The q-Heun operator of big q-Jacobi type and the q-Heun algebra

Abstract

The q-Heun operator of the big q-Jacobi type on the exponential grid is defined. This operator is the most general second-order q-difference operator that maps polynomials of degree n to polynomials of degree \(n+1\). It is tridiagonal in bases made out of either q-Pochhammer or big q-Jacobi polynomials and is bilinear in the operators of the q-Hahn algebra. The extension of this algebra that includes the q-Heun operator as generator is described. Biorthogonal Pastro polynomials are shown to satisfy a generalized eigenvalue problem or equivalently to be in the kernel of a special linear pencil made out of two q-Heun operators. The special case of the q-Heun operator associated to the little q-Jacobi polynomials is also treated.

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References

  1. 1.

    Baseilhac, P., Martin, X., Vinet, L., Zhedanov, A.: Little and big q-Jacobi polynomials and the Askey-Wilson algebra. arXiv:1806.02656

  2. 2.

    Genest, V., Ismail, M.E.H., Vinet, L., Zhedanov, A.: Tridiagonalization of the hypergeometric operator and the Racah-Wilson algebra. arXiv:1506.07803

  3. 3.

    Granovskii, Y.I., Zhedanov, A.: Exactly solvable problems and their quadratic algebras. DonFTI, Preprint (1989)

  4. 4.

    Granovskii, Ya A., Lutzenko, I.M., Zhedanov, A.: Mutual integrability, quadratic algebras, and dynamical symmetry. Ann. Phys. 217, 1–20 (1992)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Grünbaum, F.A., Vinet, L., Zhedanov, A.: Tridiagonalization and the Heun equation. J. Math. Phys. 58, 031703 (2017). arXiv:1602.04840

    MathSciNet  Article  Google Scholar 

  6. 6.

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

  7. 7.

    Hahn, W.: On linear geometric difference equations with accessory parameters. Funkcial. Ekvac. 14, 73–78 (1971)

    MathSciNet  MATH  Google Scholar 

  8. 8.

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

    MATH  Google Scholar 

  9. 9.

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

    Article  Google Scholar 

  10. 10.

    Koekoek, R., Lesky, P.A., Swarttouw, R.F.: Hypergeometric Orthogonal Polynomials and Their q-analogues, 1st edn. Springer, New York (2010)

    Google Scholar 

  11. 11.

    Magnus, A., Ndayiragije, F., Ronveaux, A.: Heun differential equation satisfied by some classical biorthogonal rational functions. (to be published). https://perso.uclouvain.be/alphonse.magnus/num3/biorthclassCanterb2017.pdf

  12. 12.

    Nomura, K., Terwilliger, P.: Linear transformations that are tridiagonal with respect to both eigenbases of a Leonard pair. Linear Algebr. Appl. 420, 198–207 (2007). arXiv:math/0605316

    MathSciNet  Article  Google Scholar 

  13. 13.

    Pastro, P.I.: Orthogonal polynomials and some q-beta integrals of Ramanujan. J. Math. Anal. Appl. 112, 517–540 (1985)

    MathSciNet  Article  Google Scholar 

  14. 14.

    Ruijsenaars, S.N.M.: Integrable \(BC_N\) analytic difference operators: hidden parameter symmetries and eigenfunctions. In: New Trends in Integrability and Partial Solvability, NATO Sci. Ser. II Math. Phys. Chem., vol. 132, pp. 217–261. Kluwer Acad. Publ., Dordrecht (2004)

  15. 15.

    Takemura, K.: On \(q\)-deformations of Heun equation. arxiv:1712.09564

  16. 16.

    Terwilliger, P.: Two linear transformations each tridiagonal with respect to an eigenbasis of the other. Linear Algebr Appl. 330, 149–203 (2001)

    MathSciNet  Article  Google Scholar 

  17. 17.

    van Diejen, J.F.: Integrability of difference CalogeroMoser systems. J. Math. Phys. 35, 2983–3004 (1994)

    MathSciNet  Article  Google Scholar 

  18. 18.

    Vinet, L., Zhedanov, A.: Spectral transformations of the laurent biorthogonal polynomials, II. Pastro Polynomials Can. Math. Bull 44(3), 337–345 (2001)

    MathSciNet  Article  Google Scholar 

  19. 19.

    Vinet, L., Zhedanov, A.: The Heun operator of Hahn type. arXiv:1808.00153

  20. 20.

    Zhedanov, A.S.: Hidden symmetry of Askey-Wilson polynomials. Theor. Math. Phys. 89, 1146–1157 (1991)

    MathSciNet  Article  Google Scholar 

  21. 21.

    Zhedanov, A.: The classical Laurent biorthogonal polynomials. J. Comput. Appl. Math. 98, 121–147 (1998)

    MathSciNet  Article  Google Scholar 

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Acknowledgements

PB and AZ would wish to acknowledge the hospitality of the CRM during the course of this work.

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Correspondence to Alexei Zhedanov.

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PB is supported by the CNRS. The research of LV is funded in part by a discovery Grant from the Natural Sciences and Engineering Research Council (NSERC) of Canada. Work of AZ is supported by the National Science Foundation of China (Grant No. 11771015).

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Baseilhac, P., Vinet, L. & Zhedanov, A. The q-Heun operator of big q-Jacobi type and the q-Heun algebra. Ramanujan J 52, 367–380 (2020). https://doi.org/10.1007/s11139-018-0106-8

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Keywords

  • Heun operator
  • q-orthogonal polynomials
  • Askey–Wilson algebra

Mathematics Subject Classification

  • 34D45
  • 39A13