§35.1 Special Notation

Defines:: $\mathop{\mathrm{etr}\/}\nolimits\!\left(\mathbf{X}\right)$ : exponential of trace
Keywords:: Bessel functions, Gaussian hypergeometric function, beta function, confluent hypergeometric functions, functions, functions of matrix argument, gamma function, generalized hypergeometric functions, hypergeometric function, hypergeometric functions of matrix argument, multivariate beta function, multivariate gamma function, polynomials
Permalink:: http://dlmf.nist.gov/35.1

(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.
$j,k$	nonnegative integers.
$m$	positive integer.
$\left[a\right]_{{\kappa}}$	partitional shifted factorial (§35.4(i)).
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.
$\trace{\mathbf{X}}$	trace of $\mathbf{X}$ .
$\mathop{\mathrm{etr}\/}\nolimits\!\left(\mathbf{X}\right)$	$\mathop{\exp\/}\nolimits\!\left(\trace{\mathbf{X}}\right)$ .
$\|\mathbf{X}\|$	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 $\mathbf{X}$ .
$d\mathbf{X}$	$\prod _{{1\leq j\leq k\leq m}}dx_{{j,k}}$ .
${\boldsymbol{\Omega}}$	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.
$\mathbf{Z}$	complex symmetric matrix.
$\mathbf{U},\mathbf{V}$	real and complex parts of $\mathbf{Z}$ .
$f(\mathbf{X})$	complex-valued function with $\mathbf{X}\in{\boldsymbol{\Omega}}$ .
$\mathbf{O}(m)$	space of orthogonal matrices.
$\mathbf{H}$	orthogonal matrix.
$d\mathbf{H}$	normalized Haar measure on $\mathbf{O}(m)$ .
$\mathop{Z_{{\kappa}}\/}\nolimits\!\left(\mathbf{T}\right)$	zonal polynomials.

The main functions treated in this chapter are the multivariate gamma and beta functions, respectively $\mathop{\Gamma _{{m}}\/}\nolimits\!\left(a\right)$ and $\mathop{\mathrm{B}_{{m}}\/}\nolimits\!\left(a,b\right)$ , and the special functions of matrix argument: Bessel (of the first kind) $\mathop{A_{{\nu}}\/}\nolimits\!\left(\mathbf{T}\right)$ and (of the second kind) $\mathop{B_{{\nu}}\/}\nolimits\!\left(\mathbf{T}\right)$ ; confluent hypergeometric (of the first kind) $\mathop{{{}_{{1}}F_{{1}}}\/}\nolimits\!\left(a;b;\mathbf{T}\right)$ or $\displaystyle\mathop{{{}_{{1}}F_{{1}}}\/}\nolimits\!\left({a\atop b};\mathbf{T}\right)$ and (of the second kind) $\mathop{\Psi\/}\nolimits\!\left(a;b;\mathbf{T}\right)$ ; Gaussian hypergeometric $\mathop{{{}_{{2}}F_{{1}}}\/}\nolimits\!\left(a_{1},a_{2};b;\mathbf{T}\right)$ or $\displaystyle\mathop{{{}_{{2}}F_{{1}}}\/}\nolimits\!\left({a_{1},a_{2}\atop b};\mathbf{T}\right)$ ; generalized hypergeometric $\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits\!\left(a_{1},\dots,a_{p};b_{1},\dots,b_{q};\mathbf{T}\right)$ or $\displaystyle\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits\!\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};\mathbf{T}\right)$ .

An alternative notation for the multivariate gamma function is $\Pi _{m}(a)=\mathop{\Gamma _{{m}}\/}\nolimits\!\left(a+\tfrac{1}{2}(m+1)\right)$ (Herz (1955, p. 480)). Related notations for the Bessel functions are $\mathcal{J}_{{\nu+\frac{1}{2}(m+1)}}(\mathbf{T})=\mathop{A_{{\nu}}\/}\nolimits\!\left(\mathbf{T}\right)/\mathop{A_{{\nu}}\/}\nolimits\!\left(\boldsymbol{{0}}\right)$ (Faraut and Korányi (1994, pp. 320–329)), $K_{m}(0,\dots,0,\nu\mathpunct{|}\mathbf{S},\mathbf{T})=|\mathbf{T}|^{\nu}\mathop{B_{{\nu}}\/}\nolimits\!\left(\mathbf{S}\mathbf{T}\right)$ (Terras (1988, pp. 49–64)), and $\mathcal{K}_{\nu}(\mathbf{T})=|\mathbf{T}|^{\nu}\mathop{B_{{\nu}}\/}\nolimits\!\left(\mathbf{S}\mathbf{T}\right)$ (Faraut and Korányi (1994, pp. 357–358)).