@ARTICLE{10.21494/ISTE.OP.2022.0886, 

TITLE={A new generalization of the Genocchi numbers and its consequence on the Bernoulli polynomials}, 
  
AUTHOR={Bakir Farhi, }, 
	
JOURNAL={Advances in Pure and Applied Mathematics}, 
	
VOLUME={13}, 

NUMBER={Issue 4 (September 2022)}, 
	
YEAR={2022}, 

URL={http://openscience.fr/A-new-generalization-of-the-Genocchi-numbers-and-its-consequence-on-the}, 
	
DOI={10.21494/ISTE.OP.2022.0886}, 
	
ISSN={1869-6090}, 
	
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.}}