$\Phi_k(M)=\displaystyle\sum_{n=0}^M(-1)^nn^k$ can be written by Euler polynomials and $F_k(N)$ can be written by Bernoulli polynomials. See this paper 


Duke Math. J.
Volume 19, Number 3 (1952), 475-481.
Note on irreducibility of the Bernoulli and Euler polynomials

L. Carlitz



 http://projecteuclid.org/euclid.dmj/1077477373