Explicit Congruences for Euler Polynomials

Abstract

In this paper we establish some explicit congruences for Euler polynomials modulo a general positive integer. As a consequence, if a,m ∈ ℤ and 2 ∤ m then

$$ \frac{{m^{k + 1} }} {2}E_k \left( {\frac{{x + a}} {m}} \right) - \frac{{\left( { - 1} \right)^a }} {2}E_k \left( x \right) \in \mathbb{Z}\left[ x \right] for every k = 0,1,2,..., $$

which may be regarded as a refinement of a multiplication formula.

Supported by the National Science Fund for Distinguished Young Scholars (No. 10425103) and the Key Program of NSF (No. 10331020) in China.