q-Legendre based Gould–Hopper polynomials and q-operational methods

The generalization of the monomiality principle for q-special polynomials has just been explained and demonstrated. This extension is used to study the monomiality features of the number of q-special polynomials, such as q-Appell polynomials, q- Gould–Hopper polynomials, two variables q-Hermite, q-Laguerre and q-Legendre polynomials. Additionally, several kinds of hybrid q-special polynomials and their monomiality features are studied, such as two-variable q-Laguerre-Appell polynomials, two-variable based q-Hermite-Appell polynomials and q-Gould–Hopper–Appell polynomials. This study seeks to generate the q-Legendre–Gould–Hopper polynomials and then describe their attributes by extending the idea of monomiality for q-polynomials. Furthermore, we propose operational representations, expansion formulae and new families of these polynomials with the aid of q-operational methods and extension for monomiality principle of q-polynomials.