Polynomials defined by three-term recursion relations and satisfying a second recursion relation: connection with discrete integrability, remarkable (often Diophantine) factorizations

In this paper (as in previous ones) we identify and investigate polynomials p(n)((nu)) (x) featuring at least one additional parameter nu besides their argument x and the integer n identifying their degree. They are orthogonal (provided the parameters they generally feature fit into appropriate ranges) inasmuch as they are defined via standard three-term linear recursion relations; and they are interesting inasmuch as they obey a second linear recursion relation involving shifts of the parameter nu and of their degree n, and as a consequence, for special values of the parameter nu, also remarkable factorizations, often having a Diophantine connotation. The main focus of this paper is to relate our previous machinery to the standard approach to discrete integrability, and to identify classes of polynomials featuring these remarkable properties.