Skip to main content
SpringerLink
Log in

Constructing Bispectral Orthogonal Polynomials from the Classical Discrete Families of Charlier, Meixner and Krawtchouk

  • Published:
Constructive Approximation Aims and scope

Abstract

Given a sequence of polynomials \((p_n)_n\), an algebra of operators \({\mathcal A}\) acting in the linear space of polynomials, and an operator \(D_p\in {\mathcal A}\) with \(D_p(p_n)=np_n\), we form a new sequence of polynomials \((q_n)_n\) by considering a linear combination of \(m+1\) consecutive \(p_n\): \(q_n=p_n+\sum _{j=1}^m\beta _{n,j}p_{n-j}\). Using the concept of \(\mathcal {D}\)-operator, we determine the structure of the sequences \(\beta _{n,j}, j=1,\ldots ,m,\) so that the polynomials \((q_n)_n\) are eigenfunctions of an operator in the algebra \({\mathcal A}\). As an application, from the classical discrete families of Charlier, Meixner, and Krawtchouk, we construct orthogonal polynomials \((q_n)_n\) which are also eigenfunctions of higher-order difference operators.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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. Álvarez-Nodarse, R., Durán, A.J.: Using \({\cal D}\)-operators to construct orthogonal polynomials satisfying higher order \(q\)-difference equations (submitted). arXiv:1309.3296

  2. Bavinck, H., van Haeringen, H.: Difference equations for generalizations of Meixner polynomials. J. Math. Anal. Appl. 184, 453–463 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  3. Bavinck, H., Koekoek, R.: On a difference equation for generalizations of Charlier polynomials. J. Approx. Theory 81, 195–206 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  4. Chihara, T.: An Introduction to Orthogonal Polynomials, Mathematics and its Applications, vol. 13. Gordon and Breach Science Publishers, New York (1978)

    Google Scholar 

  5. Christoffel, E.B.: Über die Gaussische Quadratur und eine Verallgemeinerung derselben. J. Reine Angew. Math. 55, 61–82 (1858)

    Article  MATH  Google Scholar 

  6. Duistermaat, J.J., Grünbaum, F.A.: Differential equations in the spectral parameter. Commun. Math. Phys. 103, 177–240 (1986)

    Article  MATH  Google Scholar 

  7. Durán, A.J.: The Stieltjes moments problem for rapidly decreasing functions. Proc. Am. Math. Soc. 107, 731–741 (1989)

    Article  MATH  Google Scholar 

  8. Durán, A.J.: Orthogonal polynomials satisfying higher order difference equations. Constr. Approx. 36, 459–486 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  9. Durán, A.J.: Using \({\cal D}\)-operators to construct orthogonal polynomials satisfying higher order difference or differential equations. J. Approx. Theory 174, 10–53 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  10. Durán, A.J., de la Iglesia, M.D.: Constructing Krall-Hahn orthogonal polynomials (submitted). arXiv:1407.7569

  11. Geronimus, YaL: On the polynomials orthogonal with respect to a given number sequence. Zap. Mat. Otdel. Khar’kov Univers. i Nil Mat. i Mehan. 17, 3–18 (1940)

    MathSciNet  Google Scholar 

  12. Geronimus, YaL: On the polynomials orthogonal with respect to a given number sequence. and a theorem by W. Hahn, lzv. Akad. Nauk SSSR 4, 215–228 (1940)

    MathSciNet  Google Scholar 

  13. Grünbaum, F.A., Haine, L.: Orthogonal polynomials satisfying differential equations: the role of the Darboux transformation. In: Levi, D., Vinet, L., Winternitz, P. (eds.) Symmetries and Integrability of Differential Equations, CRM Proceedings of Lecture Notes, vol. 9, pp. 143–154. Amer. Math. Soc. Providence, RI (1996)

  14. Grünbaum, F.A., Haine, L.: Bispectral Darboux transformations: an extension of the Krall polynomials. Int. Math. Res. Not. 8, 359–392 (1997)

