Let $f(z)\in\mathbb{C}[z]$ have all its zeros on the line $\Re(z)=\alpha$ for some $\alpha\in\mathbb{R}$. It is an elementary fact (equivalent to Lemma 9.13 here) that if $u\in\mathbb{C}$ and $|u|=1$, then all the zeros of $g(z):=f(z+1)-uf(z-1)$ also lie on the line $\Re(z)=\alpha$. Moreover, $\deg g(z)=\deg f(z)$ unless $u=1$.

Let $u_1,u_2,\dots$ be any sequence of complex numbers on the circle $|z|=1$ such that no $u_i=1$. Let $n\geq 1$, and set $f_{1,n}(z)= z^n$. For $j\geq 1$ define inductively $f_{j+1,n}(z)=f_{j,n}(z+1)-u_jf_{j,n}(z-1)$. What can we say about the limiting behavior of $f_{m,n}(z)$ as $m,n\to\infty$? Will it always approach some scaled version of $\cosh(z)$? For one instance of this behavior, see Theorem 11.7 of the above link.