On properties of the coefficients of the complicated and exotic formal solutions of the sixth Painlev\'e equation

Abstract

It is known, that among the formal solutions of the sixth Painlev\'e equation there met series with integer power exponents of the independent variable $x$ with coefficients in form of formal Laurent series (with finite main parts) in $\log^{-1} x$ (complicated expansions), or in $x^{{\rm i}\,\theta}$, where ${\rm i}=\sqrt{-1},$ $\theta\in\mathbb{R},$ $\theta\neq 0$ (exotic expansions). These coefficients can be computed consecutively. Here we research analytic properties of the series, that are the coefficients of the complicated and exotic formal solutions of the sixth Painlev\'e equation.