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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. [ABSTRACT FROM AUTHOR]

The article aims at introducing a generalized 2D extension of the q -Bessel polynomials. The generating equation, series expansion and determinant form for the generalized 2 D q -Bessel polynomials are established. An example providing the continuous 2 D q -Hermite--Bessel polynomials is considered. The recurrence relation and other properties for the continuous 2 D q -Hermite--Bessel polynomials are derived. The graphs and plots depicting the behaviour of the continuous 2 D Hermite polynomials, continuous 2 D Hermite--Bessel polynomials and continuous 2 D q -Hermite--Bessel polynomials are drawn. The distribution of zeros for these polynomials is also presented. [ABSTRACT FROM AUTHOR]

The main objective of this article is to introduce the 2 D q -Bessel functions by means of the generating function and to frame these q -special functions into the context of the representation of the 2-dimensional Euclidian quantum algebra ε q (2). The quantum algebra ε q (2) and its representation present an algebraic derivation for an explicit expression of the 2 D q -Bessel functions in terms of the product of the q -Bessel functions of the first and second kinds. The series expansion, differential recurrence relations and functional identities for these functions are derived. Further, the q -Hermite—Bessel functions are introduced and their series expansions are derived. These expansions are used to derive the hypergeometric representation and generating equation for the q -Hermite—Bessel functions. [ABSTRACT FROM AUTHOR]

The significance of multi-variable special polynomials has been identified both in mathematical and applied frameworks. The article aims to focus on a new class of 3-variable Legendre-truncated-exponential-based Sheffer sequences and to investigate their properties by means of Riordan array techniques. The quasi-monomiality of these sequences is studied within the context of Riordan arrays. These sequences are expressed in determinant forms by utilizing the relation between the Sheffer sequences and Riordan arrays. Certain special Sheffer sequences are used to construct the Legendre-truncated-exponential-based Chebyshev polynomials. The same polynomials are used as base to introduce the hybrid classes involving the exponential polynomials and the Euler polynomials of higher order as illustrative examples of the aforementioned general class of polynomials. The shapes are shown and zeros are computed for these sequences by using mathematical software. The main purpose is to demonstrate the advantage of using numerical investigation and computations to discover fascinating pattern of scattering of zeros through graphical representations. [ABSTRACT FROM AUTHOR]

In this article, the 2-variable general polynomials are taken as base with Apostol type polynomials to introduce a family of 2-variable Apostol type polynomials. These polynomials are framed within the context of monomiality principle and their properties are established. Certain summation formulae for these polynomials are also derived. Examples of some members belonging to this family are considered and numbers related to some mixed special polynomials are also explored. [ABSTRACT FROM AUTHOR]