# multivariate

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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: 35.9 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 $p\leq 2$ and $q\leq 1$. … 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). …
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 $\Gamma_{m}\left(a\right)$ 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)).
##### 5: 35.6 Confluent Hypergeometric Functions of Matrix Argument
35.6.2 $\Psi\left(a;b;\mathbf{T}\right)=\frac{1}{\Gamma_{m}\left(a\right)}\int_{% \boldsymbol{\Omega}}\operatorname{etr}\left(-\mathbf{T}\mathbf{X}\right)\left|% \mathbf{X}\right|^{a-\frac{1}{2}(m+1)}\*{\left|\mathbf{I}+\mathbf{X}\right|}^{% b-a-\frac{1}{2}(m+1)}\,\mathrm{d}{\mathbf{X}},$ $\Re\left(a\right)>\frac{1}{2}(m-1)$, $\mathbf{T}\in{\boldsymbol{\Omega}}$.
35.6.3 $L^{(\gamma)}_{\nu}\left(\mathbf{T}\right)=\frac{\Gamma_{m}\left(\gamma+\nu+% \frac{1}{2}(m+1)\right)}{\Gamma_{m}\left(\gamma+\frac{1}{2}(m+1)\right)}\*{{}_% {1}F_{1}}\left({-\nu\atop\gamma+\frac{1}{2}(m+1)};\mathbf{T}\right),$ $\Re\left(\gamma\right),\Re\left(\gamma+\nu\right)>-1$.
35.6.4 ${{}_{1}F_{1}}\left({a\atop b};\mathbf{T}\right)=\frac{1}{\mathrm{B}_{m}\left(a% ,b-a\right)}\int\limits_{\boldsymbol{{0}}<\mathbf{X}<\mathbf{I}}\operatorname{% etr}\left(\mathbf{T}\mathbf{X}\right)\left|\mathbf{X}\right|^{a-\frac{1}{2}(m+% 1)}\left|\mathbf{I}-\mathbf{X}\right|^{b-a-\frac{1}{2}(m+1)}\,\mathrm{d}{% \mathbf{X}},$ $\Re\left(a\right),\Re\left(b-a\right)>\frac{1}{2}(m-1)$.
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 $\lim_{a\to\infty}\Gamma_{m}\left(a\right)\Psi\left(a+\nu;\nu+\tfrac{1}{2}(m+1)% ;a^{-1}\mathbf{T}\right)=B_{\nu}\left(\mathbf{T}\right).$
##### 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 $A_{\nu}\left(\mathbf{T}\right)=\dfrac{1}{\Gamma_{m}\left(\nu+\frac{1}{2}(m+1)% \right)}{{}_{0}F_{1}}\left({-\atop\nu+\frac{1}{2}(m+1)};-\mathbf{T}\right),$ $\mathbf{T}\in\boldsymbol{\mathcal{S}}$.
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 ${\int_{\boldsymbol{\Omega}}\operatorname{etr}\left(-\mathbf{T}\mathbf{X}\right% )\left|\mathbf{X}\right|^{\gamma-\frac{1}{2}(m+1)}\*{{}_{p}F_{q}}\left({a_{1},% \dots,a_{p}\atop b_{1},\dots,b_{q}};-\mathbf{X}\right)\,\mathrm{d}{\mathbf{X}}% }=\Gamma_{m}\left(\gamma\right)\left|\mathbf{T}\right|^{-\gamma}{{}_{p+1}F_{q}% }\left({a_{1},\dots,a_{p},\gamma\atop b_{1},\dots,b_{q}};-\mathbf{T}^{-1}% \right),$ $\Re\left(\gamma\right)>\frac{1}{2}(m-1)$.
##### 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) $R_{-a}\left(\mathbf{b};\mathbf{z}\right)$
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
19.16.14 $R_{F}\left(x,y,z\right)=R_{-\frac{1}{2}}\left(\tfrac{1}{2},\tfrac{1}{2},\tfrac% {1}{2};x,y,z\right),$