Certain advancements in multidimensional q-hermite polynomials

Abstract

In the realm of specialized functions, the allure of q-calculus beckons to many scholars, captivating them with its prowess in shaping models of quantum computing, noncommutative probability, combinatorics, functional analysis, mathematical physics, approximation theory, and beyond. This study explores a new idea called the multidimensional q-Hermite polynomials, using different q-calculus techniques. Numerous properties and novel findings regarding these polynomials are derived, encompassing their generating function, series representations, recurrence relations, q-differential formula, and operational principles. Further, we proved that these polynomials are quasi-monomial in q-aspect. As the applications, these findings are subsequently employed to address connection between the multidimensional q-Hermite polynomials and the two-variable q-Legendre polynomials for the first time. Various characterizations are examined, as well the graphical representations of the two-variable q-Legendre polynomials are provided by the surface plots and graphs of distribution of zeros for the q-Legendre polynomials with some specific set of parameters are shown using Mathematica. Our investigations shed light on the intricate nature of these polynomials, elucidating their behaviour and facilitating deeper understanding within the realm of q-calculus.