multivariate

SearchAdvancedHelp

(0.001 seconds)

1—10 of 37 matching pages

1: 35.3 Multivariate Gamma and Beta Functions

§35.3 Multivariate Gamma and Beta Functions

►

§35.3(i) Definitions

… ►

§35.3(ii) Properties

… ►

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:

pMML, png, TeX

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:

pMML, png, TeX

See also:

Annotations for §35.3(ii), §35.3 and Ch.35

…

2: 35.9 Applications

§35.9 Applications

►In multivariate statistical analysis based on the multivariate normal distribution, the probability density functions of many random matrices are expressible in terms of generalized hypergeometric functions of matrix argument ${}_p{}^{}F_q^{}$, with $p\le 2$ and $q\le 1$. … ►For other statistical applications of ${}_p{}^{}F_q^{}$ functions of matrix argument see Perlman and Olkin (1980), Groeneboom and Truax (2000), Bhaumik and Sarkar (2002), Richards (2004) (monotonicity of power functions of multivariate statistical test criteria), Bingham et al. (1992) (Procrustes analysis), and Phillips (1986) (exact distributions of statistical test criteria). … ►For applications of the integral representation (35.5.3) see McFarland and Richards (2001, 2002) (statistical estimation of misclassification probabilities for discriminating between multivariate normal populations). …

3: Robb J. Muirhead

… ►Muirhead’s main research interests are multivariate statistical analysis, statistical modeling, Bayesian statistics, and pharmaceutical statistics. His book Aspects of Multivariate Statistical Theory was published by John Wiley & Sons in 1982. …

4: 35.1 Special Notation

… ► ►►

$a,b$	complex variables.
…

►The main functions treated in this chapter are the multivariate gamma and beta functions, respectively ${\mathrm{\Gamma }}_m\left(a\right)$ and ${\mathrm{B}}_m(a,b)$, and the special functions of matrix argument: Bessel (of the first kind) $A_{\nu }\left(𝐓\right)$ and (of the second kind) $B_{\nu }\left(𝐓\right)$; confluent hypergeometric (of the first kind) ${}_1{}^{}F_1^{}(a;b;𝐓)$ or ${}_1{}^{}F_1^{}({\displaystyle \genfrac{}{}{0pt}{}{a}{b}};𝐓)$ and (of the second kind) $\mathrm{\Psi }(a;b;𝐓)$; Gaussian hypergeometric ${}_2{}^{}F_1^{}(a_1,a_2;b;𝐓)$ or ${}_2{}^{}F_1^{}({\displaystyle \genfrac{}{}{0pt}{}{a_1,a_2}{b}};𝐓)$; generalized hypergeometric ${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;𝐓)$ or ${}_p{}^{}F_q^{}({\displaystyle \genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q}};𝐓)$. ►An alternative notation for the multivariate gamma function is ${\mathrm{\Pi }}_m(a)={\mathrm{\Gamma }}_m\left(a+\frac{1}{2}(m+1)\right)$ (Herz (1955, p. 480)). Related notations for the Bessel functions are ${𝒥}_{\nu +\frac{1}{2}(m+1)}(𝐓)=A_{\nu }\left(𝐓\right)/A_{\nu }\left(𝟎\right)$ (Faraut and Korányi (1994, pp. 320–329)), $K_m(0,\mathrm{\dots },0,\nu |𝐒,𝐓)={\left|𝐓\right|}^{\nu }{B}_{\nu }\left(𝐒𝐓\right)$ (Terras (1988, pp. 49–64)), and ${𝒦}_{\nu }(𝐓)={\left|𝐓\right|}^{\nu }{B}_{\nu }\left(𝐒𝐓\right)$ (Faraut and Korányi (1994, pp. 357–358)).

5: 35.6 Confluent Hypergeometric Functions of Matrix Argument

… ►

35.6.2 $$\mathrm{\Psi }(a;b;𝐓)=\frac{1}{{\mathrm{\Gamma }}_m\left(a\right)}{\int }_{𝛀}\mathrm{etr}\left(-𝐓𝐗\right){\left|𝐗\right|}^{a-\frac{1}{2}(m+1)}{\left|𝐈+𝐗\right|}^{b-a-\frac{1}{2}(m+1)}d𝐗,$$ $\mathrm{\Re }\left(a\right)>\frac{1}{2}(m-1)$, $𝐓\in 𝛀$.

