Cylindrical Dyck paths and the Mazorchuk–Turowska equation

We classify all solutions (p, q) to the equation $$p(u)q(u)=p(u+\beta )q(u+\alpha )$$p(u)q(u)=p(u+β)q(u+α) where p and q are complex polynomials in one indeterminate u, and $$\alpha $$α and $$\beta $$β are fixed but arbitrary complex numbers. This equation is a special case of a system of equations which ensures that certain algebras defined by generators and relations are non-trivial. We first give a necessary condition for the existence of non-trivial solutions to the equation. Then, under this condition, we use combinatorics of generalized Dyck paths to describe all solutions and a canonical way to factor each solution into a product of irreducible solutions.