalink:: http://dlmf.nist.gov/35.3.E5
Encodings:: TeX, pMML, png

35.3.6 $\mathop{\Gamma _{{m}}\/}\nolimits\!\left(a,\dots,a\right)=\mathop{\Gamma _{{m}}\/}\nolimits\!\left(a\right).$

Symbols:: $\mathop{\Gamma _{{m}}\/}\nolimits\!\left(a\right)$ : multivariate gamma function, $a$ : complex variable and $m$ : positive integer
Permalink:: http://dlmf.nist.gov/35.3.E6
Encodings:: TeX, pMML, png

35.3.7 $\mathop{\mathrm{B}_{{m}}\/}\nolimits\!\left(a,b\right)=\frac{\mathop{\Gamma _{{m}}\/}\nolimits\!\left(a\right)\mathop{\Gamma _{{m}}\/}\nolimits\!\left(b\right)}{\mathop{\Gamma _{{m}}\/}\nolimits\!\left(a+b\right)}.$

Symbols:: $\mathop{\mathrm{B}_{{m}}\/}\nolimits\!\left(a,b\right)$ : multivariate beta function, $\mathop{\Gamma _{{m}}\/}\nolimits\!\left(a\right)$ : 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

35.3.8 $\mathop{\mathrm{B}_{{m}}\/}\nolimits\!\left(a,b\right)=\int _{{\boldsymbol{\Omega}}}|\mathbf{X}|^{{a-\frac{1}{2}(m+1)}}|\mathbf{I}+\mathbf{X}|^{{-(a+b)}}d\mathbf{X},$ $\realpart{(a)},\realpart{(b)}>\frac{1}{2}(m-1)$ .

Symbols:: $dx$ : differential of $x$ , $\int$ : integral, $\mathop{\mathrm{B}_{{m}}\/}\nolimits\!\left(a,b\right)$ : multivariate beta function, $\realpart{}$ : real part, $a$ : complex variable, ${\boldsymbol{\Omega}}$ : space of matrices, $b$ : complex variable and $m$ : positive integer
Permalink:: http://dlmf.nist.gov/35.3.E8
Encodings:: TeX, pMML, png

§35.3 Multivariate Gamma and Beta Functions

Contents

§35.3(i) Definitions

§35.3(ii) Properties