Asymptotics of Multivariate Sequences II: Multiple Points of the Singular Variety

Asymptotics of Multivariate Sequences, Part I: Smooth Points of the Singular Variety

Given a multivariate generating function F (z1, . . . , zd) = ∑ ar1,... ,rdz r1 1 · · · z rd d , we determine asymptotics for the coefficients. Our approach is to use Cauchy’s integral formula near singular points of F , resulting in a tractable oscillating integral. This paper treats the case where the singular point of F is a smooth point of a surface of poles. Companion papers will treat sin...

Asymptotics of Multivariate Sequences , Part I : Smooth Points Ofthe Singular

O / 0 40 60 22 v 1 1 J un 2 00 4 ASYMPTOTICS OF MULTIVARIATE SEQUENCES II . MULTIPLE POINTS OF THE SINGULAR VARIETY

Let F (z) = ∑ r arz r be a multivariate generating function which is meromorphic in some neighborhood of the origin of C, and let V be its set of singularities. Effective asymptotic expansions for the coefficients can be obtained by complex contour integration near points of V. In the first article in this series, we treated the case of smooth points of V. In this article we deal with multiple ...

Asymptotics of Coefficients of Multivariate Generating Functions: Improvements for Multiple Points

Let F (x) = ∑ ν∈Nd Fνx ν be a multivariate power series with complex coefficients that converges in a neighborhood of the origin. Assume F = G/H for some functions G and H holomorphic in a neighborhood of the origin. For example, F could be a rational combinatorial generating function. We derive asymptotics for the ray coefficients Fnα as n → ∞ for α in a permissible subset of d-tuples of posit...

Asymptotics of coe cients of multivariate generating functions: improvements for multiple points

Let F (x) = P ⌫2Nd F⌫x ⌫ be a multivariate power series with complex coe cients that converges in a neighborhood of the origin. Assume F = G/H for some functions G and H holomorphic in a neighborhood of the origin. We derive asymptotics for the coe cients Fr↵ as r ! 1 with r↵ 2 Nd for ↵ in a permissible subset of d-tuples of positive reals. More specifically, we give an algorithm for computing ...