ⓘ

Defines:

$\mathrm{\Psi }(a;b;𝐓)$: confluent hypergeometric function of matrix argument (second kind)

Symbols:

$\in$: 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, $a$: complex variable, $𝐈$: $m\times m$ identity matrix, $𝐓$: real symmetric $m\times m$ 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/35.6.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §35.6(i), §35.6 and Ch.35

… ►

35.6.3 $$L_{\nu }^{(\gamma )}\left(𝐓\right)=\frac{{\mathrm{\Gamma }}_m\left(\gamma +\nu +\frac{1}{2}(m+1)\right)}{{\mathrm{\Gamma }}_m\left(\gamma +\frac{1}{2}(m+1)\right)}{}_1{}^{}F_1^{}(\genfrac{}{}{0pt}{}{-\nu }{\gamma +\frac{1}{2}(m+1)};𝐓),$$ $\mathrm{\Re }\left(\gamma \right),\mathrm{\Re }\left(\gamma +\nu \right)>-1$.

ⓘ

Defines:

$L_{\nu }^{(\gamma )}\left(𝐓\right)$: Laguerre function of matrix argument

Symbols:

${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;𝐓)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};𝐓)$: generalized hypergeometric function of matrix argument, ${\mathrm{\Gamma }}_m\left(a\right)$: multivariate gamma function, $\mathrm{\Re }$: real part, $𝐓$: real symmetric $m\times m$ matrix and $m$: positive integer

Permalink:

http://dlmf.nist.gov/35.6.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §35.6(i), §35.6(i), §35.6 and Ch.35

… ►

35.6.4 $\mathrm{\Re }\left(a\right),\mathrm{\Re }\left(b-a\right)>\frac{1}{2}(m-1)$.

ⓘ

Symbols:

${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;𝐓)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};𝐓)$: generalized hypergeometric function of matrix argument, $\mathrm{etr}\left(𝐀\right)$: exponential of trace, $\int$: integral, ${\mathrm{B}}_m(a,b)$: multivariate beta function, $\mathrm{\Re }$: real part, $a$: complex variable, $𝐈$: $m\times m$ identity matrix, $𝐓$: real symmetric $m\times m$ 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

Permalink:

http://dlmf.nist.gov/35.6.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §35.6(ii), §35.6 and Ch.35

… ►

35.6.8 $${\int }_{𝛀}{\left|𝐓\right|}^{c-\frac{1}{2}(m+1)}\mathrm{\Psi }(a;b;𝐓)d𝐓=\frac{{\mathrm{\Gamma }}_m\left(c\right){\mathrm{\Gamma }}_m\left(a-c\right){\mathrm{\Gamma }}_m\left(c-b+\frac{1}{2}(m+1)\right)}{{\mathrm{\Gamma }}_m\left(a\right){\mathrm{\Gamma }}_m\left(a-b+\frac{1}{2}(m+1)\right)},$$ $\mathrm{\Re }\left(a\right)>\mathrm{\Re }\left(c\right)+\frac{1}{2}(m-1)>m-1$, $\mathrm{\Re }\left(c-b\right)>-1$.

ⓘ

Symbols:

$\mathrm{\Psi }(a;b;𝐓)$: confluent hypergeometric function of matrix argument (second kind), $\int$: integral, ${\mathrm{\Gamma }}_m\left(a\right)$: multivariate gamma function, $\mathrm{\Re }$: real part, $a$: complex variable, $𝐓$: 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, $c$: complex variable and $m$: positive integer

Permalink:

http://dlmf.nist.gov/35.6.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §35.6(ii), §35.6 and Ch.35

… ►

35.6.10 $$\underset{a\to \mathrm{\infty }}{lim}{\mathrm{\Gamma }}_m\left(a\right)\mathrm{\Psi }(a+\nu ;\nu +\frac{1}{2}(m+1);a^{-1}𝐓)=B_{\nu }\left(𝐓\right).$$

ⓘ

Symbols:

$B_{\nu }\left(𝐓\right)$: Bessel function of matrix argument (second kind), $\mathrm{\Psi }(a;b;𝐓)$: confluent hypergeometric function of matrix argument (second kind), ${\mathrm{\Gamma }}_m\left(a\right)$: multivariate gamma function, $a$: complex variable, $𝐓$: real symmetric $m\times m$ matrix and $m$: positive integer

