Hyperasymptotic solutions of higher order linear differential equations with a singularity of rank one

A sequence of re-expansions is developed for the remainder terms in the well-known Poincare series expansions of the solutions of homogeneous linear differential equations of higher order in the neighbourhood of an irregular singularity of rank one. These re-expansions are a series whose terms are a product of Stokes multipliers, coefficients of the original Poincare series expansions, and certain multiple integrals, the so-called hyperterminants. Each step of the process reduces the estimate of the error term by an exponentially small factor. The method of this paper is based on the Borel-Laplace transform, which makes it applicable to other problems. At the end of the paper the method is applied to integrals with saddles. Also, a powerful new method is presented to compute the Stokes multipliers. A numerical example is included.