Hermite numbers and new families of polynomials
Giuseppe Dattoli, Subuhi Khan, Ujair Ahmad
TL;DR
The paper addresses leveraging Hermite numbers and umbral calculus to streamline the study of Hermite polynomials and to develop new families via lacunary and higher-order constructions. It uses a Newton-binomial umbral representation with operators like $\hat h$ and $_m\hat h$ to derive generating functions, recurrences, differential equations, and integral transforms; it also connects these polynomials to evolution equations such as $\partial_y F=\partial_x^{m}F$ and to transform techniques like the Airy transform and Gauss-Weierstrass transform. Key contributions include explicit lacunary and higher-order Hermite polynomials, compact operator-based derivations, and integral-transform representations; demonstration of evolution operator forms and combinatorial interpretations. Significance lies in expanding the analytical and combinatorial toolkit for Hermite-type polynomials, enabling streamlined computation for higher-order and lacunary families and broadening applications in diffusion-type PDEs and optical beam theory.
Abstract
The operational calculus associated with Hermite numbers has been shown to be an effective tool for simplifying the study of special functions. Within this context, Hermite polynomials have been viewed as Newton binomials, with the consequent possibility of establishing previously unknown properties. In this article, this method is extended to study the lacunary Hermite polynomials and obtain novel results concerning their generating functions, recurrence relations, differential equations and certain integral transforms. Furthermore, we extend the idea to combinatorial interpretation of these polynomials, broadening their applicability in mathematical analysis and discrete structures.