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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=, or . 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 (a 1 ,···,a p ;b 1 ,···,b q ;s)= m [a 1 ] m ···[a p ] m Z m (s) [b 1 ] m ,···,[b q ] m |m|!·

The sum is extended over all partitions m=(m 1 ,···,m n ), [a] 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=. 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:
33C70Other hypergeometric functions and integrals in several variables