Fourth-order difference equation satisfied by the co-recursive of q-classical orthogonal polynomials

Abstract

We derive the fourth-order q-difference equation satisfied by the co-recursive of q-classical orthogonal polynomials. The coefficients of this equation are given in terms of the polynomials φ and ψ appearing in the q-Pearson difference equation D q(φρ)=ψρ defining the weight ρ of the q-classical orthogonal polynomials inside the q-Hahn tableau. Use of suitable change of variable and limit processes allow us to recover the results known for the co-recursive of the classical continuous and classical discrete orthogonal polynomials. Moreover, we describe particular situations for which the co-recursive of classical orthogonal polynomials are still classical and express these new families in terms of the starting ones.