Uniformization and transcendence of solutions for the first and second Painleve hierarchies

摘要

The first and second Painleve equation hierarchies are considered. The relations between them are given. The Backlund transformations for the second Painleve equations of higher order are presented. Some rational solutions of these equations are obtained. The indices for the first and second Painleve equations of fourth order are found. These equations are shown to pass the Painleve test. Lax pairs for the first and second Painleve equations of higher order are given. The transcendence of the solutions for the first Painleve hierarchy is discussed. (C) 1998 Elsevier Science B.V. [References: 29]