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

IF 0.9 3区物理与天体物理 Q2 MATHEMATICS

Symmetry Integrability and Geometry-Methods and Applications Pub Date : 2023-10-25 DOI:10.3842/sigma.2023.080

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

引用次数: 2

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.

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第三阶painlevleve方程Umemura多项式的构造证明

我们关注与第三阶painlevleve方程的有理解相关的Umemura多项式。我们推广了Taneda的Yablonskii-Vorob'ev多项式与第二painlev方程相关联的方法，给出了由非线性递归关系生成的有理函数确实是多项式的代数证明。我们的证明是建设性的，并且给出了关于Umemura多项式的根的信息。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Symmetry Integrability and Geometry-Methods and Applications 物理-物理：数学物理

CiteScore

1.80

自引率

0.00%

发文量

审稿时长

4-8 weeks

期刊介绍： Scope Geometrical methods in mathematical physics Lie theory and differential equations Classical and quantum integrable systems Algebraic methods in dynamical systems and chaos Exactly and quasi-exactly solvable models Lie groups and algebras, representation theory Orthogonal polynomials and special functions Integrable probability and stochastic processes Quantum algebras, quantum groups and their representations Symplectic, Poisson and noncommutative geometry Algebraic geometry and its applications Quantum field theories and string/gauge theories Statistical physics and condensed matter physics Quantum gravity and cosmology.