Poles and \(\alpha\)-points of meromorphic solutions of the first Painlevé hierarchy
The author develops the Nevanlinna theory for meromorphic solutions of the first Painlevé hierarchy. The hierarchy is a family of nonlinear ordinary differential equations \((\text{PI}_{2\nu})\): \(d_{\nu+1}[w]+ 4z= 0\), \(\nu= 1,2,\dots\), where \(d_{\nu+1}[w]\) are certain recursively determined differential polynomials of \(w\). The equation \((\text{PI}_{2\nu})\) is of \(2\nu\)th order and \((\text{PI}_2)\) essentially coincides with the first Painlevé equation (PI). We notice that these equations follow from the singular manifold equations for the mKdV hierarchy. The main result is: For any solution \(w_\nu= m_\nu(z)\) of \((\text{PI}_{2\nu})\) meromorphic on \(\mathbb{C}\), it holds \[ \varlimsup_{r\to\infty}\, {\log N(r,w_\nu)\over\log r}\geq {2\nu+ 3\over\nu+ 1}, \] where \(N(r, w_\nu)\) is the counting function for poles of \(w_\nu\). Therefore, the growth order \(\sigma(w_\nu)\) of \(w_\nu\) is not less than \((2\nu+ 3)/(\nu+ 1)\). From this result, the transcendency and a lower estimate on the frequency of \(\alpha\)-points of \(w_\nu\) follow. It is known that the growth order of any solution of (PI) is just \(5/2\), and then the author proposes the conjecture \(\sigma(w_\nu)= (2\nu+ 3)/(\nu+ 1)\). The above inequality is shown by estimating certain sums concerning the poles of \(w_\nu\).
scientific article; zbMATH DE number 2114255