$\Phi_k(M)=\displaystyle\sum_{n=0}^M(-1)^nn^k$ 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