Complicated and Exotic Expansions of Solutions to the Painlevé Equations

We consider the complicated and exotic asymptotic expansions of solutions to a polynomial ordinary differential equation (ODE). They are such series on integral powers of the independent variable, which coefficients are the Laurent series on decreasing powers of the logarithm of the independent variable and on its pure imaginary power correspondingly. We propose an algorithm for writing ODEs for these coefficients. The first coefficient is a solution of a truncated equation. For some initial equations, it is an usual or Laurent polynomial. Question: will the following coefficients be such polynomials? Here the question is considered for the third (\(P_{3}\)), fifth (\(P_5\)) and sixth (\(P_{6}\)) Painlevé equations. These 3 Painlevé equations have 8 families of complicated expansions and 4 families of exotic expansions. I have calculated several first polynomial coefficients of expansions for all these 12 families. Second coefficients in 7 of 8 families of complicated expansions are polynomials, as well in 2 families of exotic expansions, but one family of complicated and two families of exotic expansions demand some conditions for polynomiality of the second coefficient. Here we give a detailed presentation with proofs of all results.