    Article  Google Scholar 

  15. Grünbaum, F.A., Haine, L., Horozov, E.: Some functions that generalize the Krall–Laguerre polynomials. J. Comput. Appl. Math. 106, 271–297 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  16. Grünbaum, F.A., Yakimov, M.: Discrete bispectral Darboux transformations from Jacobi operators. Pac. J. Math. 204, 395–431 (2002)

    Article  MATH  Google Scholar 

  17. Haine, L., Iliev, P.: Askey–Wilson type functions with bound state. Ramanujan J. 11, 285–329 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  18. Iliev, P.: Krall–Jacobi commutative algebras of partial differential operators. J. Math. Pures Appl. 96, 446–461 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  19. Iliev, P.: Krall–Laguerre commutative algebras of ordinary differential operators. Ann. Mat. Pur. Appl. 192, 203–224 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  20. Koekoek, J., Koekoek, R.: On a differential equation for Koornwinder’s generalized Laguerre polynomials. Proc. Am. Math. Soc. 112, 1045–1054 (1991)

    MATH  MathSciNet  Google Scholar 

  21. Koekoek, R.: Differential equations for symmetric generalized ultraspherical polynomials. Trans. Am. Math. Soc. 345, 47–72 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  22. Koekoek, J., Koekoek, R.: Differential equations for generalized Jacobi polynomials. J. Comput. Appl. Math. 126, 1–31 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  23. Koekoek, R., Lesky, P.A., Swarttouw, R.F.: Hypergeometric orthogonal polynomials and their \(q\)-analogues. With a foreword by Tom H. Koornwinder. Springer Monographs in Mathematics. Springer-Verlag, Berlin (2010)

  24. Krall, H.L.: On orthogonal polynomials satisfying a certain fourth order differential equation. The Pennsylvania State College Studies, No. 6 (1940)

  25. Littlejohn, L.L.: The Krall polynomials: a new class of orthogonal polynomials. Quaest. Math. 5, 255–265 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  26. Littlejohn, L.L.: An application of a new theorem on orthogonal polynomials and differential equations. Quaest. Math. 10, 49–61 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  27. Macdonald, I.G.: Symmetric Functions and Hall Polynomials, 2nd edn. Oxford University Press, New York (1995)

    MATH  Google Scholar 

  28. Nikiforov, A.F., Suslov, S.K., Uvarov, V.B.: Classical orthogonal polynomials of a discrete variable. Springer, Berlin (1991)

    Book  MATH  Google Scholar 

  29. Szegö, G.: Orthogonal Polynomials, vol. XXIII, 4th edn. American Mathematical Society, Colloquium Publications, Providence, R.I. (1975)

    MATH  Google Scholar 

  30. Zhedanov, A.: Rational spectral transformations and orthogonal polynomials. J. Comput. Appl. Math. 85, 67–86 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  31. Zhedanov, A.: A method of constructing Krall’s polynomials. J. Comput. Appl. Math. 107, 1–20 (1999)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Departamento de Análisis Matemático, Universidad de Sevilla, APDO, P. O. Box 1160, 41080 , Seville, Spain

    Antonio J. Durán & Manuel D. de la Iglesia

Authors
  1. Antonio J. Durán
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Manuel D. de la Iglesia
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Manuel D. de la Iglesia.

Additional information

Communicated by Tom H. Koornwinder.

Partially supported by MTM2012-36732-C03-03 (Ministerio de Economía y Competitividad), FQM-262, FQM-4643, FQM-7276 (Junta de Andalucía) and Feder Funds (European Union).

The authors would like to thank an anonymous referee for his/her comments and suggestions.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Durán, A.J., de la Iglesia, M.D. Constructing Bispectral Orthogonal Polynomials from the Classical Discrete Families of Charlier, Meixner and Krawtchouk. Constr Approx 41, 49–91 (2015). https://doi.org/10.1007/s00365-014-9251-5

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00365-014-9251-5

Keywords

Mathematics Subject Classification

Search

Navigation