Linear partial divided-difference equation satisfied by multivariate orthogonal polynomials on quadratic lattices

TL;DR

This paper derives a fourth-order partial divided-difference equation for bivariate Racah polynomials on quadratic lattices, providing explicit recurrence relations, polynomial families, and connections to other bivariate orthogonal polynomials.

Contribution

It introduces a new fourth-order partial divided-difference equation for bivariate Racah polynomials and explicitly determines the associated recurrence matrices and polynomial families.

Findings

Explicit matrix coefficients for recurrence relations are obtained.

The monic bivariate Racah polynomials are constructed from these relations.

Connections between different families of bivariate Racah polynomials are established.

Abstract

In this paper, a fourth-order partial divided-difference equation on quadratic lattices with polynomial coefficients satisfied by bivariate Racah polynomials is presented. From this equation we obtain explicitly the matrix coefficients appearing in the three-term recurrence relations satisfied by any bivariate orthogonal polynomial solution of the equation. In particular, we provide explicit expressions for the matrices in the three-term recurrence relations satisfied by the bivariate Racah polynomials introduced by Tratnik. Moreover, we present the family of monic bivariate Racah polynomials defined from the three-term recurrence relations they satisfy, and we solve the connection problem between two different families of bivariate Racah polynomials. These results are then applied to other families of bivariate orthogonal polynomials, namely the bivariate Wilson, continuous dual Hahn and continuous Hahn, the latter two through limiting processes. The fourth-order partial divided-difference equations on quadratic lattices are shown to be of hypergeometric type in the sense that the divided-difference derivatives of solutions are themselves solution of the same type of divided-difference equations.