Asymptotics for solutions of smooth recurrence equations

It is shown that convergent solutions of a smooth recurrence equation whose gradient satisfies a certain "nonunimodularity" condition can be approximated by an asymptotic expansion. The lemma used to show this has some features in common with Poincaré's theorem on homogeneous linear recurrence equations. An application to the study of polynomials orthogonal with respect to the weight function <!-- MATH $\exp ( - {x^6}/6)$ --> is given.