Factorization of fourth-order differential equations for perturbed classical orthogonal polynomials

We factorize the fourth-order differential equations satisfied by the Laguerre–Hahn orthogonal polynomials obtained from some perturbations of classical orthogonal polynomials such as: the rth associated (for generic r), the general co-recursive, the general co-recursive associated, the general co-dilated and the general co-modified classical orthogonal polynomials. Moreover, we find four linearly independent solutions of the fourth-order differential equations, and show that the factorization obtained for modifications of classical orthogonal polynomials is still valid, with some minor changes when the polynomial family modified is semi-classical. Finally, we extend the validity of the results obtained for the associated classical orthogonal polynomials with integer order of association from integers to reals.