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Zbl 0626.33010
Gross, Kenneth I.; Richards, Donald St.P.
Special functions of matrix argument. I: Algebraic induction, zonal polynomials, and hypergeometric functions. (English)
[J] Trans. Am. Math. Soc. 301, 781-811 (1987). ISSN 0002-9947; ISSN 1088-6850/e

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. \par 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.
[P.W.Karlsson]

MSC 2000:: *33C80 Connections of theory of special functions with groups and algebras
22E30 Analysis on real and complex Lie groups
22E45 Analytic repres.of Lie and linear algebraic groups over real fields
43A85 Analysis on homogeneous spaces
43A90 Spherical functions (abstract harmonic analysis)

Keywords: special functions of matrix argument; zonal polynomials; Pochhammer symbols

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Zentralblatt MATH Berlin [Germany]