Acta Applicandae Mathematicae

, Volume 153, Issue 1, pp 81–100 | Cite as

Matrix Pearson Equations Satisfied by Koornwinder Weights in Two Variables

  • Francisco Marcellán
  • Misael Marriaga
  • Teresa E. Pérez
  • Miguel A. Piñar


We consider Koornwinder’s method for constructing orthogonal polynomials in two variables from orthogonal polynomials in one variable. If semiclassical orthogonal polynomials in one variable are used, then Koornwinder’s construction generates semiclassical orthogonal polynomials in two variables. We consider two methods for deducing matrix Pearson equations for weight functions associated with these polynomials, and consequently, we deduce the second order linear partial differential operators for classical Koornwinder polynomials.


Orthogonal polynomials in two variables Koornwinder weights Partial differential equations Matrix Pearson equations 

Mathematics Subject Classification (2010)

42C05 33C50 



The authors thank to the anonymous referees for their careful revision of the manuscript. Their comments and suggestions contributed to improve the presentation.


  1. 1.
    Álvarez de Morales, M., Fernández, L., Pérez, T.E., Piñar, M.A.: Semiclassical orthogonal polynomials in two variables. J. Comput. Appl. Math. 207, 323–330 (2007) MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Álvarez de Morales, M., Fernández, L., Pérez, T.E., Piñar, M.A.: A semiclassical perspective on multivariate orthogonal polynomials. J. Comput. Appl. Math. 214, 447–456 (2008) MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Bochner, S.: Über Sturm-Liouvillesche polynomsysteme. Math. Z. 29, 730–736 (1929) MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Chihara, T.S.: An Introduction to Orthogonal Polynomials. Mathematics and Its Applications, vol. 13. Gordon & Breach, New York (1978) MATHGoogle Scholar
  5. 5.
    Dunkl, C.F., Xu, Y.: Orthogonal Polynomials of Several Variables, 2nd edn. Encyclopedia of Mathematics and Its Applications, vol. 155. Cambridge University Press, Cambridge (2014) CrossRefMATHGoogle Scholar
  6. 6.
    Fernández, L., Pérez, T.E., Piñar, M.A.: Classical orthogonal polynomials in two variables: a matrix approach. Numer. Algorithms 39, 131–142 (2005) MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Fernández, L., Pérez, T.E., Piñar, M.A.: On Koornwinder classical orthogonal polynomials in two variables. J. Comput. Appl. Math. 236, 3817–3826 (2012) MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Hendriksen, E., van Rossum, H.: Semi-classical orthogonal polynomials. In: Brezinski, C., Draux, A., Magnus, A.P., Maroni, P., Ronveaux, A. (eds.) Polynômes Orthogonaux et Applications, Proceedings Bar-le-Duc 1984. Lecture Notes in Math., vol. 1171, pp. 354–361. Springer, Berlin (1985) Google Scholar
  9. 9.
    Horn, R.A., Johnson, C.R.: Matrix Analysis, 2nd edn. Cambridge University Press, Cambridge (2012) CrossRefGoogle Scholar
  10. 10.
    Koornwinder, T.H.: Two-variable analogues of the classical orthogonal polynomials. In: Askey, R. (ed.) Theory and Application of Special Functions, pp. 435–495. Academic Press, New York (1975) CrossRefGoogle Scholar
  11. 11.
    Krall, H.L., Sheffer, I.M.: Orthogonal polynomials in two variables. Ann. Mat. Pura Appl. (4) 76, 325–376 (1967) MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Lee, J.K.: Bivariate version of the Hahn-Sonine theorem. Proc. Am. Math. Soc. 128(8), 2381–2391 (2000) MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Maroni, P.: Prolégomènes à l’étude des polynômes semiclassiques. Ann. Mat. Pura Appl. (4) 149, 165–184 (1987) MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Maroni, P.: Une théorie algébrique des polynômes orthogonaux. Application aux polynômes orthogonaux semi-classiques. In: Brezinski, C., Gori, L., Ronveaux, A. (eds.) Orthogonal Polynomials and Their Applications, Erice, 1990. IMACS Ann. Comput. Appl. Math., vol. 9, pp. 95–130. Baltzer, Basel (1991) Google Scholar
  15. 15.
    Szegő, G.: Orthogonal Polynomials, 4th edn. Amer. Math. Soc. Colloq. Publ., vol. 23. Am. Math. Soc., Providence RI (1975) MATHGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2017

Authors and Affiliations

  • Francisco Marcellán
    • 1
  • Misael Marriaga
    • 2
  • Teresa E. Pérez
    • 3
  • Miguel A. Piñar
    • 3
  1. 1.Instituto de Ciencias Matemáticas (ICMAT) and Departamento de MatemáticasUniversidad Carlos III de MadridLeganésSpain
  2. 2.Departamento de MatemáticasUniversidad Carlos III de MadridLeganésSpain
  3. 3.IEMath—Math. Institute and Departamento de Matemática Aplicada, Facultad de CienciasUniversidad de GranadaGranadaSpain

Personalised recommendations