Asymptotics of Some Convolutional Recurrences

We study the asymptotic behavior of the terms in sequences satisfying recurrences of the form an = an−1 + n−d k=d f(n, k)akan−k where, very roughly speaking, f(n, k) behaves like a product of reciprocals of binomial coefficients. Some examples of such sequences from map enumerations, Airy constants, and Painlevé I equations are discussed in detail. 1 Main results There are many examples in the literature of sequences defined recursively using a convolution. It often seems difficult to determine the asymptotic behavior of such sequences. In this note we study the asymptotics of a general class of such sequences. We prove subexponential growth by using an iterative method that may be useful for other recurrences. By subexponential growth we mean that, for every constant D > 1, an = o(D ) as n → ∞. Thus our motivation for this note is both the method and the applications we give. Let d > 0 be a fixed integer and let f(n, k) ≥ 0 be a function that behaves like a product of some powers of reciprocals of binomial coefficients, in a general sense to be specified in Theorem 1. We deal with the sequence an for n ≥ d where ad, ad+1, · · · , a2d−1 ≥ 0 are arbitrary and, when n ≥ 2d, an = an−1 + n−d