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Symmetries in the fourth Painlevé equation and Okamoto polynomials. (English) Zbl 0932.34088
The fourth Painlevé equation P IV is known to have the symmetry of the affine Weyl group of type A 2 (1) with respect to the Bäcklund transformations. Here, a new representation of P 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 IV is given in terms of this representation. Through the symmetric form, it is shown that P 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 IV , called Okamoto polynomials, are expressed in terms of the 3-reduced Schur functions.

34M55Painlevé and other special equations; classification, hierarchies
33C47Other special orthogonal polynomials and functions