R$_{II}$ type three term relations for bivariate polynomials orthogonal with respect to varying weights

TL;DR

This paper investigates bivariate orthogonal polynomials with respect to varying weights, establishing R_{II} type three-term relations and developing a construction method based on Koornwinder's approach.

Contribution

It introduces R_{II} type three-term relations for bivariate orthogonal polynomials under varying weights and proposes a new construction method using Koornwinder's technique.

Findings

Orthogonal polynomials satisfy R_{II} type relations for each variable.

A construction method for bivariate orthogonal systems with varying weights is developed.

Several examples and special cases are analyzed.

Abstract

Given a bivariate weight function defined on the positive quadrant of $\mathbb{R}^2$, we study polynomials in two variables orthogonal with respect to varying measures obtained by special modifications of this weight function. In particular, the varying weight functions are given by the multiplication of $x_1^{-n}x_2^{-n}$ times the original weight function. Apart from the question of the existence and construction of such kind of orthogonal polynomials, we show that the systems of bivariate polynomials orthogonal with respect to this kind of varying weights satisfy R$_{II}$ type three term relations, one for every variable. A method to construct bivariate orthogonal systems with respect to varying weights based in the Koornwinder's method is developed. Finally, several examples and particular cases have been analysed.