Studies on the Painlevé equations. III: Second and fourth Painlevé equations, \(P_{II}\) and \(P_{IV}\) (Q1074195)

[Part II is to appear in Jap. J. Math., Part I in Ann. Mat. Pura Appl., IV. Ser.] In this paper, which is the third part of the series of papers ''Studies on the Painlevé equations'', we study the second and the fourth Painlevé equations by means of the method of birational canonical transformations. We associate with each equation the nonautonomous Hamiltonian system (H), called the Painlevé system. The group of birational canonical transformations of (H) is investigated by the use of the notion of the affine Weyl group; we attach the root system (R) to each (H). We consider also the families of particular solutions of the Painlevé systems, written in terms of the Airy functions or the Hermite functions. In particular, the rational solutions of (H) are studied in detail. The \(\tau\)-functions related to (H) is the other main object of this article; it is shown that the sequence \(\{\tau_ n\); \(n\in {\mathbb{Z}}\}\) of \(\tau\)-functions of (H) satisfies the Toda equation: \(\delta^ 2\log \tau_ n=\tau_{n-1}\tau_{n+1}/\tau^ 2_ n\).