Skip to main content
SpringerLink
Log in

Connection coefficients for orthogonal polynomials: symbolic computations, verifications and demonstrations in the Mathematica language

  • Original Paper
  • Published:
Numerical Algorithms Aims and scope Submit manuscript

Abstract

We deal with the problem of obtaining closed formulas for the connection coefficients between orthogonal polynomial sequences and, also, the canonical sequence, using a recursive methodology based on symbolic computations, verifications and demonstrations in the Mathematica ® language. We present the corresponding software that is available in Netlib and, with it, we derive new formulas for the connection coefficients for some semi-classical of class 1 families.

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. Alaya, J., Maroni, P.: Symmetric Laguerre–Hahn forms of class s = 1. Integral Transforms Spec. Funct. 4(4), 301–320 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  2. Area, I., Godoy, E., Ronveaux, A., Zarzo, A.: Minimal recurrence relations for connection coefficients between classical orthogonal polynomials: discrete case. J. Comput. Appl. Math. 89, 309–325 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  3. Area, I., Godoy, E., Ronveaux, A., Zarzo, A.: Inversion problems in the q-Hahn Tableau. J. Symb. Comput. 28, 767–776 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  4. Atia, M.J.: An example of non-symmetric semi-classical form of class s = 1; generalization of a case of Jacobi sequence. Int. J. Math. Math. Sci. 24(10), 676–689 (2000)

    Article  MathSciNet  Google Scholar 

  5. Chagarra, H., Koepf, W.: On linearization and connection coefficients for generalized Hermite polynomials. J. Comput. Appl. Math. 236(1), 65–73 (2011)

    Article  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  7. Godoy, E., Area, I., Ronveaux, A., Zarzo, A.: Minimal recurrence relations for connection coefficients between classical orthogonal polynomials: continuous case. J. Comput. Appl. Math. 84(2), 257–275 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  8. Lewanowicz, S.: Quick construction of recurrence relations for the Jacobi coefficients. J. Comput. Appl. Math. 43, 355–372 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  9. Lewanowicz, S.: Recurrence relations for the connection coefficients of orthogonal polynomials of a discrete variable. J. Comput. Appl. Math. 76, 213–229 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  10. Koepf, W., Swarttouw, R.: CAOP: Computer Algebra and Orthogonal Polynomials. http://pool-serv1.mathematik.uni-kassel.de/CAOP (2007)

  11. Koepf, W., Schmersau, D.: Representations of orthogonal polynomials. J. Comput. Appl. Math. 90, 57–94 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  12. Maroni, P.: Variations around classical orthogonal polynomials. Connected problems. J. Comput. Appl. Math. 48, 133–155 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  13. Maroni, P.: Fonctions Eulériennes. Polynômes Orthogonaux Classiques. Techniques de l’Ingénieur, traité Généralités. Sciences Fondamentales (1994)

  14. Maroni, P., da Rocha, Z.: Connection coefficients for orthogonal polynomials and the canonical sequence. Preprints CMUP, Centro de Matemática da Universidade do Porto, vol. 28, pp. 1–18 (2007). http://cmup.fc.up.pt/cmup/v2/frames/publications.htm

  15. Maroni, P., da Rocha, Z.: Connection coefficients between orthogonal polynomials and the canonical sequence: an approach based on symbolic computation. Numer. Algor. 47(3), 291–314 (2008)

    Article  MATH  Google Scholar 

  16. Ronveaux, A., Zarzo, A., Godoy, E.: Recurrence relation for connection coefficients between two families of orthogonal polynomials. J. Comput. Appl. Math. 62, 67–73 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  17. Ronveaux, A., Rebillard, L.: Expansion of multivariate polynomials in products of univariate orthogonal polynomials: discrete case. J. Comput. Appl. Math. 133, 567–578 (2001)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie - CNRS, Boite courrier 187, Paris cedex 05, 75252, France

    Pascal Maroni

  2. Departamento de Matemática, Faculdade de Ciências da Universidade do Porto - CMUP, Rua do Campo Alegre n.687, Porto, 4169-007, Portugal

    Zélia da Rocha

Authors
  1. Pascal Maroni
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Zélia da Rocha
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zélia da Rocha.

Additional information

Research funded by the European Regional Development Fund through the programe COMPETE and by the Portuguese Government through the FCT Fundação para a Ciência e a Tecnologia under the project PEst-C/MAT/UI0144/2011.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Maroni, P., da Rocha, Z. Connection coefficients for orthogonal polynomials: symbolic computations, verifications and demonstrations in the Mathematica language. Numer Algor 63, 507–520 (2013). https://doi.org/10.1007/s11075-012-9634-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11075-012-9634-2

Keywords

Mathematics Subject Classifications (2010)

Search

Navigation