35 Functions of Matrix Argument Notation 35 Functions of Matrix Argument 35.2 Laplace Transform

§35.1 Special Notation

(For other notation see Notation for the Special Functions.)

All matrices are of order $m\times m$ , unless specified otherwise. All fractional or complex powers are principal values.

$a, b$	complex variables.
$j, k, p, q$	nonnegative integers.
$m$	positive integer.
${\left[a\right]_{\kappa}}$	partitional shifted factorial (§35.4(i)).
$\boldsymbol{{0}}$	zero matrix.
$\mathbf{I}$	identity matrix.
$\boldsymbol{\mathcal{S}}$	space of all real symmetric matrices.
$\mathbf{S},\mathbf{T},\mathbf{X}$	real symmetric matrices.
$\operatorname{tr}{\mathbf{X}}$	trace of $\mathbf{X}$ .
$\operatorname{etr}\left(\mathbf{X}\right)$	$\exp\left(\operatorname{tr}{\mathbf{X}}\right)$ .
$\left\|\mathbf{X}\right\|$	determinant of $\mathbf{X}$ (except when $m=1$ where it means either determinant or absolute value, depending on the context).
$\|(\mathbf{X})_{j}\|$	$j$ th principal minor of $\mathbf{X}$ .
$x_{j,k}$	$(j,k)$ th element of $\mathbf{X}$ .
$\,\mathrm{d}{\mathbf{X}}$	$\prod_{1\leq j\leq k\leq m}\,\mathrm{d}x_{j,k}$ .
${\boldsymbol{\Omega}}$	space of positive-definite real symmetric matrices.
$t_{1},\dots,t_{m}$	eigenvalues of $\mathbf{T}$ .
$\\|\mathbf{T}\\|$	spectral norm of $\mathbf{T}$ .
$\mathbf{X}>\mathbf{T}$	$\mathbf{X}-\mathbf{T}$ is positive definite. Similarly, $\mathbf{T}<\mathbf{X}$ is equivalent.
$\mathbf{Z}$	complex symmetric matrix.
$f(\mathbf{X})$	complex-valued function with $\mathbf{X}\in{\boldsymbol{\Omega}}$ .
$\mathbf{O}(m)$	space of orthogonal matrices.
$\mathbf{H}$	orthogonal matrix.
$\mathrm{d}{\mathbf{H}}$	normalized Haar measure on $\mathbf{O}(m)$ .
$Z_{\kappa}\left(\mathbf{T}\right)$	zonal polynomials.

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)).

35 Functions of Matrix Argument 35.2 Laplace Transform

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