5 research outputs found
Bivariate orthogonal polynomials in the Lyskova class
AbstractClassical orthogonal polynomials in two variables can be characterized as the polynomial solutions of a matrix second-order partial differential equation involving matrix polynomial coefficients. In this work, we study classical orthogonal polynomials in two variables whose partial derivatives satisfy again a second-order partial differential equation of the same type
Bivariate second--order linear partial differential equations and orthogonal polynomial solutions
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.Comment: 27 page
On a class of bivariate second-order linear partial difference equations and their monic orthogonal polynomial solutions
AbstractIn this paper we classify the bivariate second-order linear partial difference equations, which are admissible, potentially self-adjoint, and of hypergeometric type. Using vector matrix notation, explicit expressions for the coefficients of the three-term recurrence relations satisfied by monic orthogonal polynomial solutions are obtained in terms of the coefficients of the partial difference equation. Finally, we make a compilation of the examples existing in the literature belonging to the class analyzed in this paper, namely bivariate Charlier, Meixner, Kravchuk and Hahn orthogonal polynomials