A digression on Hermite polynomials

TL;DR

This paper provides a comprehensive survey of Hermite polynomials, highlighting their fundamental properties and applications in polynomial connection problems, probability, and graph combinatorics, including some original contributions.

Contribution

It offers a self-contained exposition with new insights in certain sections, enhancing understanding of Hermite polynomials and their diverse applications.

Findings

Summarizes key properties of Hermite polynomials

Demonstrates applications in probability and combinatorics

Includes original results in specific sections

Abstract

Orthogonal polynomials are of fundamental importance in many fields of mathematics and science, therefore the study of a particular family is always relevant. In this manuscript, we present a survey of some general results of the Hermite polynomials and show a few of their applications in the connection problem of polynomials, probability theory and the combinatorics of a simple graph. Most of the content presented here is well known, except for a few sections where we add our own work to the subject, nevertheless, the text is meant to be a self-contained personal exposition.