Characterization of Kummer hypergeometric Bernoulli polynomials and applications

Driss Drissi

doi:10.1016/j.crma.2019.10.004

Number theory/Mathematical analysis

Characterization of Kummer hypergeometric Bernoulli polynomials and applications

Presented by: the Editorial Board

Driss Drissi ¹

¹ Department of Mathematics, Rowan University, Glassboro, NJ 08028, USA

Comptes Rendus. Mathématique, Volume 357 (2019) no. 10, pp. 743-751.

Abstract
Résumé

In this paper, we present two characterizations of the sequences of Kummer hypergeometric polynomials $B_{a, b, n} (x)$ and Kummer hypergeometric polynomials of the second kind $K_{a, b, n} (x)$ , which are respectively defined by the exponential generating functions:

\begin{matrix} \frac{e^{x t}}{M (a, a + b; t)} = \sum_{n = 0}^{\infty} B_{a, b, n} (x) \frac{t^{n}}{n!} and \frac{e^{x t}}{U (a, a + b; t)} = \sum_{n = 0}^{\infty} K_{a, b, n} (x) \frac{t^{n}}{n!} \\ with M (a, b; t) = \sum_{n = 0}^{\infty} \frac{{(a)}_{n}}{{(b)}_{n}} \frac{t^{n}}{n!}, \end{matrix}

where

U (a, a + b; t)

is the Kummer hypergeometric function of the second kind.

First we construct Gauss–Weierstrass-type convolution operators $T_{w_{a, b}}$ with a well-chosen kernel (density) function for each sequence of Kummer hypergeometric polynomials and for Kummer hypergeometric polynomials of the second kind. Then we characterize Kummer hypergeometric polynomials as the only Appell polynomials having a weighted-integral mean equal to zero. Our approach is inspired by the Gauss–Weierstrass convolution transform for Hermite polynomials and the Kummer integral representation for confluent hypergeometric functions.

Dans cet article, nous présentons deux caractérisations des suites $B_{a, b, n} (x)$ et $K_{a, b, n} (x)$ de polynômes hypergéométriques de type Kummer définies par leurs fonctions génératrices :

\begin{matrix} \frac{e^{x t}}{M (a, a + b; t)} = \sum_{n = 0}^{\infty} B_{a, b, n} (x) \frac{t^{n}}{n!} et \frac{e^{x t}}{U (a, a + b; t)} = \sum_{n = 0}^{\infty} K_{a, b, n} (x) \frac{t^{n}}{n!} \\ avec M (a, b; t) = \sum_{n = 0}^{\infty} \frac{{(a)}_{n}}{{(b)}_{n}} \frac{t^{n}}{n!}, \end{matrix}

où

U (a, a + b; t)

est la fonction hypergéométrique de Kummer de seconde espèce.

Premièrement, nous construisons des opérateurs de convolution $T_{w_{a, b}}$ du type Gauss–Weierstrass pour chacune des suites de polynômes de Kummer de première et de seconde espèces. Deuxièmement, nous caractérisons les polynômes hypergéométriques de Kummer $B_{a, b, n} (x)$ comme étant les seuls polynômes ayant une moyenne intégrale pondérée égale à zero. Cette approche nous a été inspirée par la transformation de Gauss–Weierstrass pour les polynômes de Hermite et par la représentation intégrale de type Euler–Kummer pour les fonctions hypergéométriques.

Received: 2018-10-15
Accepted: 2019-10-08
Published online: 2019-10-01

DOI: 10.1016/j.crma.2019.10.004

Author's affiliations:

Driss Drissi ¹

¹ Department of Mathematics, Rowan University, Glassboro, NJ 08028, USA

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     title = {Characterization of {Kummer} hypergeometric {Bernoulli} polynomials and applications},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {743--751},
     publisher = {Elsevier},
     volume = {357},
     number = {10},
     year = {2019},
     doi = {10.1016/j.crma.2019.10.004},
     language = {en},
}

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Driss Drissi. Characterization of Kummer hypergeometric Bernoulli polynomials and applications. Comptes Rendus. Mathématique, Volume 357 (2019) no. 10, pp. 743-751. doi : 10.1016/j.crma.2019.10.004. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2019.10.004/

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