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\(( \mathcal {R}, p,q)\)-Rogers–Szegö and Hermite Polynomials, and Induced Deformed Quantum Algebras

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Orthogonal Polynomials (AIMSVSW 2018)

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Abstract

Deformed quantum algebras, namely the q-deformed algebras and their extensions to (p, q)-deformed algebras, continue to attract much attention. One of the main reasons is that these topics represent a meeting point of nowadays fast developing areas in mathematics and physics like the theory of quantum orthogonal polynomials and special functions, quantum groups, integrable systems, quantum and conformal field theories and statistics.

This contribution paper aims at characterizing the \(({\mathcal R},p,q)\)-Rogers–Szegö polynomials, and the \(({\mathcal R},p,q)\)-deformed difference equation giving rise to raising and lowering operators. These polynomials induce some realizations of generalized deformed quantum algebras, (called \(({\mathcal R},p,q)\)-deformed quantum algebras), which are here explicitly constructed. The study of continuous \(({\mathcal R},p,q)\)-Hermite polynomials is also performed. Known particular cases are recovered.

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Acknowledgements

This work is supported by TWAS Research Grant RGA No. 17-542 RG/MATHS/AF/AC_G-FR3240300147. The ICMPA-UNESCO Chair is in partnership with Daniel Iagolnitzer Foundation (DIF), France, and the Association pour la Promotion Scientifique de l’Afrique (APSA), supporting the development of mathematical physics in Africa. I am grateful to my students, Fridolin Melong and Cyrille Essossolim Haliya, who devoted their Christmas day to carefully read this manuscript and check the details.

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Authors and Affiliations

  1. International Chair in Mathematical Physics and Applications (ICMPA-UNESCO Chair), University of Abomey-Calavi, Cotonou, Republic of Benin

    Mahouton Norbert Hounkonnou

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  1. Mahouton Norbert Hounkonnou
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Correspondence to Mahouton Norbert Hounkonnou .

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Editors and Affiliations

  1. University of Yaoundé I, Yaoundé, Cameroon; African Institute for Mathematical Sciences, Limbe, Cameroon

    Mama Foupouagnigni

  2. Institute for Mathematics, University of Kassel, Kassel, Germany

    Wolfram Koepf

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Hounkonnou, M.N. (2020). \(( \mathcal {R}, p,q)\)-Rogers–Szegö and Hermite Polynomials, and Induced Deformed Quantum Algebras. In: Foupouagnigni, M., Koepf, W. (eds) Orthogonal Polynomials. AIMSVSW 2018. Tutorials, Schools, and Workshops in the Mathematical Sciences . Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-36744-2_16

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