Series with Hermite Polynomials and Applications

TL;DR

This paper derives new series transformation formulas involving Hermite polynomials and applies them to various mathematical sequences and numbers, revealing new identities and relationships.

Contribution

It introduces novel series transformation formulas with Hermite polynomials and explores their applications to harmonic numbers, Lucas sequences, and Stirling numbers.

Findings

New series involving Hermite polynomials and harmonic numbers

Series connecting Hermite and Laguerre polynomials

Identities involving Hermite polynomials and Stirling numbers

Abstract

We obtain a series transformation formula involving the classical Hermite polynomials. We then provide a number of applications using appropriate binomial transformations. Several of the new series involve Hermite polynomials and harmonic numbers, Lucas sequences, exponential and geometric numbers. We also obtain a series involving both Hermite and Laguerre polynomials, and a series with Hermite polynomials and Stirling numbers of the second kind.