Applications of Hermite transforms in computer algebra

Let L be a linear differential operator with polynomial coefficients. We show that there is an automorphism of differential operators D-alpha and an integral transform H-alpha (called the Hermite transform) on functions for which (DalphaL)f(x) = 0 implies LHalpha(f)(x) = 0. We present an algorithm that computes the Hermite transform of a rational function and use it to find n + 1 linearly independent solutions of Ly = 0 when (DalphaL)y = 0 has a rational solution with n distinct finite poles. (C) 2003 Elsevier Inc. All rights reserved.