Closed form general solution of the hypergeometric matrix differential equation

Abstract

In this paper, the hypergeometric matrix differential equation z(1 − z) W′' − zAW′ + W′( C − z( B + I)) − AWB = 0 is studied. First it is proved that if matrix C is invertible and no negative integer is one of its eigenvalues, then the hypergeometric matrix function F( A, B; C; z) is an analytic solution in the unit disc. If, apart from the above hypothesis on C, matrices A and B commute with C, then a closed form general solution is expressed in terms of F( A, B; C; z) and F(A + I − C, B + I − C; 2I − C; z)z I − C in Ω(δ) = z ϵ D 0, 0 < ¦z¦ < δ , where D 0 is the complex plane cut along the negative real axis, and δ > 0 is a positive number determined in terms of the data.