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

Abstract

Let $F(x)= \sum_{\nu\in\NN^d} F_\nu x^\nu$ 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. We derive asymptotics for the coefficients $F_{r\alpha}$ as $r \to \infty$ with $r\alpha \in \NN^d$ for $\alpha$ in a permissible subset of $d$-tuples of positive reals. More specifically, we give an algorithm for computing arbitrary terms of the asymptotic expansion for $F_{r\alpha}$ when the asymptotics are controlled by a transverse multiple point of the analytic variety $H = 0$. This improves upon earlier work by R. Pemantle and M. C. Wilson. We have implemented our algorithm in Sage and apply it to obtain accurate numerical results for several rational combinatorial generating functions.