Characterization theorem for Laguerre–Hahn orthogonal polynomials on non-uniform lattices

A characterization theorem for Laguerre–Hahn orthogonal polynomials on non-uniform lattices is stated and proved. This theorem proves the equivalence between the Riccati equation for the formal Stieltjes function, linear first-order difference relations for the orthogonal polynomials as well as for the associated polynomials of the first kind, and linear first-order difference relations for the functions of the second kind.