On Fourth-order Difference Equations for Orthogonal Polynomials of a Discrete Variable: Derivation, Factorization and Solutions

Abstract

We derive and factorize the fourth-order difference equations satisfied by orthogonal polynomials obtained from some modifications of the recurrence coefficients of classical discrete orthogonal polynomials such as: the associated, the general co-recursive, co-recursive associated, co-dilated and the general co-modified classical orthogonal polynomials. Moreover, we find four linearly independent solutions of these fourth-order difference equations, and show how the results obtained for modified classical discrete orthogonal polynomials can be extended to modified semi-classical discrete orthogonal polynomials. Finally, we extend the validity of the results obtained for the associated classical discrete orthogonal polynomials with integer order of association from integers to reals.

On Factorization and Solutions ofq-difference Equations Satisfied by some Classes of Orthogonal Polynomials

“…(10) for each case are given in Refs. [11,12,21] for the first five cases and in Ref. [27] for the anti-associated.…”

“…The proofs are similar to the previous ones and are also similar to those given in Refs. [11,12] for the continuous and discrete cases. However, one will need the following relations [11,12] linking the modified families to the starting ones.…”

“…Among these perturbations belong [21] (see also Refs. [11,12] for more details and intermediate relations):…”

“…Following the above mentioned works [11,12], we derive in this paper the fourth-order q-difference equations satisfied by the above mentioned modifications of the three-term recurrence coefficients of the q-classical orthogonal polynomials. We also factorize the corresponding q-difference operators and find a basis of solutions for each case.…”

“…Also, the factorization and the solution of the fourth-order differential and difference equations satisfied by some modifications of classical orthogonal polynomials of continuous and discrete variables were obtained recently in Refs. [11,12].…”