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Asymptotics of Some Convolutional Recurrences

By Edward A. Bender, Adri B. Olde Daalhuis, Zhicheng Gao, L. Bruce Richmond and Nicholas Wormald


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 Research supported by NSERC Research supported by NSERC Research supported by NSERC and Canada Research Chair Program the electronic journal of combinatorics 17 (2010), #R1 1subexponential growth by using an iterative method that may be useful for other recurrences

Year: 2010
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