Abstract
In this paper we consider a matrix Hypergeometric differential equation, which are special matrix functions and solution of a specific second order linear differential equation. The aim of this work is to extend a well known theorem on Hypergeometric function in the complex plane to a matrix version, and we show that the asymptotic expansions of Hypergeometric function in the complex plane ” that are given in the literature are special members of our main result. Background and motivation are discussed.
Highlights
Generalization and extension of scalar special functions to matrix special functions have been developed in the past two decades The Gamma matrix function, whose eigenvalues are all in the right open half-plane is presented and investigated by L
Other classical orthogonal polynomials as Laguerre and Chebyshev have been extended to orthogonal matrix polynomials, and some results have been studied in L
The main our goal is that, some cases of the asymptotic expansions of 2F1(a, b; c; z) have been provided in the literature, they are all limited by a narrow domain of validity in the complex plane of the variable
Summary
Generalization and extension of scalar special functions to matrix special functions have been developed in the past two decades The Gamma matrix function, whose eigenvalues are all in the right open half-plane is presented and investigated by L. Relations between the Beta, Gamma and the Hypergeometric matrix function are given in L. Overcoming this restriction, we provide new asymptotic expansion for the matrix hypergeometric function .The order of presentation in this article is as follows. A matrix is called holomorphic if every entry of it is a holomorphic function.
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