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


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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  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].


  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)

    Article  MathSciNet  Google Scholar 

  2. Floreanini, R., Vinet, L.: Quantum algebras and q-special functions. Ann. Phys. 221, 53–70 (1993)

    Article  MathSciNet  Google Scholar 

  3. Floreanini, R., Vinet, L.: On the quantum group and quantum algebra approach to q-special functions. Lett. Math. Phys. 27, 179–190 (1993)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  5. Granovskii, Ya.I., Zhedanov, A.S.: Linear covariance algebra for \(SL_q(2)\). J. Phys. A 26, L357 (1993)

  6. Grünbaum, F.A., Vinet, L., Zhedanov, A.: Algebraic Heun operator and band-time limiting. arXiv:1711.07862

  7. Ismail, M.E.H., Koelink, E.: The J-matrix method. Adv. Appl. Math. 56, 379–395 (2011)

    Article  Google Scholar 

  8. Ismail, M.E.H., Koelink, E.: Spectral analysis of certain Schrödinger operators. SIGMA 8, 61–79 (2012)

    MATH  Google Scholar 

  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

    Article  MathSciNet  Google Scholar 

  10. Koekoek, R., Swarttouw, R.: The Askey-scheme of hypergeometric orthogonal polynomials and its \(q\)-analogue. arXiv:math.CA/9602214v1

  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. Koornwinder, T.H.: Representations of the twisted \(SU(2)\) quantum group and some q-hypergeometric orthogonal polynomials. Indag. Math. 51, 97–117 (1989)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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. 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)

    MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  17. Noumi, M., Mimachi, K.: Quantum \(2\)-spheres and big \(q\)-Jacobi polynomials. Commun. Math. Phys. 128, 521–531 (1990)

    Article  MathSciNet  Google Scholar 

  18. Sklyanin, E.K.: Some algebraic structures connected with the Yang–Baxter equation. Representations of quantum algebras. Funct. Anal. Appl. 17, 273–284 (1983)

    Article  MathSciNet  Google Scholar 

  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

    MATH  Google Scholar 

  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

    Article  MathSciNet  Google Scholar 

  21. Tsujimoto, S., Vinet, L., Zhedanov, A.: Tridiagonal representations of the q-oscillator algebra and Askey/Wilson polynomials. J. Phys. A 50, 235202 (2017)

    Article  MathSciNet  Google Scholar 

  22. Vaksman, L.L., Soibelman, Ya.S.: Function algebra on the quantum group \(SU(2)\). Funk. Anal. Priloz. 22, 1–14 (1988)

  23. Zhedanov, A.S.: Hidden symmetry of Askey–Wilson polynomials. Teoret. Mat. Fiz. 89, 190–204 (1991)

    MathSciNet  MATH  Google Scholar 

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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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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).

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  • Askey–Wilson algebra
  • Tridiagonalization
  • Orthogonal polynomials

Mathematics Subject Classification

  • 81R50
  • 81R10
  • 81U15
  • 39A70
  • 33D50
  • 39A13