35 Functions of Matrix ArgumentProperties35.2 Laplace Transform35.4 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

1. §35.3(i) Definitions
2. §35.3(ii) Properties

§35.3(i) Definitions

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Defines:

${\mathrm{\Gamma }}_m\left(a\right)$: 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		$${\mathrm{\Gamma }}_m\left(a\right)={\int }_{𝛀}\mathrm{etr}\left(-𝐗\right){\left|𝐗\right|}^{a-\frac{1}{2}(m+1)}d𝐗,$$
$\mathrm{\Re }\left(a\right)>\frac{1}{2}(m-1)$.
ⓘ Symbols: $\mathrm{etr}\left(𝐀\right)$: exponential of trace, $\int$: integral, ${\mathrm{\Gamma }}_m\left(a\right)$: multivariate gamma function, $\mathrm{\Re }$: real part, $a$: complex variable, $𝐗$: real symmetric $m\times m$ matrix, $\left|𝐗\right|$: determinant of $𝐗$, $d$: differential of matrix, $𝛀$: 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		$${\mathrm{\Gamma }}_m\left(s_1,\mathrm{\dots },s_m\right)={\int }_{𝛀}\mathrm{etr}\left(-𝐗\right){\left|𝐗\right|}^{s_m-\frac{1}{2}(m+1)}\prod _{j=1}^{m-1}{|{(𝐗)}_j|}^{s_j-s_{j+1}}d𝐗,$$
$s_j\in ℂ$, $\mathrm{\Re }\left(s_j\right)>\frac{1}{2}(j-1)$, $j=1,\mathrm{\dots },m$.
ⓘ Symbols: $ℂ$: complex plane, $\in$: element of, $\mathrm{etr}\left(𝐀\right)$: exponential of trace, $\int$: integral, ${\mathrm{\Gamma }}_m\left(a\right)$: multivariate gamma function, $\mathrm{\Re }$: real part, $𝐗$: real symmetric $m\times m$ matrix, $\left|𝐗\right|$: determinant of $𝐗$, $|{(𝐗)}_j|$: $j$th principal minor of $𝐗$, $d$: differential of matrix, $𝛀$: 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
$\mathrm{\Re }\left(a\right),\mathrm{\Re }\left(b\right)>\frac{1}{2}(m-1)$.
ⓘ Defines: ${\mathrm{B}}_m(a,b)$: multivariate beta function Symbols: $\int$: integral, $\mathrm{\Re }$: real part, $a$: complex variable, $𝐈$: $m\times m$ identity matrix, $𝐗$: real symmetric $m\times m$ matrix, $\left|𝐗\right|$: determinant of $𝐗$, $d$: differential of matrix, $b$: complex variable, : less-than for matrices, $m$: positive integer and $𝟎$: $m\times m$ 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		$${\mathrm{\Gamma }}_m\left(a\right)={\pi }^{m(m-1)/4}\prod _{j=1}^m\mathrm{\Gamma }\left(a-\frac{1}{2}(j-1)\right).$$
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, ${\mathrm{\Gamma }}_m\left(a\right)$: 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 See also: Annotations for §35.3(ii), §35.3 and Ch.35

35.3.5		$${\mathrm{\Gamma }}_m\left(s_1,\mathrm{\dots },s_m\right)={\pi }^{m(m-1)/4}\prod _{j=1}^m\mathrm{\Gamma }\left(s_j-\frac{1}{2}(j-1)\right).$$
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, ${\mathrm{\Gamma }}_m\left(a\right)$: multivariate gamma function, $j$: nonnegative integer and $m$: positive integer Permalink: http://dlmf.nist.gov/35.3.E5 Encodings: TeX, pMML, png See also: Annotations for §35.3(ii), §35.3 and Ch.35

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

35.3.7		$${\mathrm{B}}_m(a,b)=\frac{{\mathrm{\Gamma }}_m\left(a\right){\mathrm{\Gamma }}_m\left(b\right)}{{\mathrm{\Gamma }}_m\left(a+b\right)}.$$
ⓘ Symbols: ${\mathrm{B}}_m(a,b)$: multivariate beta function, ${\mathrm{\Gamma }}_m\left(a\right)$: 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 See also: Annotations for §35.3(ii), §35.3 and Ch.35

35.3.8		$${\mathrm{B}}_m(a,b)={\int }_{𝛀}{\left|𝐗\right|}^{a-\frac{1}{2}(m+1)}{\left|𝐈+𝐗\right|}^{-(a+b)}d𝐗,$$
$\mathrm{\Re }\left(a\right),\mathrm{\Re }\left(b\right)>\frac{1}{2}(m-1)$.
ⓘ Symbols: $\int$: integral, ${\mathrm{B}}_m(a,b)$: multivariate beta function, $\mathrm{\Re }$: real part, $a$: complex variable, $𝐈$: $m\times m$ identity matrix, $𝐗$: real symmetric $m\times m$ matrix, $\left|𝐗\right|$: determinant of $𝐗$, $d$: differential of matrix, $𝛀$: 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 Transform35.4 Partitions and Zonal Polynomials

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