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Zbl 0932.34088
Noumi, Masatoshi; Yamada, Yasuhiko
Symmetries in the fourth Painlevé equation and Okamoto polynomials. (English)
[J] Nagoya Math. J. 153, 53-86 (1999). ISSN 0027-7630

The fourth Painlevé equation $P_{\text{IV}}$ is known to have the symmetry of the affine Weyl group of type $A^{(1)}_2$ with respect to the Bäcklund transformations. Here, a new representation of $P_{\text{IV}}$, called the symmetric form, is introduced by taking the three fundamental invariant divisors as the dependent variables. A complete description of the symmetry of $P_{\text{IV}}$ is given in terms of this representation. Through the symmetric form, it is shown that $P_{\text{IV}}$ is obtained as a similarity reduction of the 3-reduced modified KP hierarchy. It is proved in particular that the special polynomials for rational solutions to $P_{\text{IV}}$, called Okamoto polynomials, are expressed in terms of the 3-reduced Schur functions.
[Masatoshi Noumi (Kobe)]

MSC 2000:: *34M55 Painlevé and other special equations
33C47 Other special orthogonal polynomials and functions

Keywords: fourth Painlevé equation; Bäcklund transformations; KP hierarchy; Okamoto polynomials; Schur functions

Cited in: Zbl 1042.82019 Zbl 1052.33504

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