Bivariate second-order linear partial differential equations and orthogonal polynomial solutions |
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| Authors: | I Area E Godoy A Ronveaux A Zarzo |
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| Institution: | 1. Departamento de Matemática Aplicada II, E.E. Telecomunicación, Universidade de Vigo, 36310 Vigo, Spain;2. Departamento de Matemática Aplicada II, E.E. Industrial, Universidade de Vigo, 36310 Vigo, Spain;3. Departement de Mathématique, Université Catholique de Louvain, Bâtiment Marc de Hemptinne, Chemin du Cyclotron 2, B-1348 Louvain-la-Neuve, Belgium;4. Instituto Carlos I de Física Teórica y Computacional, Facultad de Ciencias, Universidad de Granada, Spain;5. Departamento de Matemática Aplicada, E.T.S. Ingenieros Industriales, Universidad Politécnica de Madrid, Spain |
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| Abstract: | 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. As illustration, these results are applied to a two parameter monic Appell polynomials. Finally, the non-monic case is briefly discussed. |
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