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Characterizations of the Symmetric \(T_{(\theta , q)}\)-Classical Orthogonal q-Polynomials

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Abstract

In this paper, we give two characterizations for symmetric q-Dunkl-classical orthogonal polynomials. The first one is related to a spectral problem for a second-order linear q-difference differential operator. The second one is given by a distribution equation of Pearson type fulfilled by their corresponding linear functionals. Then, we show that the \(q^2\)-analogue of generalized Hermite and the \(q^2\)-analogue of generalized Gegenbauer polynomials are, up a dilation, the only symmetric q-Dunkl-classical orthogonal polynomials.

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Acknowledgements

The authors thank the valuable comments and suggestions by the referee. They have contributed to improve the presentation of this manuscript. The work of the third author (FM) has been supported by Agencia Estatal de Investigación (AEI) of Spain, grant PGC2018-096504-B-C33, and Fondos Europeos de Desarrollo Regional (FEDER).

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Authors and Affiliations

  1. Department of Mathematics, College of Sciences, Gabès University, Gabes, Tunisia

    B. Bouras & Y. Habbachi

  2. Departamento de Matemáticas, Universidad Carlos III de, Madrid, Spain

    F. Marcellán

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  1. B. Bouras
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  2. Y. Habbachi
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  3. F. Marcellán
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Bouras, B., Habbachi, Y. & Marcellán, F. Characterizations of the Symmetric \(T_{(\theta , q)}\)-Classical Orthogonal q-Polynomials. Mediterr. J. Math. 19, 66 (2022). https://doi.org/10.1007/s00009-022-01986-8

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  • DOI: https://doi.org/10.1007/s00009-022-01986-8

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