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

  • Pascal Baseilhac
  • Xavier Martin
  • Luc VinetEmail author
  • Alexei Zhedanov
Article

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 \(\mathfrak U_q(sl(2))\).

Keywords

Askey–Wilson algebra Tridiagonalization Orthogonal polynomials 

Mathematics Subject Classification

81R50 81R10 81U15 39A70 33D50 39A13 

Notes

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.

References

  1. 1.
    Baseilhac, P., Genest, V.X., Vinet, L., Zhedanov, A.: An embedding of the Bannai–Ito algebra in \(\cal{U}(osp\mathit{(1, 2)}) \) and \(-1\) polynomials. Lett. Math. Phys. 108, 1623–1634 (2018)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Floreanini, R., Vinet, L.: Quantum algebras and q-special functions. Ann. Phys. 221, 53–70 (1993)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Floreanini, R., Vinet, L.: On the quantum group and quantum algebra approach to q-special functions. Lett. Math. Phys. 27, 179–190 (1993)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Genest, V.X., Ismail, M.E.H., Vinet, L., Zhedanov, A.: Tridiagonalization of the hypergeometric operator and the Racah–Wilson algebra. Proc. Am. Math. Soc. 144, 5217–5226 (2016)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Granovskii, Ya.I., Zhedanov, A.S.: Linear covariance algebra for \(SL_q(2)\). J. Phys. A 26, L357 (1993)Google Scholar
  6. 6.
    Grünbaum, F.A., Vinet, L., Zhedanov, A.: Algebraic Heun operator and band-time limiting. arXiv:1711.07862
  7. 7.
    Ismail, M.E.H., Koelink, E.: The J-matrix method. Adv. Appl. Math. 56, 379–395 (2011)CrossRefGoogle Scholar
  8. 8.
    Ismail, M.E.H., Koelink, E.: Spectral analysis of certain Schrödinger operators. SIGMA 8, 61–79 (2012)zbMATHGoogle Scholar
  9. 9.
    Ito, T., Terwilliger, P., Weng, C.-W.: The quantum algebra \(\mathfrak{U}_q(sl(2))\) and its equitable presentation. J. Algebra 298, 284–301 (2006). arXiv:math/0507477 MathSciNetCrossRefGoogle Scholar
  10. 10.
    Koekoek, R., Swarttouw, R.: The Askey-scheme of hypergeometric orthogonal polynomials and its \(q\)-analogue. arXiv:math.CA/9602214v1
  11. 11.
    Koelink, H.T., Van der Jeugt, J.: Convolutions for orthogonal polynomials from Lie and quantum algebra representations. SIAM J. Math. Anal. 29, 794–822 (1998). arXiv:q-alg/9607010
  12. 12.
    Koornwinder, T.H.: Representations of the twisted \(SU(2)\) quantum group and some q-hypergeometric orthogonal polynomials. Indag. Math. 51, 97–117 (1989)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Koornwinder, T.H.: Askey–Wilson polynomials as zonal spherical functions on the SU(2) quantum group. SIAM J. Math. Anal. 24, 795–813 (1993)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Koornwinder, T.H.: q-Special functions, a tutorial, representations of Lie groups and quantum groups. In: Baldoni, V.., Picardello, M.A. (eds.) Longman Scientific and Technical, pp. 46–128 (1994). arXiv:math/9403216
  15. 15.
    Masuda, T., Mimachi, Y., Nakagami, Y., Noumi, M., Ueno, K.: Representations of quantum groups and a q-analogue of orthogonal polynomials. C. R. Acad. Sci. Paris I 307, 559–564 (1988)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Masuda, T., Mimachi, K., Nakagami, Y., Noumi, M., Ueno, K.: Representations of the quantum group \(SU_q(2)\) and the little q-Jacobi polynomials. J. Funct. Anal. 99, 357–386 (1991)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Noumi, M., Mimachi, K.: Quantum \(2\)-spheres and big \(q\)-Jacobi polynomials. Commun. Math. Phys. 128, 521–531 (1990)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Sklyanin, E.K.: Some algebraic structures connected with the Yang–Baxter equation. Representations of quantum algebras. Funct. Anal. Appl. 17, 273–284 (1983)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Terwilliger, P.: The universal Askey–Wilson alebra and the equitable presentation of \(\mathfrak{U}_q(sl(2))\). SIGMA 7, 099 (2011). arXiv:1107.3544 zbMATHGoogle Scholar
  20. 20.
    Terwilliger, P.: The Lusztig automorphism of \(\mathfrak{U}_q(sl(2))\) from the equitable point of view. J. Algebra Appl. 16, 1750235 (2017). arXiv:1509.08956 MathSciNetCrossRefGoogle Scholar
  21. 21.
    Tsujimoto, S., Vinet, L., Zhedanov, A.: Tridiagonal representations of the q-oscillator algebra and Askey/Wilson polynomials. J. Phys. A 50, 235202 (2017)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Vaksman, L.L., Soibelman, Ya.S.: Function algebra on the quantum group \(SU(2)\). Funk. Anal. Priloz. 22, 1–14 (1988)Google Scholar
  23. 23.
    Zhedanov, A.S.: Hidden symmetry of Askey–Wilson polynomials. Teoret. Mat. Fiz. 89, 190–204 (1991)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Institut Denis-Poisson CNRS/UMR 7013 - Université de Tours - Université d’OrléansToursFrance
  2. 2.Centre de Recherches MathématiquesUniversité de MontréalMontrealCanada
  3. 3.Department of Mathematics, Information SchoolRenmin University of ChinaBeijingChina

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