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Symmetry, Integrability and Geometry: Methods and Applications, 2023, Volume 19, 080, 20 pp.
DOI: https://doi.org/10.3842/SIGMA.2023.080 (Mi sigma1975) 		 
This article is cited in 1 scientific paper (total in 1 paper)

A Constructive Proof for the Umemura Polynomials of the Third Painlevé Equation

Peter A. Clarksona, Chun-Kong Lawb, Chia-Hua Linb

a School of Mathematics, Statistics and Actuarial Science, University of Kent, Canterbury, CT2 7NF, UK
b Department of Applied Mathematics, National Sun Yat-sen University, Kaohsiung, Taiwan 804, Taiwan
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DOI: https://doi.org/10.3842/SIGMA.2023.080
Abstract: We are concerned with the Umemura polynomials associated with rational solutions of the third Painlevé equation. We extend Taneda's method, which was developed for the Yablonskii–Vorob'ev polynomials associated with the second Painlevé equation, to give an algebraic proof that the rational functions generated by the nonlinear recurrence relation which determines the Umemura polynomials are indeed polynomials. Our proof is constructive and gives information about the roots of the Umemura polynomials.
Keywords: Umemura polynomials; third Painlevé equation; recurrence relation.
Funding agency Grant number
National Center for Theoretical Sciences
CKL is partially supported by National Science and Technology Council (formerly Ministry of Science and Technology), Taiwan.
Received: June 29, 2023; in final form October 17, 2023; Published online October 25, 2023
Document Type: Article
MSC: 33E17, 34M55, 65Q30
Language: English
Citation: Peter A. Clarkson, Chun-Kong Law, Chia-Hua Lin, “A Constructive Proof for the Umemura Polynomials of the Third Painlevé Equation”, SIGMA, 19 (2023), 080, 20 pp.
Citation in format AMSBIB
\Bibitem{ClaLawLin23}
\by Peter~A.~Clarkson, Chun-Kong~Law, Chia-Hua~Lin
\paper A Constructive Proof for the Umemura Polynomials of the Third Painlev\'e Equation
\jour SIGMA
\yr 2023
\vol 19
\papernumber 080
\totalpages 20
\mathnet{http://mi.mathnet.ru/sigma1975}
\crossref{https://doi.org/10.3842/SIGMA.2023.080}
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    		Symmetry, Integrability and Geometry: Methods and Applications
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