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Little and big q-Jacobi polynomials and the Askey–Wilson algebra

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Abstract

The little and big q-Jacobi polynomials are shown to arise as basis vectors for representations of the Askey–Wilson algebra. The operators that these polynomials respectively diagonalize are identified within the Askey–Wilson algebra generated by twisted primitive elements of \(\mathfrak U_q(sl(2))\). The little q-Jacobi operator and a tridiagonalization of it are shown to realize the equitable embedding of the Askey–Wilson algebra into \(\mathfrak U_q(sl(2))\).

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Notes

  1. We choose \(c_0=\overline{c}_1=\mu _0=0\) in (2.7).

  2. We choose \(\mu _0=c_0=0\) in (2.7).

  3. Here \(q^2\) is q of [10].

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Acknowledgements

We thank Paul Terwilliger for comments. L.V. would like to express his gratitude for the hospitality extended to him by the Institut Denis-Poisson of the Université François-Rabelais de Tours as Chercheur Invité where most of this research was carried out.

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Authors and Affiliations

  1. Institut Denis-Poisson CNRS/UMR 7013 - Université de Tours - Université d’Orléans, Parc de Grammont, 37200, Tours, France

    Pascal Baseilhac & Xavier Martin

  2. Centre de Recherches Mathématiques, Université de Montréal, CNRS/UMI 3457 Centre-Ville Station, P.O. Box 6128, Montreal, QC, H3C 3J7, Canada

    Luc Vinet & Alexei Zhedanov

  3. Department of Mathematics, Information School, Renmin University of China, Beijing, 100872, China

    Alexei Zhedanov

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  1. Pascal Baseilhac
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  2. Xavier Martin
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  3. Luc Vinet
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  4. Alexei Zhedanov
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Correspondence to Luc Vinet.

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The research of L.V. is funded in part by a discovery Grant from the Natural Sciences and Engineering Research Council (NSERC) of Canada. P.B. is supported by C.N.R.S. Work of A.Z. is supported by the National Science Foundation of China (Grant No. 11771015).

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Baseilhac, P., Martin, X., Vinet, L. et al. Little and big q-Jacobi polynomials and the Askey–Wilson algebra. Ramanujan J 51, 629–648 (2020). https://doi.org/10.1007/s11139-018-0080-1

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  • DOI: https://doi.org/10.1007/s11139-018-0080-1

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