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Hypergeometric function of a matrix argument ## Summary

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 α

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 $\beta =2$ .