On differential properties for bivariate orthogonal polynomials

Morales, María Álvarez de; Fernández, Lidia; Pérez, Teresa E.; Piñar, Miguel A.

doi:10.1007/s11075-007-9113-3

Cited by 1 publication

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References 17 publications

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“…One possibility to avoid this problem is to consider graded lexicographical order and use the matrix vector representation, first introduced by Kowalski [18,19] and afterwards considered by Xu [34,35]. In fact, using this point of view, in [2] the authors proved some structure and orthogonality relations for the successive partial derivatives of the vector orthogonal polynomials associated with a quasi-definite moment functional which satisfies a Pearson-type partial differential equation.…”

Section: Introductionmentioning

confidence: 99%

Bivariate second-order linear partial differential equations and orthogonal polynomial solutions

Area

Godoy

Ronveaux

et al. 2012

Journal of Mathematical Analysis and Applications

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In this paper we construct the main algebraic and differential properties and the weight functions of orthogonal polynomial solutions of bivariate second-order linear partial differential equations, which are admissible potentially self-adjoint and of hypergeometric type. General formulae for all these properties are obtained explicitly in terms of the polynomial coefficients of the partial differential equation, using vector matrix notation. Moreover, Rodrigues representations for the polynomial eigensolutions and for their partial derivatives of any order are given. Finally, as illustration, these results are applied to specific Appell and Koornwinder orthogonal polynomials, solutions of the same partial differential equation.

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