This work uses a first order vector differential equation whose solution involves hypergeometric functions of F-p+1(p) type. The factorizability of the solution of this equation is shown in yet another work of this meeting. We do not use the vector differential equation directly. Instead, we focus on its matrix counterpart for the propagator matrix of the solution. We use the factorizations developed around the regular singular points, 0, and 1, of the equation. Since these are two expressions for the same entity, that is the propagator matrix, they must be linearly dependent through a matrix coefficient. Work concentrates on the evaluation of this matrix coefficient. By this way, the solution valid inside the unit disk centered at the origin of the complex plane of the independent variable is expressed in terms of the solutions valid inside the unit disk centered at the point where the independent variable is 1. (c) 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.