# §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. nonnegative integers. positive integer. partitional shifted factorial (§35.4(i)). zero matrix. identity matrix. space of all real symmetric matrices. real symmetric matrices. trace of $\mathbf{X}$. $\exp\left(\operatorname{tr}{\mathbf{X}}\right)$. determinant of $\mathbf{X}$ (except when $m=1$ where it means either determinant or absolute value, depending on the context). $j$th principal minor of $\mathbf{X}$. $(j,k)$th element of $\mathbf{X}$. $\prod_{1\leq j\leq k\leq m}\,\mathrm{d}x_{j,k}$. space of positive-definite real symmetric matrices. eigenvalues of $\mathbf{T}$. spectral norm of $\mathbf{T}$. $\mathbf{X}-\mathbf{T}$ is positive definite. Similarly, $\mathbf{T}<\mathbf{X}$ is equivalent. complex symmetric matrix. complex-valued function with $\mathbf{X}\in{\boldsymbol{\Omega}}$. space of orthogonal matrices. orthogonal matrix. normalized Haar measure on $\mathbf{O}(m)$. 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)).