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Expansion of multivariable polynomials in products of orthogonal polynomials in one variable. (English) Zbl 1025.40003

Summary: 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 with polynomial coefficients are also examined giving new results on harmonic polynomials based on the Appell property of Hermite polynomials.

MSC:

40A30 Convergence and divergence of series and sequences of functions
33C50 Orthogonal polynomials and functions in several variables expressible in terms of special functions in one variable
05E35 Orthogonal polynomials (combinatorics) (MSC2000)

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