I'm wondering what the solutions of complex linear difference equations like \begin{equation} f(z+\eta_1)+f(z+\eta_2)+\ldots+f(z+\eta_n)=0,\ \ \ \eta_1 \cdots\eta_n \in \mathbf{C} \end{equation} look like. It seems easy when $\eta_i$ are all integers, but I don't know the general case.

If we cannot get the exact form of solutions, can we get the order?

remark. the order of a meromorphic function is define to be $\rho(f) = \overline{\lim}\frac{\log T(r,f)}{\log r}$, where $T(r,f)$ represents the Nevanlinna characteristic of $f(z)$. If $f(z)$ is an entire function, $\rho(f) = \overline{\lim} \frac{\log\log M(r,f)}{\log r}$ where $M(r,f) = \sup_{|z| < r} f(z)$.