Value distribution of the Painlevé transcendents
The order of growth of the first Painlevé transcendents is studied. For a regular point \(z_0\neq 0\) of a fixed solution \(w(z)\) to the first Painlevé differential equation \(w''= z+6w^2\), the author defines the main working tool, the so-called scaling unit \[ r(z_0)= \min\{|w(z_0)|^{-1/2}, |w'(z_0)|^{-1/3}, |z_0|^{-1/4}\}. \] With help of this notion, some form of the theorem on analytic dependence on parameters and initial values is proved. Using these results, the author obtains the following asymptotic relations for the counting function \(N(r,w)= O(r^{5/2})\), for the Nevanlinna characteristic function \(T(r,w)= O(r^{5/2})\) and the proximity function \(m(r,w)= O(\log r)\), where \(w(z)\) is any first Painlevé transcendent. These relations yield, that the first Painlevé transcendent has the order of growth \[ \lambda(\omega)= \limsup_{r\to\infty} {\log T(r,w)\over\log r}={5\over 2}. \]
scientific article; zbMATH DE number 1819979
scientific article