Asymptotics of coefficients of multivariate generating functions: improvements for multiple points

Details Asymptotics of coefficients of multivariate generating functions: improvements for multiple points Asymptotics of coefficients of multivariate generating functions: improvements for multiple points

2010-09-28

arxiv.org/abs/1009.5715v2

Mathematics

Combinatorics

To appear in Online Journal of Analytic Combinatorics in 2012

Scientific paper

Let $F(x)= \sum_{\nu\in\nats^d} F_\nu x^\nu$ 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 $F_{n \alpha}$ as $n\to\infty$ for $\alpha$ in a permissible subset of $d$-tuples of positive integers. More specifically, we give an algorithm for computing arbitrary terms of the asymptotic expansion for $F_{n\alpha}$ when the asymptotics are controlled by a transverse multiple point of the analytic variety $H = 0$. We have implemented our algorithm in Sage and apply it to several examples. This improves upon earlier work on analytic combinatorics in several variables by R. Pemantle and M. C. Wilson.

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