Cylindrical Dyck paths and the Mazorchuk–Turowska equation

@article{Hartwig2015CylindricalDP,
  title={Cylindrical Dyck paths and the Mazorchuk–Turowska equation},
  author={Jonas Torbj{\"o}rn Hartwig and Daniele Rosso},
  journal={Journal of Algebraic Combinatorics},
  year={2015},
  volume={44},
  pages={223-247}
}
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… 
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