Constructive Approximation

, Volume 39, Issue 1, pp 223–254 | Cite as

The Relationship Between Semiclassical Laguerre Polynomials and the Fourth Painlevé Equation



We discuss the relationship between the recurrence coefficients of orthogonal polynomials with respect to a semiclassical Laguerre weight and classical solutions of the fourth Painlevé equation. We show that the coefficients in these recurrence relations can be expressed in terms of Wronskians of parabolic cylinder functions that arise in the description of special function solutions of the fourth Painlevé equation.


Semiclassical orthogonal polynomials Recurrence coefficients Painlevé equations Wronskians Parabolic cylinder functions Hamiltonians 

Mathematics Subject Classification (2010)

34M55 33E17 33C47 42C05 33C15 



We thank the London Mathematical Society for support through a “Research in Pairs” grant. PAC thanks Ana Loureiro, Paul Nevai, James Smith, and Walter van Assche for their helpful comments and illuminating discussions. We also thank the referees for helpful suggestions and additional references.


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© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.School of Mathematics, Statistics and Actuarial ScienceUniversity of KentCanterburyUK
  2. 2.Department of Mathematics and Applied MathematicsUniversity of PretoriaPretoriaSouth Africa

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