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Mathematics   > Home   > Advances in Pure and Applied Mathematics   > Issue 4 (September 2022)   > Article

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

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


Bakir Farhi
Université de Bejaia
Algeria



Published on 21 October 2022   DOI : 10.21494/ISTE.OP.2022.0886

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