Hermite numbers and new families of polynomials

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.

• The paper introduces a new framework using Hermite numbers to define lacunary Hermite polynomials, providing novel generating functions and recurrence relations.
• It extends the analysis to higher-order Hermite polynomials, applying Newton binomial representations and third-order heat equation models.
• The research establishes connections with applied fields like optics and signal processing, proposing new algorithms for solving nonlinear equations.

Hermite Numbers and New Families of Polynomials

Introduction

This paper illustrates how Hermite numbers, typically used in operational calculus, serve as an instrument for exploring special polynomial functions. Specifically, the research extends known results of Hermite numbers and polynomials by introducing lacunary Hermite polynomials. It analyzes their generating functions, recurrence relations, differential equations, and integral transforms. The operational calculus associated with Hermite numbers simplifies the study of special functions by treating Hermite polynomials as analogous to Newton binomials. This approach unveils new identities and applications, particularly in combinatorial analysis.

Lacunary Hermite Polynomials

The paper first emphasizes the significance of Hermite polynomials expressed as $$H_{n}(x,y) = n!\sum_{r=0}^{\infty} \frac{x^{n-2r}y^{r}}{r!(n-2r)!}$$. It introduces lacunary Hermite polynomials and derives results concerning their associated generating functions and recurrence relations. The Hermite numbers of order $2$ are essential to this formulation, expressed using the umbral operator $\hat{h}$. The operator $\hat{h}$ expedites computations such as determining the Gaussian function as an ordinary exponential, $e^{i\hat{h}x}\varphi_{0} = e^{-x^{2}}$. The paper proposes integrals involving Hermite numbers and showcases their use in simplifying repeated derivatives of composite functions.

Extension to Higher Order Hermite Polynomials

The analysis extends to higher-order Hermite polynomials, particularly introducing the third-order $2$-variable Hermite polynomials $H_{n}^{(3)}(x,y)$ and their Hermite numbers $_{3}{h}_{r}$. The paper utilizes the Newton binomial representation $H_{n}^{(3)}(x,y)=(x+\sqrt[3]{y}\;{_3{\hat{h})^{n}\;{_3{\varphi}_{0}}}$ to explain the evolution operator associated with these polynomials and details their connection to the third-order heat equation. Applications are made using Airy functions to represent operators by reducing higher-order derivatives.

Further Generalizations and Applications

For $m>3$, the paper extends Hermite polynomial properties by constructing a multi-variable form of Hermite numbers and redefining them using gamma functions and Dirac delta functions. This allows for the derivation of $m$-th order Hermite polynomials, which simplify to Newton multinomials. Moreover, it explores how exponential operators with higher-order derivatives reduce to integral and differential equations involving shift operators.

Conclusion

The paper concludes by asserting the utility of Hermite numbers in umbral calculus and demonstrating applications to super-Gaussian beams in optics. It speculates on future impacts in designing algorithms for solving nonlinear equations using the developed framework. The research opens avenues not only in pure mathematics but also in applied mathematics domains, including signal processing and optical physics. By providing theoretical groundwork and mathematical constructs, this paper lays a foundation for further explorations in combinatorial interpretations and advanced mathematical applications.