A NEW CHARACTERIZATION OF SYMMETRIC DUNKL AND \(q\)-DUNKL-CLASSICAL ORTHOGONAL POLYNOMIALS

Yahia Habbachi     (Faculty of Sciences, University of Gabes, Erriadh City 6072, Zrig, Gabes, Tunisia)

Abstract


In this paper, we consider the following \(\mathcal{L}\)-difference equation
$$\Phi(x) \mathcal{L}P_{n+1}(x)=(\xi_nx+\vartheta_n)P_{n+1}(x)+\lambda_nP_{n}(x),\quad n\geq0,$$where \(\Phi\) is a monic polynomial (even), \(\deg\Phi\leq2\), \(\xi_n,\,\vartheta_n,\,\lambda_n,\,n\geq0\), are complex numbers and \(\mathcal{L}\) is either the Dunkl operator \(T_\mu\) or the the \(q\)-Dunkl operator \(T_{(\theta,q)}\). We show that if \(\mathcal{L}=T_\mu\), 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 \(\mathcal{L}=T_{(\theta,q)}\), then the \(q^2\)-analogue of generalized Hermite and the \(q^2\)-analogue of generalized Gegenbauer polynomials are, up a dilation, the only orthogonal polynomials sequences satisfying the \(\mathcal{L}\)-difference equation.


Keywords


Orthogonal polynomials, Dunkl operator, \(q\)-Dunkl operator.

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References


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DOI: http://dx.doi.org/10.15826/umj.2023.2.009

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