This work focuses on the construction of a first order vector differential equation whose solution is related to the hypergeometric functions of type F-p(p). The differential equation is constructed in such a way that the derivative of the unknown vector has unit matrix coefficient while it is multiplied by a matrix coefficient which is the sum of two constant matrices one of which is multiplied by the reciprocal of the independent variable. The purpose is to construct series solutions in vector form and then by combining these solutions to get the evolution matrix and its adjoint for the solution. We try to factorize these matrices into an infinite product. The truncation of this product can be used to get approximations to the hypergeometric functions of the type F-p(p). (c) 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.