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Mathématiques   > Accueil   > Avancées en Mathématiques Pures et Appliquées   > Numéro 4 (Septembre 2022)   > Article

Une nouvelle géneralisation des nombres de Genocchi et conséquence sur les polynômes de Bernoulli

A new generalization of the Genocchi numbers and its consequence on the Bernoulli polynomials


Bakir Farhi
Université de Bejaia
Algeria



Publié le 21 octobre 2022   DOI : 10.21494/ISTE.OP.2022.0886

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This paper presents a new generalization of the Genocchi numbers and the Genocchi theorem. As consequences, we obtain some important families of integer-valued polynomials those are closely related to the Bernoulli polynomials. Denoting by $$${(B_n)}_{n \in \mathbb{N}}$$$ the sequence of the Bernoulli numbers and by $$${(B_n(X))}_{n \in \mathbb{N}}$$$ the sequence of the Bernoulli polynomials, we especially obtain that for any natural number $$$n$$$, the reciprocal polynomial of the polynomial $$$\big(B_n(X) - B_n\big)$$$ is integer-valued.

This paper presents a new generalization of the Genocchi numbers and the Genocchi theorem. As consequences, we obtain some important families of integer-valued polynomials those are closely related to the Bernoulli polynomials. Denoting by $$${(B_n)}_{n \in \mathbb{N}}$$$ the sequence of the Bernoulli numbers and by $$${(B_n(X))}_{n \in \mathbb{N}}$$$ the sequence of the Bernoulli polynomials, we especially obtain that for any natural number $$$n$$$, the reciprocal polynomial of the polynomial $$$\big(B_n(X) - B_n\big)$$$ is integer-valued.

Genocchi numbers Bernoulli numbers Bernoulli polynomials formal power series integer-valued polynomials

Genocchi numbers Bernoulli numbers Bernoulli polynomials formal power series integer-valued polynomials