Lacunary generating functions of Hermite polynomials and symbolic methods

We employ an umbral formalism to reformulate the theory of Hermite polynomials and the derivation of the associated lacunary generating functions

Identities involving 3-variable Hermite polynomials arising from umbral method

In this paper, we employ an umbral method to reformulate the 3-variable Hermite polynomials and introduce the 4-parameter 3-variable Hermite polynomials. We also obtain some new properties for these polynomials. Moreover, some special cases are discussed and some concluding remarks are also given. © 2020, The Author(s)

On an Umbral Point of View of the Gaussian and Gaussian-like Functions

The theory of Gaussian functions is reformulated using an umbral point of view. The symbolic method we adopt here allows an interpretation of the Gaussian in terms of a Lorentzian image function. The formalism also suggests the introduction of a new point of view of trigonometry, opening a new interpretation of the associated special functions. The Erfi ( x ) , is, for example, interpreted as the “sine” of the Gaussian trigonometry. The possibilities offered by the Umbral restyling proposed here are noticeable and offered by the formalism itself. We mention the link between higher-order Gaussian trigonometric functions, Hermite polynomials, and the possibility of introducing new forms of distributions with longer tails than the ordinary Gaussians. The possibility of framing the theoretical content of the present article within a redefinition of the hypergeometric function is eventually discussed