Generalized q-difference equations for general q-polynomials with double q-binomial coefficients

TL;DR

This paper develops generalized q-difference equations for q-polynomials with double q-binomial coefficients, extending existing theories and deriving new generating functions and formulas in the field of q-calculus.

Contribution

It introduces a new framework of q-difference equations involving multiple variables for generalized q-polynomials, extending prior work and deriving novel generating functions.

Findings

Derived q-difference equations with seven variables.

Established Rogers and extended Rogers formulas for generalized q-polynomials.

Obtained mixed generating functions using q-difference equations.

Abstract

In this paper, we use the generalized q-polynomials with double q-binomial coefficients and homogeneous q-operators [J. Difference Equ. Appl. 20 (2014), 837--851.] to construct q-difference equations with seven variables, which generalize recent works of Jia et al [Symmetry 2021, 13, 1222.]. In addition, we derive Rogers formulas, extended Rogers formulas and Srivastava--Agarwal type bilinear generating functions for generalized q-polynomials, which generalize generating functions for Cigler's polynomials [J. Difference Equ. Appl. 24 (2018), 479--502.]. Finally, we also derive mixed generating functions using q-difference equations.