Permalink:

http://dlmf.nist.gov/35.6.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §35.6(iii), §35.6 and Ch.35

…

6: 19.15 Advantages of Symmetry

… ►Elliptic integrals are special cases of a particular multivariate hypergeometric function called Lauricella’s $F_D$ (Carlson (1961b)). …

7: 35.8 Generalized Hypergeometric Functions of Matrix Argument

… ►

35.8.4 $$A_{\nu }\left(𝐓\right)=\frac{1}{{\mathrm{\Gamma }}_m\left(\nu +\frac{1}{2}(m+1)\right)}{}_0{}^{}F_1^{}(\genfrac{}{}{0pt}{}{-}{\nu +\frac{1}{2}(m+1)};-𝐓),$$ $𝐓\in 𝓢$.

ⓘ

Symbols:

$A_{\nu }\left(𝐓\right)$: Bessel function of matrix argument (first kind), ${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;𝐓)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};𝐓)$: generalized hypergeometric function of matrix argument, $\in$: element of, ${\mathrm{\Gamma }}_m\left(a\right)$: multivariate gamma function, $𝓢$: space of real symmetric $m\times m$ matrices, $𝐓$: real symmetric $m\times m$ matrix and $m$: positive integer

Permalink:

http://dlmf.nist.gov/35.8.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §35.8(ii), §35.8 and Ch.35

… ►

35.8.5 $${}_3{}^{}F_2^{}(\genfrac{}{}{0pt}{}{a_1,a_2,a_3}{b_1,b_2};𝐈)=\frac{{\mathrm{\Gamma }}_m\left(b_2\right){\mathrm{\Gamma }}_m\left(c\right)}{{\mathrm{\Gamma }}_m\left(b_2-a_3\right){\mathrm{\Gamma }}_m\left(c+a_3\right)}{}_3{}^{}F_2^{}(\genfrac{}{}{0pt}{}{b_1-a_1,b_1-a_2,a_3}{b_1,c+a_3};𝐈),$$ $\mathrm{\Re }\left(b_2\right),\mathrm{\Re }\left(c\right)>\frac{1}{2}(m-1)$.

ⓘ

Symbols:

${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;𝐓)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};𝐓)$: generalized hypergeometric function of matrix argument, ${\mathrm{\Gamma }}_m\left(a\right)$: multivariate gamma function, $\mathrm{\Re }$: real part, $a$: complex variable, $𝐈$: $m\times m$ identity matrix, $b$: complex variable, $c$: complex variable and $m$: positive integer

Permalink:

http://dlmf.nist.gov/35.8.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §35.8(iii), §35.8(iii), §35.8 and Ch.35

… ►

35.8.6 $${}_3{}^{}F_2^{}(\genfrac{}{}{0pt}{}{a_1,a_2,a_3}{b_1,b_2};𝐈)=\frac{{\mathrm{\Gamma }}_m\left(b_1-a_1\right){\mathrm{\Gamma }}_m\left(b_1-a_2\right)}{{\mathrm{\Gamma }}_m\left(b_1\right){\mathrm{\Gamma }}_m\left(b_1-a_1-a_2\right)}\frac{{\mathrm{\Gamma }}_m\left(b_1-a_3\right){\mathrm{\Gamma }}_m\left(b_1-a_1-a_2-a_3\right)}{{\mathrm{\Gamma }}_m\left(b_1-a_1-a_3\right){\mathrm{\Gamma }}_m\left(b_1-a_2-a_3\right)}.$$

ⓘ

Symbols:

${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;𝐓)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};𝐓)$: generalized hypergeometric function of matrix argument, ${\mathrm{\Gamma }}_m\left(a\right)$: multivariate gamma function, $a$: complex variable, $𝐈$: $m\times m$ identity matrix, $b$: complex variable and $m$: positive integer

Permalink:

http://dlmf.nist.gov/35.8.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §35.8(iii), §35.8(iii), §35.8 and Ch.35

… ►

35.8.7 $${}_3{}^{}F_2^{}(\genfrac{}{}{0pt}{}{a_1,a_2,a_3}{b_1,b_2};𝐈)=\frac{{\mathrm{\Gamma }}_m\left(b_1\right){\mathrm{\Gamma }}_m\left(b_2\right)\mathrm{\Gamma }\left(c\right)}{{\mathrm{\Gamma }}_m\left(a_1\right){\mathrm{\Gamma }}_m\left(c+a_2\right)\mathrm{\Gamma }\left(c+a_3\right)}{}_3{}^{}F_2^{}(\genfrac{}{}{0pt}{}{b_1-a_1,b_2-a_2,c}{c+a_2,c+a_3};𝐈),$$ $\mathrm{\Re }\left(b_1\right)$, $\mathrm{\Re }\left(b_2\right)$, $\mathrm{\Re }\left(c\right)>\frac{1}{2}(m-1)$.

