Painlevé Equations through Symmetry

Generating Function Associated with the Hankel Determinant Formula for the Solutions of the Painleve IV Equation

“…(We note that the convention of composition of transformations is di¤erent from what is often used in generating complex exact solutions from simple ones. See the section A.4 of [15] for details. )…”

“…P IV (1.1) has been studied extensively from various points of view. In particular, it is well-known that it admits symmetries of a‰ne Weyl group of type A ð1Þ 2 as a group of Bäcklund transformations [15,17,18] (see also [13,1,3]). Moreover, it is known that P IV admits two classes of classical solutions for special values of parameters: transcendental classical solutions which are expressed by the parabolic cylinder function [18] and rational solutions [10,14].…”

“…Those solutions are expressed by logarithmic derivatives of certain characteristic polynomials with integer coe‰cients, which are generated by the Toda equation. Those polynomials are called the Okamoto polynomials [15,17,18,4].…”

“…In fact, it has been shown that the Okamoto polynomials are nothing but a specialization of the 3-core Schur functions by using the Jacobi-Trudi type determinant formula [12,15,17]. Also, there is another determinant formula which expresses the Okamoto polynomials in terms of the Hankel determinant.…”

“…In Section 2, we give a brief review of the symmetries and t functions for P IV through the theory of the symmetric form of P IV [15]. We then construct the Hankel determinant formula by applying the formula for the Toda equation [11] in Section 3.…”