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

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


Bakir Farhi
Université de Bejaia
Algeria



Published on 16 May 2022   DOI :

Abstract

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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