A Constructive Proof for the Umemura Polynomials of the Third Painlev\'e Equation

TL;DR

This paper provides a constructive algebraic proof that the Umemura polynomials related to the third Painlevé equation are indeed polynomials, extending methods used for the second Painlevé equation and offering insights into their roots.

Contribution

It extends Taneda's method to prove the polynomial nature of Umemura polynomials for the third Painlevé equation, providing a constructive approach and root information.

Findings

Confirmed the polynomial nature of Umemura polynomials

Extended Taneda's method to third Painlevé equation

Provided root-related insights into the polynomials

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.