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Special functions of matrix argument. I: Algebraic induction, zonal polynomials, and hypergeometric functions. (English) Zbl 0626.33010
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.
Reviewer: P.W.Karlsson

33C80Connections of hypergeometric functions with groups and algebras
22E30Analysis on real and complex Lie groups
22E45Analytic representations of Lie and linear algebraic groups over real fields
43A85Analysis on homogeneous spaces
43A90Spherical functions (abstract harmonic analysis)
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