Mathematical Research Letters
Volume 4 (1997)
Exponentially small corrections to divergent asymptotic expansions of solutions of the fifth Painlevé equation
Pages: 741 – 759
We calculate the leading term of asymptotics for the coefficients of certain divergent asymptotic expansions for the solutions of the fifth Painlevé equation (P5) by using the isomonodromy deformation method and the Borel transform. Unexpectedly, these asymptotics appear to be periodic functions of the coefficients of P5. We also show the relation of our results with some other facts already known in the theory of the Painlevé equations established by other methods: (1) a connection formula for the third Painlevé equation; (2) a condition for existence of rational solutions for P5; (3) and a numerical study of the tau-function for P5.