Linear Partial Divided-Difference Equation Satisfied by Multivariate Orthogonal Polynomials on Quadratic Lattices

¹ African Institute for Mathematical Sciences, AIMS-Cameroon P.O. Box 608, Limbé Crystal Gardens, South West Region, Cameroon
² Department of Mathematics, Faculty of Sciences, University of Yaounde I, Yaoundé, Cameroon
³ Department of Mathematics, Higher Teachers' Training College, University of Yaounde I, Cameroon
⁴ Departamento de Matemática Aplicada II, E.E. Industrial, Universidade de Vigo Campus Lagoas-Marcosende, 36310 Vigo, Spain
⁵ Departamento de Matemática Aplicada II, E.E. Aeronáutica e do Espazo, Universidade de Vigo Campus As Lagoas s/n, 32004 Ourense, Spain

^* Corresponding author. E-mail: area@uvigo.es

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 result, we recover the difference equation satisfied by the bivariate Racah polynomials given by Geronimo and Iliev. Moreover, 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.