Series Expansions of Painleve Transcendents near the Point at Infinity

For the Painleve equations (I) through (V) near the point at infinity we present several families of two-parameter solutions. Our solutions are expressed by asymptotic series with coefficients polynomial in exponential terms, and also by convergent power series in exponential terms with coefficients expanded into asymptotic series. Both expressions are valid without a restriction on integration constants. We propose a direct method to derive asymptotic solutions, which is also applicable to more general nonlinear equations. As applications of our results, for general solutions of the third and the fifth Painleve equations, we give estimates for the number of a-points including poles in given sectors.