Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 19 (2023), 080, 20 pages      arXiv:1609.00495      https://doi.org/10.3842/SIGMA.2023.080

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

Peter A. Clarkson a, Chun-Kong Law b and Chia-Hua Lin b
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

Received June 29, 2023, in final form October 17, 2023; Published online October 25, 2023

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.

Key words: Umemura polynomials; third Painlevé equation; recurrence relation.

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