{"title":"第三阶painlevleve方程Umemura多项式的构造证明","authors":"Peter A. Clarkson, Chun-Kong Law, Chia-Hua Lin","doi":"10.3842/sigma.2023.080","DOIUrl":null,"url":null,"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.","PeriodicalId":49453,"journal":{"name":"Symmetry Integrability and Geometry-Methods and Applications","volume":null,"pages":null},"PeriodicalIF":0.9000,"publicationDate":"2023-10-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"2","resultStr":"{\"title\":\"A Constructive Proof for the Umemura Polynomials of the Third Painlevé Equation\",\"authors\":\"Peter A. Clarkson, Chun-Kong Law, Chia-Hua Lin\",\"doi\":\"10.3842/sigma.2023.080\",\"DOIUrl\":null,\"url\":null,\"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.\",\"PeriodicalId\":49453,\"journal\":{\"name\":\"Symmetry Integrability and Geometry-Methods and Applications\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.9000,\"publicationDate\":\"2023-10-25\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"2\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Symmetry Integrability and Geometry-Methods and Applications\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.3842/sigma.2023.080\",\"RegionNum\":3,\"RegionCategory\":\"物理与天体物理\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Symmetry Integrability and Geometry-Methods and Applications","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.3842/sigma.2023.080","RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
A Constructive Proof for the Umemura Polynomials of the Third Painlevé Equation
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.
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.