In the theory of elliptic functions we have the well-known Weierstrass $\zeta$-function (which is, by the way, not elliptic but satisfies identities $$ \zeta(z+2\omega_i)=\zeta(z)+2\zeta(\omega_i), $$ where $2\omega_1, 2\omega_2$ are periods of its derivative $\zeta'(z)$).
My question is: is it possible to construct a non-constant polynomial in $z$ as a ratio of polynomials in expressions of the type $\zeta(kz+\delta)$ (where $k$ is a non-zero integer and $\delta$ is an arbitrary complex number) and its derivatives in $z$?
Motivation comes from the problem of discretization of commuting differential operators but it is probably irrelevant here.