A Constructive Proof for the Umemura Polynomials of the Third Painlev\'e Equation
Classical Analysis and ODEs
2023-10-26 v6 Exactly Solvable and Integrable Systems
Abstract
We are concerned with the Umemura polynomials associated with rational solutions of the third Painlev\'e equation. We extend Taneda's method, which was developed for the Yablonskii-Vorob'ev polynomials associated with the second Painlev\'e 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.
Cite
@article{arxiv.1609.00495,
title = {A Constructive Proof for the Umemura Polynomials of the Third Painlev\'e Equation},
author = {Peter A. Clarkson and Chun-Kong Law and Chia-Hua Lin},
journal= {arXiv preprint arXiv:1609.00495},
year = {2023}
}