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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 , F ) S = S(n,\,\mathbf {F})       of all      n ×  n n \times n       Hermitian matrices over the division algebra      F \mathbf {F}      . The theory depends intrinsically upon the representation theory of the general linear group      G = G L ( n , F ) G = GL(n,\,\mathbf {F})       of invertible      n ×  n n \times n       matrices over      F \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 G      , to decompose the space of polynomial functions on      S 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.

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