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

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.

Keywords

• Sheffer and Appell sequences
• Bernoulli
• $d$-orthogonal polynomials%22%2C%22default_operator%22%3A%22AND%22%7D%7D%7D"> Euler and Hermite <span class="mathjax-formula">$d$</span>-orthogonal polynomials