Báo cáo toán học: "Asymptotics of Some Convolutional Recurrences"

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Department of Mathematic dành cho các bạn yêu thích môn toán học đề tài:Asymptotics of Some Convolutional Recurrences. | Asymptotics of Some Convolutional Recurrences Edward A. Bender Adri B. Olde Daalhuis Department of Mathematics Maxwell Institute and School of Mathematics University of California San Diego The University of Edinburgh La Jolla CA 92093-0112 Edinburgh Eh9 3JZ UK ebender@ Zhicheng Gao t School of Mathematics and Statistics Carleton University Ottawa Ontario K1S5B6 zgao@ L. Bruce Richmond and Nicholas Wormald Department of Combinatorics and Optimization University of Waterloo Waterloo Ontario N2L3G1 lbrichmond@ nwormald@ Submitted Apr 7 2009 Accepted Dec 14 2009 Published Jan 5 2010 Abstract We study the asymptotic behavior of the terms in sequences satisfying recurrences of the form an an-1 Efc 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 Painleve 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 1 Research supported by NSERC Research supported by NSERC and Canada Research Chair Program THE ELECTRONIC JOURNAL OF COMBINATORICS 17 2010 R1 1 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 Dn as n TO. 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 n d an an 1 f