Properties of Certain Classes of Semiclassical Orthogonal Polynomials

  • Kerstin JordaanEmail author
Conference paper
Part of the Tutorials, Schools, and Workshops in the Mathematical Sciences book series (TSWMS)


In this lecture we discuss properties of orthogonal polynomials for weights which are semiclassical perturbations of classical orthogonality weights. We use the moments, together with the connection between orthogonal polynomials and Painlevé equations to obtain explicit expressions for the recurrence coefficients of polynomials associated with a semiclassical Laguerre and a generalized Freud weight. We analyze the asymptotic behavior of generalized Freud polynomials in two different contexts. We show that unique, positive solutions of the nonlinear difference equation satisfied by the recurrence coefficients exist for all real values of the parameter involved in the semiclassical perturbation but that these solutions are very sensitive to the initial conditions. We prove properties of the zeros of semiclassical Laguerre and generalized Freud polynomials and determine the coefficients an,n+j in the differential-difference equation
$$\displaystyle x\frac {d}{dx}P_n(x)=\sum _{k=-1}^{0}a_{n,n+k}P_{n+k}(x), $$
where Pn(x) are the generalized Freud polynomials. Finally, we show that the only monic orthogonal polynomials \(\{P_n\}_{n=0}^{\infty }\) that satisfy
$$\displaystyle \pi (x)\mathcal {D}_{q}^2P_{n}(x)=\sum _{j=-2}^{2}a_{n,n+j}P_{n+j}(x),\; x=\cos \theta ,\;~ a_{n,n-2}\neq 0,~ n=2,3,\dots , $$
where π(x) is a polynomial of degree at most 4 and \(\mathcal {D}_{q}\) is the Askey–Wilson operator, are Askey–Wilson polynomials and their special or limiting cases, using this relation to derive bounds for the extreme zeros of Askey–Wilson polynomials.


Orthogonal polynomials Zeros Laguerre polynomials Freud polynomials Askey–Wilson polynomials Jacobi polynomial 

Mathematics Subject Classification (2000)

