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 the closed form expression of certain multivariate integrals, especially ones appearing 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 of $_{p}F_{q}^{(\alpha )}(a_{1},\ldots ,a_{p};b_{1},\ldots ,b_{q};X)$

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 (number theory) of $k$, $(a_{i})_{\kappa }^{(\alpha )}$ is the Generalized Pochhammer symbol, and $C_{\kappa }^{(\alpha )}(X)$ is the ``C" normalization of the Jack function.

References

• K. I. Gross and D. St. P. Richards, "Total positivity, spherical series, and hypergeometric functions of matrix argument", J. Approx. Theory, 59, no. 2, 224–246, 1989.

• Koev, Plamen and Edelman, Alan, "The efficient evaluation of the hypergeometric function of a matrix argument",

Mathematics of Computation, 75, no. 254, 833-846, 2006.

• Muirhead, Robb, Aspects of Multivariate Statistical Theory, John Wiley & Sons, Inc., New York, 1984.

External links

• Software for computing the hypergeometric function of a matrix argument