UNIFORMIZATION AND TRANSCENDENCE OF SOLUTIONS FOR THE FIRST AND 2ND PAINLEVE HIERARCHIES

The first and second Painleve equation hierarchies are considered. The relations between them are given. The Backlund transformations for th e second Painleve equations of higher order are presented. Some ration al solutions of these equations are obtained. The indices for the firs t and second Painleve equations of fourth order are found. These equat ions 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) 1 998 Elsevier Science B.V.