On an Umbral point of view to the Gaussian and Gaussian like functions

TL;DR

This paper reviews Gaussian functions using umbral symbolic methods, introduces quasi-Gaussian functions with longer tails, and explores their connection to hypergeometric functions, offering a novel perspective on Gaussian-like distributions.

Contribution

It presents a new umbral approach to Gaussian functions, introduces quasi-Gaussian functions, and links them to hypergeometric functions, expanding the theoretical understanding of Gaussian-like distributions.

Findings

Introduction of quasi-Gaussian functions with longer tails

Connection established between quasi-Gaussian functions and hypergeometric functions

New perspective on Gaussian functions using umbral symbolic methods

Abstract

In this note we review the theory of Gaussian functions by exploiting a point of view based on symbolic methods of umbral nature. We introduce quasi-Gaussian functions, which are close to Gaussian distribution but have a longer tail. Their use and their link with hypergeometric function is eventually presented.