  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.
