*(English)*Zbl 0626.33010

Generalizing the classical hypergeometric function, the authors dfine ${}_{p}{F}_{q}({\alpha}_{1},\xb7\xb7\xb7,{\alpha}_{p};{\beta}_{1},\xb7\xb7\xb7,{\beta}_{q};s)$, where s is an element of the space S(n,$\mathbb{F})$ of all $n\times n$ Hermitian matrices over the division algebra $\mathbb{F}$, while the parameters are suitably restricted complex numbers, by an infinite series whose terms involve zonal polynomials and generalized Pochhammer symbols.

The main results obtained are: A convergence theorem with the cases $p\u22daq+1$, a Laplace transformation formula, and an Euler-type integral representation; all are reminiscent of classical results. Moreover, it is found that ${}_{0}{F}_{0}(;;s)=exptrs$, and that

where ${\Delta}$ is a determinant function. Most of the paper is, however, concerned with preparations that must precede the definition of ${}_{p}{F}_{q}$. Thus, a survey of the representation theory of the general linear group GL(n,$\mathbb{F})$ is given; and the above-mentioned concepts, as well as a generalized gamma function, are introduced and discussed at some length. Further results will appear in a subsequent paper.