The Properties of Multiple Orthogonal Polynomials with Mathematica

Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 171)


In this paper some computational aspects of studying various properties of multiple orthogonal polynomials are presented. The results were obtained using the symbolic and numerical computations in Mathematica ( This paper is mainly based on papers Filipuk et al., J Phys A: Math Theor 46:205–204, 2013, [1], Van Assche et al., J Approx Theory 190:1–25, 2015, [2], Zhang and Filipuk, Symmetry Integr Geom Methods Appl 10:103, 2014, [3] (joint with W. Van Assche and L. Zhang). We also perform the Painlevé analysis of certain nonlinear differential equation related to multiple Hermite polynomials and show the existence of two types of polar expansions, which might be useful to obtain relations for zeros of these polynomials.


Multiple orthogonal polynomials Recurrence coefficients Wronskians Linear differential equations Multiple Hermite polynomials Symbolic computations in Mathematica 


  1. 1.
    Filipuk, G., Van Assche, W., Zhang, L.: Ladder operators and differential equations for multiple orthogonal polynomials. J. Phys. A: Math. Theor. 46, 205204 (2013)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Van Assche, W., Filipuk, G., Zhang, L.: Multiple orthogonal polynomials associated with an exponential cubic weight. J. Approx. Theory 190, 1–25 (2015)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Zhang, L., Filipuk, G.: On certain Wronskians of multiple orthogonal polynomials. Symmetry Integr. Geom. Methods Appl. 10, 103 (2014)MathSciNetMATHGoogle Scholar
  4. 4.
    Chihara, T.S.: An Introduction to Orthogonal Polynomials. Gordon and Breach, New York (1978)MATHGoogle Scholar
  5. 5.
    Ismail, M.E.H.: Classical and Quantum Orthogonal Polynomials in One Variable. Encyclopedia of Mathematics and its Applications vol. 98. Cambridge University Press, Cambridge (2005)Google Scholar
  6. 6.
    Szegő, G.: Orthogonal Polynomials. AMS Colloquium Publications, vol. 23. American Mathematical Society, Providence (1975)Google Scholar
  7. 7.
    Nikishin, E.M., Sorokin, V.N.: Rational Approximations and Orthogonality. Translations of Mathematical Monographs vol. 92. American Mathematical Society, Providence (1991)Google Scholar
  8. 8.
    Van Assche, W.: Padé and Hermite-Padé approximation and orthogonality. Surv. Approx. Theory 2, 61–91 (2006)MathSciNetMATHGoogle Scholar
  9. 9.
    Kuijlaars, A.: Multiple orthogonal polynomials in random matrix theory. In: Bhatia, R. (ed.) Proceedings of the International Congress of Mathematicians, vol. 3, pp. 1417–1432. World Scientific, Singapore (2010)Google Scholar
  10. 10.
    Van Assche, W.: Nearest neighbor recurrence relations for multiple orthogonal polynomials. J. Approx. Theory 163, 1427–1448 (2011)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Chen, Y., Ismail, M.E.H.: Ladder operators and differential equations for orthogonal polynomials. J. Phys. A: Math. G. 30, 7817–7829 (1997)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Kudryashov, N.A., Demina, M.V.: Relations between zeros of special polynomials associated with the Painlevé equations. Phys. Lett. A 368, 227–234 (2007)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media Singapore 2016

Authors and Affiliations

  1. 1.Institute of Mathematics of the Polish Academy of SciencesWarsawPoland
  2. 2.Faculty of Mathematics, Informatics and MechanicsUniversity of WarsawWarsawPoland

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