Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: Asymptotics of coefficients of multivariate generating functions: improvements for smooth points. | Asymptotics of coefficients of multivariate generating functions improvements for smooth points Alexander Raichev and Mark C. Wilson Department of Computer Science University of Auckland Private Bag 92019 Auckland New Zealand raichev mcw @ Submitted May 20 2008 Accepted Jul 4 2008 Published Jul 14 2008 Mathematics Subject Classihcation 05A15 05A16 Keywords higher-order terms multivariate singularity analysis Abstract Let 523eNd F3x3 be a multivariate power series. For example 52 F3x3 could be a generating function for a combinatorial class. Assume that in a neighbourhood of the origin this series represents a nonentire function F G Hp where G and H are holomorphic and p is a positive integer. Given a direction a 2 N for which the asymptotics are controlled by a smooth point of the singular variety H 0 we compute the asymptotics of Fna as n 1. We do this via multivariate singularity analysis and give an explicit uniform formula for the full asymptotic expansion. This improves on earlier work of R. Pemantle and the second author and allows for more accurate numerical approximation as demonstrated by our our examples on lattice paths quantum random walks and nonoverlapping patterns . 1 Introduction Let 523gNd Fjx3 be a multivariate power series. For example 52 Fjx3 could be a generating function for a combinatorial class. In PW02 PW04 Pemantle and Wilson derived asymptotic expansions for the coefficients Fj as d 1 for large classes of series that arise often in applications. In this article we further their program of asymptotics of coefficients of multivariate generating functions. Assume that in a neighbourhood of the origin the power series 52 1 F3X3 is the Maclaurin series of a nonentire function F G Hp where G and H are holomorphic and p is a positive integer. For example F could be a rational function. Using multivariate singularity analysis we derive the asymptotics of Fna for a 2 Nị and n 1 in the case that these asymptotics are controlled by
