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The main purpose of this article is to construct a new class of multivariate Legendre-Hermite-Apostol type Frobenius-Euler polynomials. A number of significant analytical characterizations of these polynomials using various generating function techniques are provided in a methodical manner. These enactments involve explicit relations comprising Hurwitz-Lerch zeta functions and λ -Stirling numbers of the second kind, recurrence relations, and summation formulae. The symmetry identities for these polynomials are established by connecting generalized integer power sums, double power sums and Hurwitz-Lerch zeta functions. In the end, these polynomials are also characterized Svia an algebraic matrix based approach. [ABSTRACT FROM AUTHOR]

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 schemata for applications of the integral transforms of mathematical physics to recurrence relations, differential, integral, integro-differential equations and in the theory of special functions has been developed. The article aims to introduce and present operational representations for a new class of extended Hermite-Apostol type Frobenius-Genocchi polynomials via integral transforms. The recurrence relations and some identities involving these polynomials are established. The article concludes by establishing a determinant form and quasi-monomial properties for the Hermite-Apostol type Frobenius-Genocchi polynomials and for their extended forms. [ABSTRACT FROM AUTHOR]

The quantum groups nowadays attract a considerable interest of mathematicians and physicists. The theory of q-special functions has received a group-theoretic interpretation using the techniques of quantum groups and quantum algebras. This paper focuses on introducing the q-Tricomi functions and 2D q-Tricomi functions through the generating function and series expansion and for the first time establishing a connecting relation between the q-Tricomi and q-Bessel functions. The behavior of these functions is described through shapes, and the contrast between them is observed using mathematical software. Further, the problem of framing the q-Tricomi and 2D q-Tricomi functions in the context of the irreducible representation (ω) of the two-dimensional quantum algebra ℰq(2) is addressed, and certain relations involving these functions are obtained. 2-Variable 1-parameter q-Tricomi functions and their relationship with the 2-variable 1-parameter q-Bessel functions are also explored. [ABSTRACT FROM AUTHOR]

In this article, the operational definitions and integral representations are combined to introduce new families of the generalized Legendre and generalized Legendre-Appell polynomials. The explicit summation formulae, determinant definitions and recurrence relations for the generalized Legendre-Appell polynomials are derived by making use of the integral transforms and appropriate operational rules. An analogous study of these results for the generalized Legendre-Bernoulli, Legendre-Euler and Legendre-Genocchi polynomials is presented. Several identities for these polynomials are also derived by employing appropriate operational definitions. [ABSTRACT FROM AUTHOR]