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Special Function Solutions of Painlevé Equations: Theory, Asymptotics and Applications

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Orthogonal Polynomials: Current Trends and Applications

Part of the book series: SEMA SIMAI Springer Series ((SEMA SIMAI,volume 22))

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Abstract

In this paper we review the construction of special function solutions of the Painlevé differential equations. We motivate their study using the theory of orthogonal polynomials, in particular deformation of classical weight functions, as well as unitarily invariant ensembles in random matrix theory. The asymptotic behavior of these Painlevé functions can be studied in at least two different regimes, using the Riemann-Hilbert approach and the classical saddle point method for integrals.

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Acknowledgements

The author gratefully acknowledges financial support from the EPSRC grant “Painlevé equations: analytical properties and numerical computation”, reference EP/P026532/1. The author wishes to thank the organisers of the VII Iberoamerican Workshop in Orthogonal Polynomials and Applications (EIBPOA2018), at Universidad Carlos III de Madrid (Spain), for their invitation to deliver a plenary talk on material related to the content of this paper in July 2018, and the referee for useful comments and corrections on the first version of this paper.

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  1. School of Mathematics, Statistics and Actuarial Science, University of Kent, Canterbury, UK

    Alfredo Deaño

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  1. Alfredo Deaño
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  1. Departamento de Matemáticas, Universidad Carlos III de Madrid, Leganés, Spain

    Francisco Marcellán

  2. Departamento de Física y Matemáticas, Universidad de Alcalá, Alcalá de Henares, Madrid, Spain

    Edmundo J. Huertas

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Deaño, A. (2021). Special Function Solutions of Painlevé Equations: Theory, Asymptotics and Applications. In: Marcellán, F., Huertas, E.J. (eds) Orthogonal Polynomials: Current Trends and Applications. SEMA SIMAI Springer Series, vol 22. Springer, Cham. https://doi.org/10.1007/978-3-030-56190-1_4

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