$\Phi_k(M)=\displaystyle\sum_{n=0}^M(-1)^nn^k$ can be written by Euler polynomials and $F_k(N)$ can be written by Bernoulli polynomials.
More precisely,
$$\sum_{i=0}^{n-1}i^p=\frac{1}{p+1}\sum_{k=0}^p\binom{p+1}{k}B_kn^{p+1-k}$$
where $B_k$ are Bernoulli numbers and can be defined by following generating function
$$\frac{x}{e^x-1}=\sum_{k=0}^\infty B_k\frac{x^k}{k!}$$
See this paper for irreducibility for such polynomials
Duke Math. J. Volume 19, Number 3 (1952), 475-481. Note on irreducibility of the Bernoulli and Euler polynomials
L. Carlitz