Skip to main content
SpringerLink
Log in

Characterization of Laguerre polynomials as orthogonal polynomials connected by the Laguerre degree raising shift operator

  • Published:
The Ramanujan Journal Aims and scope Submit manuscript

Abstract

We introduce the notion of \(\mathcal {R}_{\mu }\)-classical orthogonal polynomials, where \(\mathcal {R}_{\mu }\) is the degree raising shift operator for the sequence of Laguerre polynomials of parameter \(\mu \). Then we show that the Laguerre polynomials \(L^{(\mu )}_n(x), \ \mu \ne -m, \ m\ge 0\), are the only \(\mathcal {R}_{\mu }\)-classical orthogonal polynomials.

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. Abdelkarim, F., Maroni, P.: The \(D_w\)-classical orthogonal polynomials. Result. Math. 32, 1–28 (1997)

    Article  MATH  Google Scholar 

  2. Aloui, B., Marcellán, F., Sfaxi, R.: Classical orthogonal polynomials with respect to a lowering operator generalizing the Laguerre operator. Integral Transforms Spec. Funct. 24(8), 636–648 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  3. Al-Salam, W.A.: Characterization theorems for orthogonal polynomials. In: Nevai, P. (ed.) Orthogonal Polynomials: Theory and Practice. NATO ASI Series C, vol. 294, pp. 1–24. Kluwer, Dordrecht (1990)

    Google Scholar 

  4. Askey, R.: Divided difference operators and classical orthogonal polynomials. Rocky Mt. J. Math. 19, 33–37 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bochner, S.: Über Sturm–Liouvillesche Polynomsysteme. Z. Math. 29, 6–730 (1929)

    MATH  Google Scholar 

  6. Chihara, T.S.: An Introduction to Orthogonal Polynomials. Gordon and Breach, New York (1978)

    MATH  Google Scholar 

  7. Hahn, W.: Über die Jacobischen polynome und zwei verwandte polynomklassen. Math. Zeit. 39, 634–638 (1935)

    Article  MATH  Google Scholar 

  8. Hahn, W.: Über Orthogonalpolynome, die \(q\)-Differenzengleichungen genügen. Math. Nach. 2, 4–34 (1949)

    Article  MATH  Google Scholar 

  9. Khériji, L., Maroni, P.: The \(H_q\)-classical orthogonal polynomials. Acta. Appl. Math. 71, 49–115 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  10. Koekoek, R., Lesky, P.A., Swarttouw, R.E.: Hypergeometric Orthogonal Polynomials and Their \(q\)-Analogues. Springer, Berlin (2010)

    Book  MATH  Google Scholar 

  11. Lesky, P.: Über polynomsysteme, die Sturm-Liouvilleschen differenzengleichungen genügen. Math. Zeit. 78, 439–445 (1962)

    Article  MATH  Google Scholar 

  12. Maroni, P.: Une théorie algébrique des polynômes orthogonaux Applications aux polynômes orthogonaux semi-classiques. In: Brezinski, C., et al. (ed.) Orthogonal Polynomials and their Applications. IMACS Ann. Comput. Appl. Math., vol. 9, pp. 95–130. Baltzer, Basel (1991)

  13. Maroni, P.: Variations autour des polynmes orthogonaux classiques. C. R. Acad. Sci. Paris Sr. I Math. 313, 209–212 (1991)

    MathSciNet  MATH  Google Scholar 

  14. Maroni, P.: Fonctions Eulériennes, Polynômes Orthogonaux Classiques. Tech. Ing. Traité Général. (Sci. Fondam.) A154, 1–30 (1994)

    Google Scholar 

  15. Nikoforov, A., Ouvarov, V.: Special Functions of Mathematical Physics. Birkhüser, Basel (1988)

    Book  Google Scholar 

  16. Sonine, N.J.: On the approximate computation of definite integrals and on the entire functions occurring there. Warsch. Univ. Izv. 18, 1–76 (1887)

    MATH  Google Scholar 

Download references

Acknowledgements

We wish to thank the referee for a careful reading, valuable comments for the original draft, and for mentioning Ref. [16].

Author information

Authors and Affiliations

  1. Département de Mathématiques, Facult’e des Sciences de Gabès, Cité Erriadh, 6072, Gabès, Tunisia

    Baghdadi Aloui

Authors
  1. Baghdadi Aloui
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Baghdadi Aloui.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Aloui, B. Characterization of Laguerre polynomials as orthogonal polynomials connected by the Laguerre degree raising shift operator. Ramanujan J 45, 475–481 (2018). https://doi.org/10.1007/s11139-017-9901-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11139-017-9901-x

Keywords

Mathematics Subject Classification

Search

Navigation