A NEW CHARACTERIZATION OF SYMMETRIC DUNKL AND \(q\)-DUNKL-CLASSICAL ORTHOGONAL POLYNOMIALS
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 numbe...
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Ural Branch of the Russian Academy of Sciences and Ural Federal University named after the first President of Russia B.N.Yeltsin, Krasovskii Institute of Mathematics and Mechanics
2023-12-01
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| Series: | Ural Mathematical Journal |
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| Online Access: | https://umjuran.ru/index.php/umj/article/view/485 |
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| author | Yahia Habbachi |
| author_facet | Yahia Habbachi |
| author_sort | Yahia Habbachi |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-ebb6c17bb5804fea990ee2d576e9e32a |
| institution | DOAJ |
| issn | 2414-3952 |
| language | English |
| publishDate | 2023-12-01 |
| publisher | Ural Branch of the Russian Academy of Sciences and Ural Federal University named after the first President of Russia B.N.Yeltsin, Krasovskii Institute of Mathematics and Mechanics |
| record_format | Article |
| series | Ural Mathematical Journal |
| spelling | doaj-art-ebb6c17bb5804fea990ee2d576e9e32a2025-08-20T02:51:46ZengUral Branch of the Russian Academy of Sciences and Ural Federal University named after the first President of Russia B.N.Yeltsin, Krasovskii Institute of Mathematics and MechanicsUral Mathematical Journal2414-39522023-12-019210.15826/umj.2023.2.009188A NEW CHARACTERIZATION OF SYMMETRIC DUNKL AND \(q\)-DUNKL-CLASSICAL ORTHOGONAL POLYNOMIALSYahia Habbachi0Faculty of Sciences, University of Gabes, Erriadh City 6072, Zrig, GabesIn 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.https://umjuran.ru/index.php/umj/article/view/485orthogonal polynomials, dunkl operator, \(q\)-dunkl operator. |
| spellingShingle | Yahia Habbachi A NEW CHARACTERIZATION OF SYMMETRIC DUNKL AND \(q\)-DUNKL-CLASSICAL ORTHOGONAL POLYNOMIALS Ural Mathematical Journal orthogonal polynomials, dunkl operator, \(q\)-dunkl operator. |
| title | A NEW CHARACTERIZATION OF SYMMETRIC DUNKL AND \(q\)-DUNKL-CLASSICAL ORTHOGONAL POLYNOMIALS |
| title_full | A NEW CHARACTERIZATION OF SYMMETRIC DUNKL AND \(q\)-DUNKL-CLASSICAL ORTHOGONAL POLYNOMIALS |
| title_fullStr | A NEW CHARACTERIZATION OF SYMMETRIC DUNKL AND \(q\)-DUNKL-CLASSICAL ORTHOGONAL POLYNOMIALS |
| title_full_unstemmed | A NEW CHARACTERIZATION OF SYMMETRIC DUNKL AND \(q\)-DUNKL-CLASSICAL ORTHOGONAL POLYNOMIALS |
| title_short | A NEW CHARACTERIZATION OF SYMMETRIC DUNKL AND \(q\)-DUNKL-CLASSICAL ORTHOGONAL POLYNOMIALS |
| title_sort | new characterization of symmetric dunkl and q dunkl classical orthogonal polynomials |
| topic | orthogonal polynomials, dunkl operator, \(q\)-dunkl operator. |
| url | https://umjuran.ru/index.php/umj/article/view/485 |
| work_keys_str_mv | AT yahiahabbachi anewcharacterizationofsymmetricdunklandqdunklclassicalorthogonalpolynomials AT yahiahabbachi newcharacterizationofsymmetricdunklandqdunklclassicalorthogonalpolynomials |