Unrestricting The Parameters of a Functional Equation

Good evening. I am looking into methods of generalization of Bernoulli polynomials. First, define

$$Phi_{N,k}(x)=frac{1}{N}sum_{j=0}^{N-1}omega_N^{-jk}expleft(omega_N^jxright)$$

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

$$G_{N,k}(x,z)=frac{x^kexpleft(xz+frac{omega_N^kx}{N}right)}{N^kk!Phi_{N,k}left(frac{x}{N}right)}$$

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

$$p_nleft(frac{2}{N}-zright)=(-1)^np_n(z)$$

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

$$p_nleft(frac{2}{N}+w_{2M}^Mzright)=w_{2M}^{Mn}p_n(z)$$

then perhaps there is a similar functional equation exists for all $N,kinmathbb{N}_0$ and $0le klt N$ of the form

$$p_nleft(xi_{N,k}+w_N^kzright)=w_N^{nk}p_n(z)$$

for some $xi_{N,k}inmathbb{C}$ 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 $xi_{N,k}$.