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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Special functions of matrix argument. I. Algebraic induction, zonal polynomials, and hypergeometric functions

Authors: Kenneth I. Gross and Donald St. P. Richards
Journal: Trans. Amer. Math. Soc. 301 (1987), 781-811
MSC: Primary 22E30; Secondary 22E45, 33A75, 43A85, 62H10
MathSciNet review: 882715
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Abstract: Hypergeometric functions of matrix argument arise in a diverse range of applications in harmonic analysis, multivariate statistics, quantum physics, molecular chemistry, and number theory. This paper presents a general theory of such functions for real division algebras. These functions, which generalize the classical hypergeometric functions, are defined by infinite series on the space $ S = S(n,\,\mathbf{F})$ of all $ n \times n$ Hermitian matrices over the division algebra $ \mathbf{F}$. The theory depends intrinsically upon the representation theory of the general linear group $ G = GL(n,\,\mathbf{F})$ of invertible $ n \times n$ matrices over $ \mathbf{F}$, and the theme of this work is the full exploitation of the inherent group theory. The main technique is the use of the method of ``algebraic induction'' to realize explicitly the appropriate representations of $ G$, to decompose the space of polynomial functions on $ S$, and to describe the ``zonal polynomials'' from which the hypergeometric functions are constructed. Detailed descriptions of the convergence properties of the series expansions are given, and integral representations are provided. Future papers in this series will develop the fine structure of these functions.

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Keywords: Generalized hypergeometric functions, zonal polynomials, representation theory, algebraic induction, multivariate statistics, general linear group, generalized gamma functions, Pochhammer symbols, Laplace transforms, maximal compact subgroup, invariant polynomials, positive cones, symmetric spaces, Schur functions, special functions of matrix argument
Article copyright: © Copyright 1987 American Mathematical Society