The Generalized Matrix Valued Hypergeometric Equation

The matrix valued analog of the Euler's hypergeometric differential equation was introduced by Tirao in [4]. This equation arises in the study of matrix valued spherical functions and in the theory of matrix valued orthogonal polynomials. The goal of this paper is to extend naturally the number of parameters of Tirao's equation in order to get a generalized matrix valued hypergeometric equation. We take advantage of the tools and strategies developed in [4] to identify the corresponding matrix hypergeometric functions (n)F(m). We prove that, if n = m+1, these functions are analytic for vertical bar z vertical bar < 1 and we give a necessary condition for the convergence on the unit circle vertical bar z vertical bar = 1.