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

Sergio A. Carrillo; Miguel Hurtado

doi:10.5802/crmath.172

Théorie des fonctions

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

Sergio A. Carrillo ¹ ; Miguel Hurtado ¹

¹ Programa de matemáticas, Universidad Sergio Arboleda, Calle 74 # 14-14, Bogotá, Colombia.

Comptes Rendus. Mathématique, Volume 359 (2021) no. 2, pp. 205-217.

Résumé
Abstract

L’objectif de cet article est de présenter une nouvelle récurrence simple pour les suites d’Appell et de Sheffer en termes de la fonctionnelle linéaire qui les définit, et d’expliquer comment cela équivaut à plusieurs caractérisations bien connues qui apparaissent dans la littérature. Nous donnons aussi plusieurs exemples, y compris des représentations intégrales des opérateurs inverses associés aux polynômes de Bernoulli et d’Euler, et une nouvelle représentation intégrale des polynômes d’Hermite $d$ -orthogonaux remis à l’échelle, qui généralise l’opérateur de Weierstrass associé aux polynômes d’Hermite.

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.

Reçu le : 2020-10-19
Révisé le : 2020-12-11
Accepté le : 2020-12-17
Publié le : 2021-03-17

DOI : 10.5802/crmath.172

Classification : 05A40, 11B83, 11B68
Mots clés : Sheffer and Appell sequences, Bernoulli, Euler and Hermite $d$-orthogonal polynomials

Affiliations des auteurs :

Sergio A. Carrillo ¹ ; Miguel Hurtado ¹

¹ Programa de matemáticas, Universidad Sergio Arboleda, Calle 74 # 14-14, Bogotá, Colombia.

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     author = {Sergio A. Carrillo and Miguel Hurtado},
     title = {Appell and {Sheffer} sequences: on their characterizations through functionals and examples},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {205--217},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {359},
     number = {2},
     year = {2021},
     doi = {10.5802/crmath.172},
     language = {en},
}

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Sergio A. Carrillo; Miguel Hurtado. Appell and Sheffer sequences: on their characterizations through functionals and examples. Comptes Rendus. Mathématique, Volume 359 (2021) no. 2, pp. 205-217. doi : 10.5802/crmath.172. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.172/

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