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;𝐓)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};𝐓)$: generalized hypergeometric function of matrix argument, ${\mathrm{\Gamma }}_m\left(a\right)$: multivariate gamma function, $\mathrm{\Re }$: real part, $a$: complex variable, $𝐈$: $m\times m$ identity matrix, $b$: complex variable, $c$: complex variable and $m$: positive integer

Permalink:

http://dlmf.nist.gov/35.8.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §35.8(iii), §35.8(iii), §35.8 and Ch.35

… ►

35.8.12 $${\int }_{𝛀}\mathrm{etr}\left(-𝐓𝐗\right){\left|𝐗\right|}^{\gamma -\frac{1}{2}(m+1)}{}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};-𝐗)d𝐗={\mathrm{\Gamma }}_m\left(\gamma \right){\left|𝐓\right|}^{-\gamma }{}_{p+1}{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p,\gamma }{b_1,\mathrm{\dots },b_q};-{𝐓}^{-1}),$$ $\mathrm{\Re }\left(\gamma \right)>\frac{1}{2}(m-1)$.

ⓘ

Symbols:

${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;𝐓)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};𝐓)$: generalized hypergeometric function of matrix argument, $\mathrm{etr}\left(𝐀\right)$: exponential of trace, $\int$: integral, ${\mathrm{\Gamma }}_m\left(a\right)$: multivariate gamma function, $\mathrm{\Re }$: real part, $a$: complex variable, $𝐓$: real symmetric $m\times m$ 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, $p$: nonnegative integer, $q$: nonnegative integer and $m$: positive integer

Permalink:

http://dlmf.nist.gov/35.8.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §35.8(iv), §35.8(iv), §35.8 and Ch.35

…

8: Tom H. Koornwinder

… ►Currently he is on the editorial board for Constructive Approximation, and is editor for the volume on Multivariable Special Functions in the ongoing Askey–Bateman book project. …

9: Donald St. P. Richards

… ►Richards has published numerous papers on special functions of matrix argument, harmonic analysis, multivariate statistical analysis, probability inequalities, and applied probability. …

10: 19.16 Definitions

… ►

§19.16(ii) $R_{-a}(𝐛;𝐳)$

►All elliptic integrals of the form (19.2.3) and many multiple integrals, including (19.23.6) and (19.23.6_5), are special cases of a multivariate hypergeometric function ►

19.16.8 $$R_{-a}(𝐛;𝐳)=R_{-a}(b_1,\mathrm{\dots },b_n;z_1,\mathrm{\dots },z_n),$$

ⓘ

Symbols:

$R_{-a}(b_1,\mathrm{\dots },b_n;z_1,\mathrm{\dots },z_n)$ or $R_{-a}(𝐛;𝐳)$: multivariate hypergeometric function and $n$: nonnegative integer

Referenced by:

§19.16(ii)

Permalink:

http://dlmf.nist.gov/19.16.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §19.16(ii), §19.16 and Ch.19

… ► … ►

19.16.14 $$R_F(x,y,z)=R_{-\frac{1}{2}}(\frac{1}{2},\frac{1}{2},\frac{1}{2};x,y,z),$$

ⓘ

Symbols:

$R_{-a}(b_1,\mathrm{\dots },b_n;z_1,\mathrm{\dots },z_n)$ or $R_{-a}(𝐛;𝐳)$: multivariate hypergeometric function and $R_F(x,y,z)$: symmetric elliptic integral of first kind

Referenced by:

§19.16(iii), §19.18(i), §19.18(ii), §19.19, §19.19, §19.24(ii)

Permalink:

http://dlmf.nist.gov/19.16.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §19.16(iii), §19.16 and Ch.19

…