Primary 33C45; Secondary 42C05 


  1. 1.
    W.A. Al-Salam, T.S. Chihara, Another characterization of the classical orthogonal polynomials. SIAM J. Math. Anal. 3, 65–70 (1972)MathSciNetCrossRefGoogle Scholar
  2. 2.
    S.M. Alsulami, P. Nevai, J. Szabados, W. Van Assche, A family of nonlinear difference equations: existence, uniqueness, and asymptotic behavior of positive solutions. J. Approx. Theory 193, 39–55 (2015)MathSciNetCrossRefGoogle Scholar
  3. 3.
    A. Arceo, E.J. Huertas, F. Marcellán, On polynomials associated with an Uvarov modification of a quartic potential Freud-like weight. Appl. Math. Comput. 281, 102–120 (2016)MathSciNetzbMATHGoogle Scholar
  4. 4.
    A.F. Beardon, The theorems of Stieltjes and Favard. Comput. Methods Funct. Theory 11, 247–262 (2011)MathSciNetCrossRefGoogle Scholar
  5. 5.
    S. Bochner S, Über Sturm-Liouvillesche Polynomsysteme. Math. Z. 29(1), 730–736 (1929)MathSciNetCrossRefGoogle Scholar
  6. 6.
    L. Boelen, W. Van Assche, Discrete Painlevé equations for recurrence relations of semiclassical Laguerre polynomials. Proc. Am. Math. Soc. 138, 1317–1331 (2011)CrossRefGoogle Scholar
  7. 7.
    Y. Chen, M.V. Feigin, Painlevé IV and degenerate Gaussian Unitary Ensembles. J. Phys. A Math. Gen. 39, 12381–12393 (2006)CrossRefGoogle Scholar
  8. 8.
    Y. Chen, M.E.H. Ismail, Ladder operators and differential equations for orthogonal polynomials. J. Phys. A 30, 7817–7829 (1997)MathSciNetCrossRefGoogle Scholar
  9. 9.
    T.S. Chihara, An Introduction to Orthogonal Polynomials (Gordon and Breach, New York, 1978). Reprinted by Dover Publications (2011)Google Scholar
  10. 10.
    A.S. Clarke, B. Shizgal, On the generation of orthogonal polynomials using asymptotic methods for recurrence coefficients. J. Comput. Phys. 104, 140–149 (1993)MathSciNetCrossRefGoogle Scholar
  11. 11.
    P.A. Clarkson, K. Jordaan, The relationship between semiclassical Laguerre polynomials and the fourth Painlevé equation. Constr. Approx. 39, 223–254 (2014)MathSciNetCrossRefGoogle Scholar
  12. 12.
    P.A. Clarkson, K. Jordaan, Properties of generalized Freud polynomials. J. Approx. Theory 225, 148–175 (2018)MathSciNetCrossRefGoogle Scholar
  13. 13.
    P.A. Clarkson, K. Jordaan, A. Kelil, A generalized Freud weight. Stud. Appl. Math. 136, 288–320 (2016)MathSciNetCrossRefGoogle Scholar
  14. 14.
    S. Datta, J. Griffin, A characterization of some q-orthogonal polynomials. Ramanujan J. 12(3), 425–437 (2006)MathSciNetCrossRefGoogle Scholar
  15. 15.
    K. Driver, K. Jordaan, Bounds for extreme zeros of some classical orthogonal polynomials. J. Approx. Theory 164, 1200–1204 (2012)MathSciNetCrossRefGoogle Scholar
  16. 16.
    J. Favard, Sur les polynomes de Tchebicheff. C. R. Acad. Sci. Paris 200, 2052–2053 (1935)zbMATHGoogle Scholar
  17. 17.
    G. Filipuk, W. Van Assche, L. Zhang, The recurrence coefficients of semiclassical Laguerre polynomials and the fourth Painlevé equation. J. Phys. A 45, 205201 (2012)MathSciNetCrossRefGoogle Scholar
  18. 18.
    G. Freud, On the coefficients in the recursion formulae of orthogonal polynomials. Proc. R. Irish Acad. Sect. A 76, 1–6 (1976)MathSciNetzbMATHGoogle Scholar
  19. 19.
    A.G. García, F. Marcellán, L. Salto, A distributional study of discrete classical orthogonal polynomials. J. Comput. Appl. Math. 57(1–2), 147–162 (1995)MathSciNetCrossRefGoogle Scholar
  20. 20.
    W. Hahn, Über die Jacobischen Polynome und zwei verwandte Polynomklassen. Math. Z. 39(1), 634–638 (1935)MathSciNetCrossRefGoogle Scholar
  21. 21.
    E. Hendriksen, H. van Rossum, Semi-classical orthogonal polynomials, in Polynômes Orthogonaux et Applications, ed. by C. Brezinski, A. Draux, A.P. Magnus, P. Maroni, A. Ronveaux. Lecture Notes in Mathematics, vol. 1171 (Springer, Berlin, 1985), pp. 354–361Google Scholar
