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.2010 Mathematics Subject Classification. 05A40, 11B83, 11B68. Key words and phrases. Sheffer and Appell sequences, Bernoulli, Euler and Hermite d-orthogonal polynomials. First author is supported by Ministerio de Economía y Competitividad from Spain, under the Project "Métodos asintóticos, algebraicos y geométricos en foliaciones singulares y sistemas dinámicos" (Ref.: PID2019-105621GB-I00) and Univ. Sergio Arboleda project IN.BG.086.20.002.