Three term relations for a class of bivariate orthogonal polynomials

TL;DR

This paper derives explicit matrix three-term recurrence relations for a class of bivariate orthogonal polynomials constructed from univariate orthogonal polynomials, with special cases analyzed.

Contribution

It provides explicit formulas for the matrix coefficients in three-term relations for bivariate orthogonal polynomials based on univariate recurrence relations.

Findings

Matrix coefficients are diagonal or tridiagonal and explicitly computable.

Explicit expressions are derived from univariate polynomial recurrence coefficients.

Special cases of the general relations are analyzed.

Abstract

We study matrix three term relations for orthogonal polynomials in two variables constructed from orthogonal polynomials in one variable. Using the three term recurrence relation for the involved univariate orthogonal polynomials, the explicit expression for the matrix coefficients in these three term relations are deduced. These matrices are diagonal or tridiagonal with entries computable from the one variable coefficients in the respective three term recurrence relation. Moreover, some interesting particular cases are considered.