Skip to main content
SpringerLink
Log in

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

  • Published:
The Ramanujan Journal Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

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

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

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

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

    MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

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

    Article  Google Scholar 

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

    Book  Google Scholar 

  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. 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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

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

Author information

Authors and Affiliations

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

    Pascal Baseilhac

  2. Centre de Recherches Mathématiques, Université de Montréal, P.O. Box 6128, Centre-ville Station, Montreal, Québec, H3C 3J7, Canada

    Luc Vinet

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

    Alexei Zhedanov

Authors
  1. Pascal Baseilhac
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Luc Vinet
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Alexei Zhedanov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Alexei Zhedanov.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

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

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11139-018-0106-8

Keywords

Mathematics Subject Classification

Search

Navigation