$q$-difference equation for generalized trivariate $q$-Hahn polynomials

TL;DR

This paper introduces a new family of trivariate $q$-Hahn polynomials, deriving their properties and generating functions using a specialized $q$-difference operator, expanding the theoretical framework of $q$-orthogonal polynomials.

Contribution

It presents a novel family of trivariate $q$-Hahn polynomials and derives their generating functions and $q$-difference equations using a homogeneous $q$-difference operator.

Findings

Derived extended generating functions for the polynomials.

Established Rogers and Srivastava-Agarwal type formulas.

Connected the polynomials to a specific $q$-difference operator.

Abstract

In this paper, we introduce a family of trivariate $q$-Hahn polynomials $\Psi_n^{(a)}(x,y,z|q)$ as a general form of Hahn polynomials $\psi_n^{(a)}(x|q),$ $\psi_n^{(a)}(x,y|q)$ and $F_n(x,y,z;q)$. We represent $\Psi_n^{(a)}(x,y,z|q)$ by the homogeneous $q$-difference operator $\widetilde{L}(a,b; \theta_{xy})$ introduced by Srivastava {\it et al} [H. M. Srivastava, S. Arjika and A. Sherif Kelil, {\it Some homogeneous $q$-difference operators and the associated generalized Hahn polynomials}, Appl. Set-Valued Anal. Optim. {\bf 1} (2019), pp. 187--201.] to derive: extended generating, Rogers formula, extended Rogers formula and Srivastava-Agarwal type generating functions involving $\Psi_n^{(a)}(x,y,z|q)$ by the $q$-difference equation.