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

Abstract

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.

Keywords

• umbral methods 05A40, 44A99, and 47B99
• operator theories 44A99, 47B99, and 47A62
• special functions 33C52, 33C65, 33C99, 33B10, and 33B15
• Bessel function 33C10
• hypergeometric function 33C20
• Fresnel integral 46T12