Asymptotic expansions for solutions of smooth recurrence equations

Abstract

Abstract.Let (y" : n > 1) be a convergent sequence of reals, where for each n the tuple (yn, yn+x, ... , yn+ k, X/n) satisfies one of r equations, depending on the residue class of n (mod r) , for some given k and r . Assume these equations are smooth, they have the same gradient in the first k + 1 variables, and this gradient satisfies a certain nonmodularity condition. We then show that y" has r asymptotic expansions, depending on the residue class of n (mod r), in terms of powers of 1 fn . This result enables us to discuss the asymptotic behavior of the recurrence coefficients associated with certain orthogonal polynomials. A key ingredient in the proof of the main result is a lemma involving exponential sums.