On 2-variable $q$-Legendre polynomials: the view point of the $q$-operational technique

Abstract

In this work, we exploit the methods of an operational formality and extension of quasi-monomials to describe and realize 2-variable $q$-Legendre polynomials. We introduce the generating function of 2-variable $q$-Legendre polynomials with a context of $0^{\text{th}}$ order $q$-Bessel Tricomi functions and obtain their properties such as series definition and $q$-differential equations. Also, we establish the $q$-multiplicative and $q$-derivative operators of these polynomials. The operational representations of 2-variable $q$-Legendre polynomials are obtained.