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

TL;DR

This paper investigates the analytic properties of the coefficients in the complicated and exotic formal solutions of the sixth Painlevé equation, which involve series with specific types of exponents and coefficients.

Contribution

It provides a detailed analysis of the analytic properties of coefficients in formal solutions with complex exponents and logarithmic terms for the sixth Painlevé equation.

Findings

Coefficients exhibit specific analytic behaviors.

Formal solutions include series with integer powers and complex exponents.

Results enhance understanding of solution structures for Painlevé equations.

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.