Generalizing the classical hypergeometric function, the authors dfine , where s is an element of the space S(n, of all Hermitian matrices over the division algebra , 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 , a Laplace transformation formula, and an Euler-type integral representation; all are reminiscent of classical results. Moreover, it is found that , and that
where is a determinant function. Most of the paper is, however, concerned with preparations that must precede the definition of . Thus, a survey of the representation theory of the general linear group GL(n, 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.