Higher-order Hermite numbers: Properties and applications to evolution problems

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. The proposed method is systematically applied to a variety of evolution equations. Furthermore, this idea is extended to combinatorial interpretation of these polynomials, broadening their applicability in mathematical analysis and discrete structures.