Series expansions of solutions of uxx + (2νx) ux + ε2utt = ut

摘要

We investigate series expansions of solutions of the equation (∗)uxx + 2νx−1ux + ϵ2utt = ut, in terms of either a set of polynomial solutions for this equation or the associated functions for these polynomials. Our results extend those of Rosenbloom-Widder, Cholewinski-Haimo and the second author, who have previously studied (∗) for various choices of the nonnegative parameters ν and ε. By considering (∗) as a singular perturbation of the generalized heat equation (the ε = 0 case), we show that the polynomial solutions, associated functions and fundamental singularities of these equations are related by a basic integral transform.