Linear partial $q$-difference equations on $q$-linear lattices and their bivariate $q$-orthogonal polynomial solutions

TL;DR

This paper studies orthogonal polynomial solutions to a class of linear second-order partial $q$-difference equations on $q$-linear lattices, providing explicit formulas, orthogonality relations, and limit behaviors as $q$ approaches 1.

Contribution

It introduces new explicit solutions and detailed analysis of bivariate $q$-orthogonal polynomials governed by partial $q$-difference equations, extending previous work on $q$-Jacobi polynomials.

Findings

Derived a $q$-Pearson system for orthogonality weights.

Provided explicit Rodrigues-type formulas for solutions.

Analyzed limit relations as $q \uparrow 1$.

Abstract

Orthogonal polynomial solutions of an admissible potentially self-adjoint linear second-order partial $q$-difference equation of the hypergeometric type in two variables on $q$-linear lattices are analyzed. A $q$-Pearson's system for the orthogonality weight function, as well as for the difference derivatives of the solutions are presented, giving rise to a solution of the $q$-difference equation under study in terms of a Rodrigues-type formula. The monic orthogonal polynomial solutions are treated in detail, giving explicit formulae for the matrices in the corresponding recurrence relations they satisfy. Lewanowicz and Wo\'zny [S. Lewanowicz, P. Wo\'zny, J. Comput. Appl. Math. 233 (2010) 1554--1561] have recently introduced a (non-monic) bivariate extension of big $q$-Jacobi polynomials together with a partial $q$-difference equation of the hypergeometric type that governs them. This equation is analyzed in the last section: we provide two more orthogonal polynomial solutions, namely, a second non-monic solution from the Rodrigues' representation, and the monic solution both from the recurrence relation that govern them and also explicitly given in terms of generalized bivariate basic hypergeometric series. Limit relations as $q \uparrow 1$ for the partial $q$-difference equation and for the all three $q$-orthogonal polynomial solutions are also presented.