Skip to main content
SpringerLink
Log in

Symbolic Approach to 2-Orthogonal Polynomial Solutions of a Third Order Differential Equation

  • Published:
Mathematics in Computer Science Aims and scope Submit manuscript

Abstract

In a recent work, a generic differential operator on the vectorial space of polynomial functions was presented and applied in the study of differential relations fulfilled by polynomial sequences either orthogonal or 2-orthogonal. Considering a third order differential operator that does not increase the degree of polynomials, we search for polynomial eigenfunctions with the help of symbolic computations, assuming that those polynomials constitute a 2-orthogonal polynomial sequence. Two examples are extensively described.

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. Bochner, S.: Über Sturm-Liouvillesche Polynomsysteme. Math. Zeit. 29, 730–736 (1929)

    Article  MATH  Google Scholar 

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

    MATH  Google Scholar 

  3. Coussement, E., Van Assche, W.: Some classical multiple orthogonal polynomials. J. Comput. Appl. Math. 127(1–2), 317–347 (2001)

    MathSciNet  MATH  Google Scholar 

  4. Coussement, J., Van Assche, W.: Differential equations for multiple orthogonal polynomials with respect to classical weights: raising and lowering operators. J. Phys. A Math. Gen. 39(13), 3311–3318 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  5. Douak, K.: On 2-orthogonal polynomials of Laguerre type. Int. J. Math. Math. Sci. 22(1), 29–48 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  6. Douak, K.: The relation of the d-orthogonal polynomials to the Appell polynomials. J. Comput. Appl. Math. 70(2), 279–295 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  7. Douak, K., Maroni, P.: Une Caractérisation des polynômes \(d\)-orthogonaux “classiques.” J. Approx. Theory 82, 177–204 (1995)

  8. Douak, K., Maroni, P.: Les polynômes orthogonaux “classiques” de dimension deux. Analysis 12, 71–107 (1992)

  9. Kwon, K.H., Lee, D.W., Littlejohn, L.L.: Differential equations having orthogonal polynomial solutions. J. Comput. Appl. Math. 80(1), 1–16 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  10. Srivastava, H.M., Ben Cheikh, Y.: Orthogonality of some polynomial sets via quasi-monomiality. Appl. Math. Comput. 141, 415–425 (2003)

    MathSciNet  MATH  Google Scholar 

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

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

  13. Maroni, P.: L’orthogonalité et les récurrences de polynômes d’ordre supérieur à deux, Ann. Fac. Sci. Toulouse, Math. 10(5) , No. 1, 105–139 (1989)

  14. Maroni, P.: Two-dimensional orthogonal polynomials, their associated sets and co-recursive sets. Numer. Algorithms 3, 299–312 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  15. Maroni, P.: New results about orthogonality preserving maps. J. Korean Math. Soc. 42(2), 243–254 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  16. Maroni, P., Mesquita, T.A.: Appell polynomial sequences with respect to some differential operators. Period. Math. Hungar. 72(2), 200–217 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  17. Mesquita, T. Augusta, Maroni, P.: Around operators not increasing the degree of polynomials. Integral Transforms Spec. Funct. 30(5), 383–399 (2019)

  18. Mesquita, T.A.: On a 2-orthogonal polynomial sequence via quadratic decomposition. Math. Comput. Sci. 15, 15–31 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  19. Pincherle, S.: Mémoire sur le calcul fonctionnel distributif. Math. Ann. 49, 325–382 (1897)

    Article  MathSciNet  MATH  Google Scholar 

  20. Iseghem, J.V.: Vector orthogonal relations. Vector QD-algorithm. J. Comput. Appl. Math. 19, 141–150 (1987)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The author T. A. Mesquita was partially supported by CMUP, which is financed by national funds through FCT-Fundação para a Ciência e a Tecnologia, I.P., under the project with reference UIDB/00144/2020. The author would like to thank the referee’s comments and suggestions.

Author information

Authors and Affiliations

  1. Escola Superior de Tecnologia e Gestão, Instituto Politécnico de Viana do Castelo, Rua Escola Industrial e Comercial de Nun’ Álvares, 4900-347, Viana do Castelo, Portugal

    Teresa Augusta Mesquita

  2. Centro de Matemática da Universidade do Porto, Rua do Campo Alegre, 687, 4169-007, Porto, Portugal

    Teresa Augusta Mesquita

Authors
  1. Teresa Augusta Mesquita
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Teresa Augusta Mesquita.

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

Mesquita, T.A. Symbolic Approach to 2-Orthogonal Polynomial Solutions of a Third Order Differential Equation. Math.Comput.Sci. 16, 6 (2022). https://doi.org/10.1007/s11786-022-00525-8

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11786-022-00525-8

Keywords

Mathematics Subject Classification

Search

Navigation