# Hypergeometric function of a matrix argument

In mathematics, the hypergeometric function of a matrix argument is a generalization of the classical hypergeometric series. It is a function defined by an infinite summation which can be used to evaluate certain multivariate integrals.

Hypergeometric functions of a matrix argument have applications in random matrix theory. For example, the distributions of the extreme eigenvalues of random matrices are often expressed in terms of the hypergeometric function of a matrix argument.

## Definition

Let $p\geq 0$ and $q\geq 0$ be integers, and let $X$ be an $m\times m$ complex symmetric matrix. Then the hypergeometric function of a matrix argument $X$ and parameter $\alpha >0$ is defined as

$_{p}F_{q}^{(\alpha )}(a_{1},\ldots ,a_{p};b_{1},\ldots ,b_{q};X)=\sum _{k=0}^{\infty }\sum _{\kappa \vdash k}{\frac {1}{k!}}\cdot {\frac {(a_{1})_{\kappa }^{(\alpha )}\cdots (a_{p})_{\kappa }^{(\alpha )}}{(b_{1})_{\kappa }^{(\alpha )}\cdots (b_{q})_{\kappa }^{(\alpha )}}}\cdot C_{\kappa }^{(\alpha )}(X),$ where $\kappa \vdash k$ means $\kappa$ is a partition of $k$ , $(a_{i})_{\kappa }^{(\alpha )}$ is the Generalized Pochhammer symbol, and $C_{\kappa }^{(\alpha )}(X)$ is the "C" normalization of the Jack function.

## Two matrix arguments

If $X$ and $Y$ are two $m\times m$ complex symmetric matrices, then the hypergeometric function of two matrix arguments is defined as:

$_{p}F_{q}^{(\alpha )}(a_{1},\ldots ,a_{p};b_{1},\ldots ,b_{q};X,Y)=\sum _{k=0}^{\infty }\sum _{\kappa \vdash k}{\frac {1}{k!}}\cdot {\frac {(a_{1})_{\kappa }^{(\alpha )}\cdots (a_{p})_{\kappa }^{(\alpha )}}{(b_{1})_{\kappa }^{(\alpha )}\cdots (b_{q})_{\kappa }^{(\alpha )}}}\cdot {\frac {C_{\kappa }^{(\alpha )}(X)C_{\kappa }^{(\alpha )}(Y)}{C_{\kappa }^{(\alpha )}(I)}},$ where $I$ is the identity matrix of size $m$ .

## Not a typical function of a matrix argument

Unlike other functions of matrix argument, such as the matrix exponential, which are matrix-valued, the hypergeometric function of (one or two) matrix arguments is scalar-valued.

## The parameter $\alpha$ In many publications the parameter $\alpha$ is omitted. Also, in different publications different values of $\alpha$ are being implicitly assumed. For example, in the theory of real random matrices (see, e.g., Muirhead, 1984), $\alpha =2$ whereas in other settings (e.g., in the complex case—see Gross and Richards, 1989), $\alpha =1$ . To make matters worse, in random matrix theory researchers tend to prefer a parameter called $\beta$ instead of $\alpha$ which is used in combinatorics.

The thing to remember is that

$\alpha ={\frac {2}{\beta }}.$ Care should be exercised as to whether a particular text is using a parameter $\alpha$ or $\beta$ and which the particular value of that parameter is.

Typically, in settings involving real random matrices, $\alpha =2$ and thus $\beta =1$ . In settings involving complex random matrices, one has $\alpha =1$ and $\beta =2$ .