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

TL;DR

This paper introduces a new generalization of Genocchi numbers, leading to novel integer-valued polynomial families related to Bernoulli polynomials, with implications for their properties and applications.

Contribution

It proposes a new generalization of Genocchi numbers and explores its impact on Bernoulli polynomials and related integer-valued polynomial families.

Findings

Reciprocal polynomials of Bernoulli polynomial differences are integer-valued.

New families of integer-valued polynomials related to Bernoulli polynomials.

Generalization of Genocchi numbers extends classical results.

Abstract

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.