Basic asymptotic expansions of solutions to the sixth Painlevé equation (Q844377)

The paper investigates the sixth Painlevé equation \[ \begin{multlined} y''= {(y')^2\over 2} \Biggl({1\over y}+{1\over y-1}+{1\over y-x}\Biggr)- y'\Biggl({1\over x}+ {1\over x-1}+ {1\over y-x}\Biggr)+\\ {y(y- 1)(y- x)\over x^2(x- 1)^2} \Biggl[a+ b{x\over y^2}+ c{x-1\over (y-1)^2}+ d{x(x- 1)\over (y- x)^2}\Biggr],\end{multlined} \] where \(a\), \(b\), \(c\) are complex variables. It has three singular points, \(x=0\), \(x=1\) and \(x=\infty\). As \(x\to 0\) the author investigates asymptotic expansions of the form \[ y= c_r x^r+ \sum_s c_s x^s\tag{\(*\)} \] with appropriate constraints on \(r\) and \(s\). The complex coefficients \(c_r\) and \(c_s\) vary according to three types included within the paper. The paper proves several asymptotic power expansions for solutions of the form given in \((*)\).