Skip to main content
SpringerLink
Log in

On an orthogonal polynomial sequence and its recurrence coefficients

  • Published:
Letters in Mathematical Physics Aims and scope Submit manuscript

Abstract

We provide a simple method to recognize a classical orthogonal polynomial sequence on a q-quadratic lattice defined only by the three-term recurrence relation. It is pointed out that this can be extended to all orthogonal polynomials in the q-Askey scheme.

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

Data availability

No data were used for the research described in the note.

References

  1. Alhaidari, A.D.: Open problem in orthogonal polynomials. arXiv:math-ph/1709.06081v1

  2. Álvarez-Nodarse, R., Castillo, K., Mbouna, D., Petronilho, J.: On classical orthogonal polynomials related to Hahn’s operator. Integral Transforms Spec. Funct. 31(6), 487–505 (2020)

    Article  MathSciNet  Google Scholar 

  3. Álvarez-Nodarse, R., Castillo, K., Mbouna, D., Petronilho, J.: On discrete coherent pairs of measures. J. Differ. Equ. Appl. 25 28(7), 853–868 (2022)

    Article  MathSciNet  Google Scholar 

  4. Castillo, K., Mbouna, D.: Proof of two conjectures on Askey–Wilson polynomials. Proc. Am. Math. Soc. 151(4), 1655–1661 (2023)

    MathSciNet  Google Scholar 

  5. Castillo, K., Mbouna, D., Petronilho, J.: A characterization of continuous q-Jacobi, Chebyshev of the first kind and Al-Salam Chihara polynomials. J. Math. Anal. Appl. 514, 126358 (2022)

    Article  MathSciNet  Google Scholar 

  6. Castillo, K., Mbouna, D., Petronilho, J.: On the functional equation for classical orthogonal polynomials on lattices. J. Math. Anal. Appl. 515, 126390 (2022)

    Article  MathSciNet  Google Scholar 

  7. Foupouagnigni, M., Kenfack-Nangho, M., Mboutngam, S.: Characterization theorem of classical orthogonal polynomials on nonuniform lattices: the functional approach. Integral Transforms Spec. Funct. 22, 739–758 (2011)

    Article  MathSciNet  Google Scholar 

  8. Ismail, M.E.H.: Classical and quantum orthogonal polynomials in one variable. In: Van Assche, W. (ed.) Encyclopedia of Mathematics and its Applications, vol. 98. With a foreword by Richard A. Askey. Reprint of the 2005 original. . Cambridge University Press, Cambridge (2009)

  9. Koekoek, R., Lesky, P.A., Swarttouw, R.F.: Hypergeometric Orthogonal Polynomials and Their \(q\)-Analogues. Springer Monographs in Mathematics, Springer, Berlin (2010)

    Book  Google Scholar 

  10. Koornwinder, T.H.: Charting the \(q\)-Askey scheme, hypergeometry, integrability and Lie theory. Contemp. Math. 780, 79–94 (2022)

    Article  Google Scholar 

  11. Tcheutia, D.D.: Recurrence equations and their classical orthogonal polynomial solutions on a quadratic or a q-quadratic lattice. J. Differ. Equ. Appl. 25(7), 969–993 (2019)

    Article  MathSciNet  Google Scholar 

  12. Van Assche, W.: Solution of an open problem about two families of orthogonal polynomials. SIGMA 15, 6 (2019)

    MathSciNet  Google Scholar 

  13. Verde-Star, L.: A unified construction of all the hypergeometric and basic hypergeometric families of orthogonal polynomial sequences. Linear Algebra Appl. 627, 242–274 (2021)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The author is grateful to the anonymous reviewer whose valuable comments and remarks have helped to improve considerably this manuscript. The author was partially supported by CMUP, member of LASI, which is financed by national funds through FCT—Fundacão para a Ciência e a Tecnologia, I.P., under the projects with reference UIDB/00144/2020 and UIDP/00144/2020. This work was finalized during the tenure of an ERCIM ‘Alain Bensoussan’ Fellowship Programme. This research has been also funded under Grant QUALIFICA (PROGRAMA: AYUDAS A ACCIONES COMPLEMENTARIAS DE I+D+i) by Junta de Andalucía Grant Number QUAL21 005 USE.

Author information

Authors and Affiliations

  1. Edificio Celestino Mutis - CITUS II, Institute of Mathematics of the University of Seville (IMUS), Av. de la Reina Mercedes, s/n, 41012, Seville, Spain

    D. Mbouna

  2. Institute of Mathematics, University of Warsaw, ul. Banacha 2, 02-097, Warsaw, Poland

    D. Mbouna

Authors
  1. D. Mbouna
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to D. Mbouna.

Ethics declarations

Conflict of interest

I confirm that there is not any conflict of interest for this note.

Additional information

Publisher's Note

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

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Mbouna, D. On an orthogonal polynomial sequence and its recurrence coefficients. Lett Math Phys 114, 54 (2024). https://doi.org/10.1007/s11005-024-01801-3

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11005-024-01801-3

Keywords

Mathematics Subject Classification

Search

Navigation