Properties of Multivariable Hermite Polynomials in Correlation with Frobenius–Genocchi Polynomials

Abstract

The evolution of a physical system occurs through a set of variables, and the problems can be treated based on an approach employing multivariable Hermite polynomials. These polynomials possess beneficial properties exhibited in functional and differential equations, recurring and explicit relations as well as symmetric identities, and summation formulae, among other examples. In view of these points, comprehensive schemes have been developed to apply the principle of monomiality from mathematical physics to various mathematical concepts of special functions, the development of which has encompassed generalizations, extensions, and combinations of other functions. Accordingly, this paper presents research on a novel family of multivariable Hermite polynomials associated with Frobenius–Genocchi polynomials, deriving the generating expression, operational rule, differential equation, and other defining characteristics for these polynomials. Additionally, the monomiality principle for these polynomials is verified, as well as establishing the series representations, summation formulae, operational and symmetric identities, and recurrence relations satisfied by these polynomials. This proposed scheme aims to provide deeper insights into the behavior of these polynomials and to uncover new connections between these polynomials, to enhance understanding of their properties.

On q-Hermite Polynomials with Three Variables: Recurrence Relations, q-Differential Equations, Summation and Operational Formulas

“…Hermite polynomials can also be found in the field of signal processing as Hermitian wavelets in the wavelet transform analysis probability, similar to the Edgeworth series as well as their relation to Brownian motion, combinatorics as a manifestation of an Appell series observing the umbral calculus and numerical computation. For further information concerning Hermite polynomials and their applications, the interested reader may consult the research papers [1][2][3][4][5][6][7][8][9][10][11].…”