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Properties of Some of Two-Variable Orthogonal Polynomials

  • Gradimir V. MilovanovićEmail author
  • Güner Öztürk
  • Rabia Aktaş
Article
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Abstract

The present paper deals with various recurrence relations, generating functions and series expansion formulas for two families of orthogonal polynomials in two variables, given Laguerre–Laguerre Koornwinder polynomials and Laguerre–Jacobi Koornwinder polynomials in the limit cases. Several families of bilinear and bilateral generating functions are derived. Furthermore, some special cases of the results presented in this study are indicated.

Keywords

Jacobi polynomials Laguerre polynomials Generating function Bilinear generating function Bilateral generating function 

Mathematics Subject Classification

MSC 33C45 33C50 42C05 

Notes

Acknowledgements

The authors are deeply grateful to the anonymous referees for their comments and constructive suggestions for improvements of this paper. The first author was supported in part by the Serbian Academy of Sciences and Arts (No. \(\Phi \)-96) and by the Serbian Ministry of Education, Science and Technological Development (No. #OI 174015).

References

  1. 1.
    Agahanov, S.A.: A method of constructing orthogonal polynomials of two variables for a certain class of weight functions. Vestnik Leningrad Univ. 20, 5–10 (1965). (Russian)MathSciNetGoogle Scholar
  2. 2.
    Aktaş, R., Altın, A., Taşdelen, F.: A note on a family of two-variable polynomials. J. Comput. Appl. Math. 235, 4825–4833 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Altın, A., Aktaş, R., Erkuş-Duman, E.: On a multivariable extension for the extended Jacobi polynomials. J. Math. Anal. Appl. 353, 121–133 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Dunkl, C.F., Xu, Y.: Orthogonal Polynomials of Several Variables. Cambridge Univ. Press, New York (2001)CrossRefzbMATHGoogle Scholar
  5. 5.
    Fernandez, L., Perez, T.E., Pinar, M.A.: On Koornwinder classical orthogonal polynomials in two variables. J. Comput. Appl. Math. 236, 3817–3826 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Koornwinder, T.H.: Two variable analogues of the classical orthogonal polynomials. In: Askey, R.A. (ed.) Theory and application of special functions. In: Proceedings of an Advanced Seminar, The University of Wisconsin-Madison, 1975 March 31–April 2, pp. 435–495. Academic Press, New York (1975)Google Scholar
  7. 7.
    Malave, P.B., Bhonsle, B.R.: On generating functions for certain two-variable orthogonal polynomials. Pure Appl. Math. Sci. 11, 9–12 (1980)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Malave, P.B., Bhonsle, B.R.: Some generating functions of two variable analogue of Jacobi polynomials of class II. Ganita 31, 29–37 (1980)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Marriaga, M., Pérez, T.E., Piñar, M.A.: Three term relations for a class of bivariate orthogonal polynomials. Mediterr. J. Math. 14, 54 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Mastroianni, G., Milovanović, G.V.: Interpolation Processes-Basic Theory and Applications. Springer, Berlin (2008)CrossRefzbMATHGoogle Scholar
  11. 11.
    Rainville, E.D.: Special Functions. The Macmillan Company, New York (1960)zbMATHGoogle Scholar
  12. 12.
    Srivastava, H.M., Manocha, H.L.: A Treatise on Generating Functions. Halsted Press (Ellis Horwood Limited, Chichester), Wiley, New York (1984)zbMATHGoogle Scholar
  13. 13.
    Suetin, P.K.: Orthogonal Polynomials in Two Variables. Gordon and Breach Science Publishers, Moscow (1988)zbMATHGoogle Scholar
  14. 14.
    Szegö, G.: Orthogonal Polynomials, Vol. 23, 4th ed., Amer. Math. Soc. Colloq. Publ. (1975)Google Scholar

Copyright information

© Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2019

Authors and Affiliations

  1. 1.Serbian Academy of Sciences and ArtsBeogradSerbia
  2. 2.Faculty of Science and MathematicsUniversity of NišNišSerbia
  3. 3.Department of Mathematics, Faculty of ScienceSelçuk UniversitySelcuklu, KonyaTurkey
  4. 4.Department of Mathematics, Faculty of ScienceAnkara UniversityTandoğan, AnkaraTurkey

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