On nth order Euler polynomials of degree n that are Eisenstein

For m an even positive integer and p an odd prime, we show that the generalized Euler polynomial 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. (c) 2023 Royal Dutch Mathematical Society (KWG). Published by Elsevier B.V. All rights reserved.