Expansion of multivariable polynomials in products of orthogonal polynomials in one variable

摘要

We propose an approach to develop multivariable polynomials in multiple series of orthogonal polynomials in one variable. The action of a partial differential operator on a series of products of classical orthogonal polynomials is first analyzed and permits the generation of recurrence relations for the expansion coefficients like in the one variable case. Polynomial solutions of linear partial differential equations (PDEs) with polynomial coefficients are also examined giving new results on harmonic polynomials based on the Appell property of Hermite polynomials.