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Zbl 0731.33015
Gross, Kenneth I.; Richards, Donald St.P.
Hypergeometric functions on complex matrix space.
(English)
[J] Bull. Am. Math. Soc., New Ser. 24, No.2, 349-355 (1991). ISSN 0273-0979; ISSN 1088-9485/e

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 $n\times n$ Hermitian matrices over ${\bbfF}={\bbfR}$, ${\bbfC}$ or ${\bbfH}$. Let $a\sb 1,...,a\sb p$ and $b\sb 1,...,b\sb q$ be complex parameters. The hypergeometric function ${}\sb pF\sb q$ of matrix argument is defined on S by $$\sb pF\sb q(a\sb 1,...,a\sb p;b\sb 1,...,b\sb q;s)=\sum\sb{m}\frac{[a\sb 1]\sb m...[a\sb p]\sb mZ\sb m(s)}{[b\sb 1]\sb m,...,[b\sb q]\sb m\vert m\vert !}.$$ The sum is extended over all partitions $m=(m\sb 1,...,m\sb n)$, $[a]\sb m$ is the generalized truncated factorial for the matrix space, and $Z\sb m$ is the zonal polynomial associated with m. \par In this note one considers the case ${\bbfF}={\bbfC}$. 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. \par 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.
[J.Faraut (Paris)]
MSC 2000:
*33C70 Other hypergeometric functions and integrals in several variables

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