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Rational Solutions of the Fifth Painleve Equation. Generalised Laguerre Polynomials

Clarkson, Peter, Dunning, Clare (2023) Rational Solutions of the Fifth Painleve Equation. Generalised Laguerre Polynomials. Studies in Applied Mathematics, 152 (1). pp. 453-507. ISSN 0022-2526. E-ISSN 1467-9590. (doi:10.1111/sapm.12649) (KAR id:103386)

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Abstract

In this paper rational solutions of the fifth Painleve equation are discussed. There are two classes of rational solutions of the fifth Painleve equation, one expressed in terms of the generalised Laguerre polynomials, which are the main subject of this paper, and the other in terms of the generalised Umemura polynomials. Both the generalised Laguerre polynomials and the generalised Umemura polynomials can be expressed as Wronskians of Laguerre polynomials specified in terms of specific families of partitions. The properties of the generalised Laguerre polynomials are determined and various differential difference and discrete equations found. The rational solutions of the fifth Painleve equation, the associated σ-equation and the symmetric fifth Painleve system are expressed in terms of generalised Laguerre polynomials. Non-uniqueness of the solutions in special cases is established and some applications are considered. In the second part of the paper, the structure of the roots of the polynomials are investigated for all values of the parameters. Interesting transitions between root structures through coalescences at the origin are discovered, with the allowed behaviours controlled by hook data associated with the partition. The discriminants of the generalised Laguerre polynomials are found and also shown to be expressible in terms of partition data. Explicit expressions for the coefficients of a general Wronskian Laguerre polynomial defined in terms of a single partition are given.

Item Type: Article
DOI/Identification number: 10.1111/sapm.12649
Uncontrolled keywords: Painleve equation, rational solutions, Laguerre polynomials, discriminant, partition, Wronskian
Subjects: Q Science > QA Mathematics (inc Computing science)
Divisions: Divisions > Division of Computing, Engineering and Mathematical Sciences > School of Mathematics, Statistics and Actuarial Science
Funders: University of Kent (https://ror.org/00xkeyj56)
Depositing User: Clare Dunning
Date Deposited: 23 Oct 2023 08:49 UTC
Last Modified: 01 Feb 2024 14:30 UTC
Resource URI: https://kar.kent.ac.uk/id/eprint/103386 (The current URI for this page, for reference purposes)

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