Power expansions of solutions to the sixth Painlevé equation near a regular point

A. D. Bruno, I. V. Goryuchkina


Abstract: Using Power Geometry [1, 2], in the generic case we find all expansion of solutions to the sixth Painlevé equation [3, 4] near a nonsingular point of the independent variable, i.e. different from zero, one and infinity. All expansions contain integral power exponents of the local variable and have constant complex coefficients and converge. There are 5 families of such expansions. Expansions of solutions near the singular points are described in [2, 5].