  22. 22.
    M.E.H. Ismail, A generalization of a theorem of Bochner. J. Comput. Appl. Math. 159(2), 319–324 (2003)MathSciNetCrossRefGoogle Scholar
  23. 23.
    M.E.H. Ismail, in Classical and Quantum Orthogonal Polynomials in One Variable. Encyclopedia of Mathematics and its Applications, vol. 98 (Cambridge University Press, Cambridge, 2005)Google Scholar
  24. 24.
    M.E.H. Ismail, X. Li, Bounds for extreme zeros of orthogonal polynomials. Proc. Am. Math. Soc. 115, 131–140 (1992)MathSciNetCrossRefGoogle Scholar
  25. 25.
    J. Jost, Postmodern Aanalysis (Springer, Berlin, 2006)Google Scholar
  26. 26.
    T. Kasuga, R. Sakai, Orthonormal polynomials with generalized Freud-type weights. J. Approx. Theory 121, 13–53 (2003)MathSciNetCrossRefGoogle Scholar
  27. 27.
    M. Kenfack Nangho, K. Jordaan, A characterization of Askey–Wilson polynomials. Proc. Am. Math. Soc. 147(6), 2465–2480 (2019)MathSciNetCrossRefGoogle Scholar
  28. 28.
    M. Kenfack Nangho, K. Jordaan. Structure relations of classical orthogonal polynomials of the quadratic andq-quadratic variable. SIGMA, 14, 26 (2018)zbMATHGoogle Scholar
  29. 29.
    R. Koekoek, P.A. Lesky, R.F. Swarttouw, in Hypergeometric Orthogonal Polynomials and Theirq-Analogues. Springer Monographs in Mathematics (Springer, Berlin, 2010)Google Scholar
  30. 30.
    J.S. Lew, D.A. Quarles, Nonnegative solutions of a nonlinear recurrence. J. Approx. Theory 38, 357–379 (1983)MathSciNetCrossRefGoogle Scholar
  31. 31.
    F. Marcellán, R. Álvarez-Nodarse, On the “Favard theorem” and its extensions. J. Comput. Appl. Math. 127, 231–254 (2001)MathSciNetCrossRefGoogle Scholar
  32. 32.
    I.P. Natanson, Konstruktive Funktionentheorie (Akademie-Verlag, Berlin, 1955)zbMATHGoogle Scholar
  33. 33.
    P. Nevai, Géza Freud, orthogonal polynomials and Christoffel functions. A case study. J. Approx. Theory 48, 3–167 (1986)CrossRefGoogle Scholar
  34. 34.
    F.W.J. Olver, D.W. Lozier, R.F. Boisvert, C.W. Clark (eds), DLMF Handbook of Mathematical Functions (Cambridge University Press, Cambridge, 2010)Google Scholar
  35. 35.
    J. Shohat, The Relation of the classical orthogonal polynomials to the polynomials of appell. Am. J. Math. 58, 453–464 (1936)MathSciNetCrossRefGoogle Scholar
  36. 36.
    J. Shohat, Sur les polynômes orthogonaux généralises. C. R. Acad. Sci. Paris 207, 556–558 (1938)zbMATHGoogle Scholar
  37. 37.
    M.H. Stone. Linear transformations in Hilbert space and their applications to analysis. AMS Colloquium Publications, vol. 15 (American Mathematical Society, Providence, 1932)Google Scholar
  38. 38.
    T.J. Stieltjes, Recherches sur les fractions continues. Ann. Fac. Sci. Toulouse Sci. Math. Sci. Phys. 8, J1–J122 (1894); A5–A47 (1895)Google Scholar
  39. 39.
    W. Van Assche, The impact of Stieltjes work on continued fractions and orthogonal polynomials, in Collected Papers, ed. by G. van Dijk (Springer, Berlin, 1993), pp. 5–37Google Scholar
  40. 40.
    L. Vinet, A. Zhedanov, Generalized Bochner theorem: characterization of the Askey–Wilson polynomials. J. Comput Appl. Math. 211(1), 45–56 (2008)MathSciNetCrossRefGoogle Scholar
  41. 41.
    Z.T. Wen, R. Wong and S.X. Xu, Global asymptotics of orthogonal polynomials associated with a generalized Freud weight. Chin. Ann. Math. Ser. B, 39, 553–596 (2018)MathSciNetCrossRefGoogle Scholar
  42. 42.
    A. Wintner, Spektraltheorie der unendlichen Matrizen (S. Hirzel, Leipzig, 1929)zbMATHGoogle Scholar

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Authors and Affiliations

  1. 1.Department of Decision SciencesUniversity of South AfricaPretoriaSouth Africa

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