On the algebraic independence of generic Painleve transcendents
  We prove that if y" = f(y,y',t) is a generic Painleve equation from among the classes II to V then any collection of distinct solutions and their derivatives are algebraically independent over C(t). (Already proved by Nishioka for the single Painleve I equation). For generic Painleve VI we prove a slightly weaker statement. 
Joel Nagloo and Anand Pillay
math.AG math.CA math.LO