Movable poles of the solutions of Painlevé's equation of the third kind and their relation with Mathieu functions
The article deals with the Painlevé equation \(u''+(1/x)u'=4shu\), \(x>0\). This equation has a solution \(u(x)=r\ell nx+s+O(x^{2-| r|})\) where \(x\to 0\), \(| r| <2\) with the asymptotic property \[ u(x)=- 2\ell n(x-2)-\frac{x-a}{a}-\frac{b(x-a)^ 2}{a^ 2}+O(x-a)^ 3,\quad x\to a \] near a singular point. The purpose of this work is to find a formula connecting the parameters (a,b) with the initial data (r,s). This problem is solved by the method of monodromy-deformation [\textit{H. Flaschka} and \textit{A. Newell}, Comm. Math. Phys. 76, 65-116 (1980; Zbl 0439.34005)]. Theorems 5, 6 give functional equations for the parameters (a,b), (r,s) containing the solution of the Mathieu equation.
scientific article; zbMATH DE number 3998254