Matrix Pearson Equations Satisfied by Koornwinder Weights in Two Variables

Abstract

We consider Koornwinder's method for constructing orthogonal polynomials in two variables from orthogonal polynomials in one variable. If semiclassical orthogonal polynomials in one variable are used, then Koornwinder's construction generates semiclassical orthogonal polynomials in two variables. We consider two methods for deducing matrix Pearson equations for weight functions associated with these polynomials, and consequently, we deduce the second order linear partial differential operators for classical Koornwinder polynomials.

Bivariate Koornwinder–Sobolev Orthogonal Polynomials

“…n } n≥0 denotes the sequence of classical Gegenbauer polynomials, and μ m = μ + m + 1/2. These polynomials constitute a mutually orthogonal sequence with respect to the moment functional w μ defined by [11,13] as well as [4,6]). This moment functional is constructed using the method described in Sect.…”

“…where β m = β + m + 1/2, and {P (α,βm) n } n≥0 are the Jacobi univariate orthogonal polynomials associated with u α,βm . Then, they are mutually orthogonal with respect to w α,β (see [11,13] as well as [4,6]).…”

“…In [13], the authors exploited the structure of Koornwinder's weight function to obtain the coefficients of the three term relation satisfied by the bivariate polynomials when both weight functions ω 1 and ω 2 are classical. Moreover, in [11] differential properties for the weight function W (x, y) are deduced. Those properties were the extension to the Koornwinder case of the Pearson's differential equation of univariate classical weight functions.…”

Representations for parameter derivatives of some Koornwinder polynomials in two variables

“…In the recent papers [1,2], parametric derivative representations in the form of (6) for Jacobi polynomials on the triangle and a family of orthogonal polynomials with two variables on the unit disc have been studied. The present paper is devoted to obtain parametric derivatives for the polynomials on the parabolic biangle, on the square and some new examples of Koornwinder polynomials introduced in [17] (see also [18]). Although the parameter derivatives of these polynomials with respect to their some parameters are in the form of (6), there exist some other parameters such that derivatives with respect to them are not in the form of (6).…”