On Finite Classes of Two-Variable Orthogonal Polynomials

  • Esra Güldoğan
  • Rabia AktaşEmail author
  • Mohammad Masjed-Jamei
Original Paper


The purpose of this paper is to introduce several finite sets of orthogonal polynomials in two variables, and investigate some general properties of them such as recurrence relations, generating functions, differential equations, and Rodrigues type representations.


Orthogonal polynomial Weight function Differential equation Recurrence relation Generating function 

Mathematics Subject Classification

33C45 33C50 



The work of the third author has been supported by the Alexander von Humboldt Foundation under the Grant number: Ref 3.4-IRN-1128637-GF-E.


  1. 1.
    Aktaş, R.: Representations for parameter derivatives of some Koornwinder polynomials in two variables. J. Egypt. Math. Soc. 24(4), 555–561 (2016)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Aktaş, R.: A note on parameter derivatives of the Jacobi polynomials on the triangle. Appl. Math. Comp. 247, 368–372 (2014)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Aktaş, R., Altın, A., Taşdelen, F.: A note on a family of two-variable polynomials. J. Comput. Appl. Math. 235, 4825–4833 (2011)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Dunkl, C.F., Xu, Y.: Orthogonal Polynomials of Several Variables. Encyclopedia of Mathematics and Its Applications, vol. 155, 2nd edn. Cambridge University Press, Cambridge (2014)CrossRefGoogle Scholar
  5. 5.
    Fernández, L., Pérez, T.E., Piñar, M.A.: On Koornwinder classical orthogonal polynomials in two variables. J. Comput. Appl. Math. 236, 3817–3826 (2012)MathSciNetCrossRefGoogle 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)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Koepf, W., Masjed-Jamei, M.: Two classes of special functions using Fourier transforms of some finite classes of classical orthogonal polynomials. Proc. Am. Math. Soc. 135(11), 3599–3606 (2007)MathSciNetCrossRefGoogle Scholar
  8. 8.
    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
  9. 9.
    Marcellán, F., Marriaga, M., Pérez, T.E., Piñar, M.A.: On bivariate classical orthogonal polynomials. Appl. Math. Comput. 325, 340–357 (2018)MathSciNetGoogle Scholar
  10. 10.
    Marcellán, F., Marriaga, M., Pérez, T.E., Piñar, M.A.: Matrix Pearson equations satisfied by Koornwinder weights in two variables. Acta Appl. Math. 153, 81–100 (2018)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Marriaga, M., Pérez, T.E., Piñar, M.A.: Three term relations for a class of bivariate orthogonal polynomials. Mediterr. J. Math. 14(54), 26 (2017)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Masjed-Jamei, M.: Classical orthogonal polynomials with weight function \(\left( \left( ax+b\right) ^{2}+\left( cx+d\right) ^{2}\right) ^{-p}\exp \left( q\arctan \left( \frac{ax+b}{cx+d}\right) \right) \); \(-\infty <x<\infty \) and a generalization of T and F distributions. Integr. Trans. Spec. Funct. 15(2), 137–153 (2004)CrossRefGoogle Scholar
  13. 13.
    Masjed-Jamei, M.: Three finite classes of hypergeometric orthogonal polynomials and their application in functions approximation. Integr. Trans. Spec. Funct. 13(2), 169–190 (2002)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Masjed-Jamei, M., Soleyman, F., Area, I., Nieto, J.J.: Two finite q-Sturm liouville problems and their orthogonal polynomial solutions. Filomat 32(1), 231–244 (2018)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Milovanovic, G., Öztürk, G., Aktaş, R.: Properties of some of two-variable orthogonal polynomials. Bull. Malays. Math. Sci. Soc. (2019). CrossRefGoogle Scholar
  16. 16.
    Soleyman, F., Masjed-Jamei, M., Area, I.: A finite class of q-orthogonal polynomials corresponding to inverse gamma distribution. Anal. Math. Phys. 7, 479–492 (2017)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Srivastava, H.M., Manocha, H.L.: A Treatise on Generating Functions. Halsted Press, New York (1984)zbMATHGoogle Scholar
  18. 18.
    Szegö, G.: Orthogonal Polynomials, American Mathematical Society Colloquium Publications, vol. 23, 4th edn. American Mathematical Society, Providence (1975)Google Scholar

Copyright information

© Iranian Mathematical Society 2019

Authors and Affiliations

  • Esra Güldoğan
    • 1
  • Rabia Aktaş
    • 1
    Email author
  • Mohammad Masjed-Jamei
    • 2
  1. 1.Faculty of Science, Department of MathematicsAnkara UniversityAnkaraTurkey
  2. 2.Department of MathematicsK. N. Toosi University of TechnologyTehranIran

Personalised recommendations