Series expansions of solutions of u < inf> xx + ( 2ν x) u < inf> x + ε < sup> 2 u < inf> tt = u < inf> t

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We investigate series expansions of solutions of the equation (*)uxx + 2νx-1ux + ε{lunate}2utt = 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. © 1984.

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Journal of Computational and Applied Mathematics