A NEW CHARACTERIZATION OF SYMMETRIC DUNKL AND Q-DUNKL-CLASSICAL ORTHOGONAL POLYNOMIALS

Abstract

In this paper, we consider the following L-difference equation Φ(x)LPn+1(x)=(ξnx+ϑn)Pn+1(x)+λnPn(x),n≥0, where Φ is a monic polynomial (even), degΦ≤2, ξn,ϑn,λn,n≥0, are complex numbers and L is either the Dunkl operator Tμ or the the q-Dunkl operator T(θ,q). We show that if L=Tμ, then the only symmetric orthogonal polynomials satisfying the previous equation are, up a dilation, the generalized Hermite polynomials and the generalized Gegenbauer polynomials and if L=T(θ,q), then the q2-analogue of generalized Hermite and the q2-analogue of generalized Gegenbauer polynomials are, up a dilation, the only orthogonal polynomials sequences satisfying the L-difference equation.