Appell and Sheffer sequences: on their characterizations through functionals and examples

TL;DR

This paper introduces a new recurrence relation for Appell and Sheffer sequences using their defining linear functionals, demonstrating equivalence with known characterizations and providing novel examples and integral representations.

Contribution

It presents a simplified recurrence for these sequences and connects it to existing characterizations, along with new integral representations and examples.

Findings

New recurrence relation for Appell and Sheffer sequences

Equivalence with known characterizations

Novel integral representations of related polynomials

Abstract

The aim of this paper is to present a new simple recurrence for Appell and Sheffer sequences in terms of the linear functional that defines them, and to explain how this is equivalent to several well-known characterizations appearing in the literature. We also give several examples, including integral representations of the inverse operators associated to Bernoulli and Euler polynomials, and a new integral representation of the re-scaled Hermite $d$-orthogonal polynomials generalizing the Weierstrass operator related to the Hermite polynomials.