An algebraic proof for the Umemura polynomials for the third Painlevé equation

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Clarkson, Peter ; Law, Chun-Kong ; Lin, Chia-Hua (2016)
  • Publisher: arxiv.org
  • Subject: QA351 | QA372
    arxiv: Nonlinear Sciences::Exactly Solvable and Integrable Systems

We are concerned with the Umemura polynomials associated with 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 satisfied by Umemura polynomials are indeed polynomials.
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