Fourth Order q-Difference Equation for the First Associated of the q-Classical Orthogonal Polynomials

We derive the fourth order q-difference equation satisfied by the first associated of the q-classical orthogonal polynomials. The coefficients of this equation are given in terms of the polynomials oe and ø which appear in the q-Pearson difference equation D q (oe ae) = ø ae defining the weight ae of the q-classical orthogonal polynomials inside the q-Hahn tableau. Keywords: q-Orthogonal polynomials, Fourth order q-difference equation. 1991 MSC: 33C25 1 Introduction The fourth order difference equation for the associated polynomials of all classical discrete polynomials were given for all integers r (order of association) in [5], using the properties of the Stieltjes functions of the associated linear forms. On the other hand, the equation for the first associated (r = 1) of all classical discrete polynomials was obtained in [13] using a useful relation proved in [2]. In this work, mimicking the approach used in [13] we give a single fourth order q-difference equation which is vali..