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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 ${}_{p}{F}_{q}\left({\alpha }_{1},···,{\alpha }_{p};{\beta }_{1},···,{\beta }_{q};s\right)$, where s is an element of the space S(n,$𝔽\right)$ of all $n×n$ Hermitian matrices over the division algebra $𝔽$, 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 $p⋚q+1$, a Laplace transformation formula, and an Euler-type integral representation; all are reminiscent of classical results. Moreover, it is found that ${}_{0}{F}_{0}\left(;;s\right)=exptrs$, and that

${}_{1}{F}_{0}\left(\alpha ;;s\right)={\left[{\Delta }\left(1-s\right)\right]}^{-\alpha },$

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,$𝔽\right)$ 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
##### MSC:
 33C80 Connections of hypergeometric functions with groups and algebras 22E30 Analysis on real and complex Lie groups 22E45 Analytic representations of Lie and linear algebraic groups over real fields 43A85 Analysis on homogeneous spaces 43A90 Spherical functions (abstract harmonic analysis)