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Hypergeometric functions on complex matrix space. (English) Zbl 0731.33015

In this note new results about hypergeometric functions of matrix argument are presented without proof. The proofs together with a detailed study will appear in a forthcoming paper. Denote by S the space of $n×n$ Hermitian matrices over $𝔽=ℝ$, $ℂ$ or $ℍ$. Let ${a}_{1},···,{a}_{p}$ and ${b}_{1},···,{b}_{q}$ be complex parameters. The hypergeometric function ${}_{p}{F}_{q}$ of matrix argument is defined on S by

${}_{p}{F}_{q}\left({a}_{1},···,{a}_{p};{b}_{1},···,{b}_{q};s\right)=\sum _{m}\frac{{\left[{a}_{1}\right]}_{m}···{\left[{a}_{p}\right]}_{m}{Z}_{m}\left(s\right)}{{\left[{b}_{1}\right]}_{m},···,{\left[{b}_{q}\right]}_{m}|m|!}·$

The sum is extended over all partitions $m=\left({m}_{1},···,{m}_{n}\right)$, ${\left[a\right]}_{m}$ is the generalized truncated factorial for the matrix space, and ${Z}_{m}$ is the zonal polynomial associated with m.

In this note one considers the case $𝔽=ℂ$. Then the hypergeometric function can be written as a determinant whose entries are classical hypergeometric functions. From this expansion one deduces an asymptotic formula and a system of partial differential equations of which hypergeometric functions are solutions.

Further one defines the operator-valued hypergeometric function inductively by using the Laplace transform associated with the cone P of positive definite Hermitian matrices. As a special case one obtains the operator-valued Bessel function studied in the 70th by Gross and Kunze.

Reviewer: J.Faraut (Paris)
##### MSC:
 33C70 Other hypergeometric functions and integrals in several variables