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On the Umemura polynomials for the Painlevé III equation. (English) Zbl 0939.34074
Summary: A determinantal expression of Jacobi-Trudi type for the rational solutions to the Painlevé III $(\text{P}_{\text{III}})$ equation is presented. Entries of the determinant are given by Laguerre polynomials. Degeneration of this determinant expression to that for the rational solutions to $\text{P}_{\text{II}}$ is discussed by applying a coalescence procedure.

34M45Ordinary differential equations on complex manifolds
33C45Orthogonal polynomials and functions of hypergeometric type
33E99Other special functions
Full Text: DOI
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