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.

Dattoli, G., Di Palma, E., Licciardi, S. (2023). On an Umbral Point of View of the Gaussian and Gaussian-like Functions. SYMMETRY, 15(12) [10.3390/sym15122157].

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

Licciardi, Silvia
2023-01-01

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.
2023
Settore MAT/08 - Analisi Numerica
Settore MAT/03 - Geometria
Settore MAT/05 - Analisi Matematica
Settore FIS/04 - Fisica Nucleare E Subnucleare
Dattoli, G., Di Palma, E., Licciardi, S. (2023). On an Umbral Point of View of the Gaussian and Gaussian-like Functions. SYMMETRY, 15(12) [10.3390/sym15122157].
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/10447/619323
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