On [formula omitted]th order Euler polynomials of degree [formula omitted] that are Eisenstein

Abstract

For m an even positive integer and p an odd prime, we show that the generalized Euler polynomial Emp(mp)(x) is in Eisenstein form with respect to p if and only if p does not divide m(2m−1)Bm. As a consequence, we deduce that at least 1/3 of the generalized Euler polynomials En(n)(x) are in Eisenstein form with respect to a prime p dividing n and, hence, irreducible over Q.