The generalized matrix valued hypergeometric equation

TL;DR

This paper extends Tirao's matrix hypergeometric equation by adding parameters, introduces generalized matrix hypergeometric functions, and analyzes their analyticity and convergence properties.

Contribution

It generalizes the matrix hypergeometric equation to include more parameters and characterizes the properties of the resulting matrix hypergeometric functions.

Findings

Functions are analytic for |z|<1

Necessary condition for convergence on |z|=1

Extension of Tirao's matrix hypergeometric equation

Abstract

The matrix valued analog of the Euler's hypergeometric differential equation was introduced by Tirao in \cite{T2}. 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 \cite{T2} to identify the corresponding matrix hypergeometric functions ${}_nF_m$. We prove that, if n=m+1, this functions are analytic for |z|<1 and we give a necesary condition for the convergence on the unit circle |z|=1.