In this paper we consider a matrix Hypergeometric differential equation, which are special functions and solution of specific second order linear equation. The aim work is to extend well known theorem on function in the complex plane version, show that asymptotic expansions ” given literature members our main result. Background motivation discussed.

In this paper sufficient conditions are derived for the existence of unique and positive definite solutions of the matrix equations X−A1XA1− . . .−A∗mXAm = Q and X+A1XA1+ . . .+ A∗mXAm = Q. In the case there is a unique solution which is positive definite an explicit expression for this solution is given.