Good evening. I am looking into methods of generalization of Bernoulli polynomials. First, define
$$\Phi_{N,k}(x)=\frac{1}{N}\sum_{j=0}^{N-1}\omega_N^{-jk}\exp\left(\omega_N^jx\right)$$
where $\omega_N$ is the first $N$-th root of unity. The multiplicative inverse of this, $\Phi_{N,k}(x)^{-1}$ was used by Carlitz studying a special class of permutations that follow the pattern: $n-1$ rises, $1$ fall, and $k-1$ represents the number of rises after the last fall. For example, if $N=3$ and $k=2$, a possible permutation for the pattern would be
$$(14625738)$$
Now I define the following generating function
$$G_{N,k}(x,z)=\frac{x^k\exp\left(xz+\frac{\omega_N^kx}{N}\right)}{N^kk!\Phi_{N,k}\left(\frac{x}{N}\right)}$$
This generalizes Bernoulli polynomials as we get them for $N=2, k=1$ and Bernouli numbers if $z=0$. After playing around with the equations and polynomials that this particular generating function generated i determined a nice functional equation under particular parameters. So if $N$ is an even integer and $k=N/2$, then the $n$-th polynomial generated by $G_{2M,M}(x,z)$ has the property that
$$p_n\left(\frac{2}{N}-z\right)=(-1)^np_n(z)$$
This makes sense because the root of unity that helps generate this sequence of polynomial is $\omega_{2M}^M=-1$. My thought is then since this can be rewritten as
$$p_n\left(\frac{2}{N}+w_{2M}^Mz\right)=w_{2M}^{Mn}p_n(z)$$
then perhaps there is a similar functional equation exists for all $N,k\in\mathbb{N}_0$ and $0\le k\lt N$ of the form
$$p_n\left(\xi_{N,k}+w_N^kz\right)=w_N^{nk}p_n(z)$$
for some $\xi_{N,k}\in\mathbb{C}$ that depends on the parameters $N,k$.
I was hoping to ask if somebody could provide some insight into whether or not this particular problem is even solvable, i.e., finding an appropriate $\xi_{N,k}$ that will provide functional equations for all $N,k$ and if so, how to come up with $\xi_{N,k}$.
