The Fourth-order Difference Equation Satisfied by the Associated Orthogonal Polynomials of the Δ -Laguerre–Hahn Class

Abstract

Starting from the Dω-Riccati difference equation satisfied by the Stieltjes function of a linear functional, we work out an algorithm which enables us to write the general fourth-order difference equation satisfied by the associated of any integer order of orthogonal polynomials of the Δ -Laguerre–Hahn class. Moreover, in classical situations (Meixner, Charlier, Krawtchouk and Hahn), we give these difference equations explicitly; and from the Hahn difference equation, by limit processes we recover the difference equations satisfied by the associated of the classical discrete orthogonal polynomials and the differential equations satisfied by the associated of the classical continuous orthogonal polynomials.