Irreducibility of Faulhaber-like Polynomials over $\mathbb Q[x]$

Question

Motivation: Inspired by the famous Faulhaber polynomials $F_k(N)=\displaystyle\sum_{n=0}^Nn^k,$ I decided to study their alternating versions, $\Phi_k(M)=\displaystyle\sum_{n=0}^M(-1)^nn^k$.

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For $k\ge2$ and even values of $M=2N$ we have:

$$\begin{align}\Phi_k(2N)~&=~N^{(k\bmod2)+1}~(2N+1)^{(k+1)\bmod2}~{\color{blue}{P_{_k}(N)}}\\\\&=~2^{k+1}F_k(N)-F_k(2N)\end{align}$$

For $k\ge2$ and odd values of $M=2N+1$ we have:

$$\begin{align}\Phi_k(2N+1)~&=~-(N+1)^{(k\bmod2)+1}~(2N+1)^{(k+1)\bmod2}~{\color{blue}{Q_{_k}(N)}}\\\\&=~\Phi_k(2N)-(2N+1)^k~=~2^{k+1}F_k(N)-F_k(2N)-(2N+1)^k\end{align}$$

where $P_k$ and $Q_k$ are polynomials with integer coefficients of degree $k-2$ in $N$.

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My conjecture, based on computer aided verification for all values of $k\le10^3,$ is that $\color{blue}{P_k}$ and $\color{blue}{Q_k}$ are irreducible over $\color{blue}{\mathbb Q[x]}$. How could we either prove or disprove this statement ? Thank you!

Appendix: The explicit formulas for the first ten $P_k$ and $Q_k$ can be found here and here.

Answer 1

It is known that the sum $\Phi_k(M)=\displaystyle\sum_{n=0}^M(-1)^nn^k$ can be written by Euler polynomials and $F_k(N)$ can be written by Bernoulli polynomials.

Historically , Euler on page 499 of "L.Euler, Institutiones Calculi Differentialis, Petersberg,1755" (Euler archive link), introduced Euler polynomials, to evaluate the alternating sum $\Phi_k(M)$

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

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