Generalizing the classical hypergeometric function, the authors dfine ${}\sb pF\sb q(\alpha\sb 1,...,\alpha\sb p;\beta\sb 1,...,\beta\sb q;s)$, where s is an element of the space S(n,${\bbfF})$ of all $n\times n$ Hermitian matrices over the division algebra ${\bbfF}$, 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 $p\lesseqgtr q+1$, a Laplace transformation formula, and an Euler-type integral representation; all are reminiscent of classical results. Moreover, it is found that ${}\sb 0F\sb 0( ; ;s)=\exp tr s$, and that $$\sb 1F\sb 0(\alpha; ;s)=[\Delta (1-s)]\sp{-\alpha}, $$ where $\Delta$ is a determinant function. Most of the paper is, however, concerned with preparations that must precede the definition of ${}\sb pF\sb q$. Thus, a survey of the representation theory of the general linear group GL(n,${\bbfF})$ 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.