This work focuses on the construction of a first order vector differential equation whose solution is related to hypergeometric function of F-p+1(p) type. Differential equation's derivative-free terms involve two constant matrices which are multiplied by the reciprocals of the independent variable and its difference from one. The purpose is to construct the evolution operator and its adjoint for the solution. The effect of this operator on an arbitrary constant matrix describes the most general solution of the differential equation mentioned above. The construction of the evolution operator requires the tools of matrix algebraic manipulations like commutations. Factorization produces an infinite number of matrix factors. The truncation of this infinite product serves as an approximation to the hypergeometric function under consideration. (c) 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.