On the hypergeometric matrix function

This paper deals with the study of the hypergeometric function with matrix arguments F(A,B;C;z). Conditions for matrices A, B, C so that the series representation of the hypergeometric function be convergent for ¦z¦ = 1 and satisfies a matrix differential equation are given. After the study of beta and gamma matrix functions, an integral representation of F(A,B;C;z) is obtained for the case where B, C and C - B are positive stable matrices with BC = CB.