Constructive Approximation

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

The Relationship Between Semiclassical Laguerre Polynomials and the Fourth Painlevé Equation

Article

Abstract

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.

Keywords

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

Mathematics Subject Classification (2010)

34M55 33E17 33C47 42C05 33C15 

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Copyright information

© 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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