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Zbl 0939.34074
Kajiwara, Kenji; Masuda, Tetsu
On the Umemura polynomials for the Painlevé III equation. (English)
[J] Phys. Lett., A 260, No.6, 462-467 (1999). ISSN 0375-9601

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.

MSC 2000:: *34M45 Differential equations on complex manifolds
33C45 Orthogonal polynomials and functions of hypergeometric type
33E99 Special functions

Keywords: Painlevé III equations; Laguerre polynomials

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