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 Hermitian matrices over , or . Let and be complex parameters. The hypergeometric function of matrix argument is defined on S by
The sum is extended over all partitions , is the generalized truncated factorial for the matrix space, and is the zonal polynomial associated with m.
In this note one considers the case . 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.