Asymptotics of coefficients of multivariate generating functions: Improvements for smooth points

Alexander Raichev ; Mark C. Wilson

The electronic journal of combinatorics, Tome 15 (2008)

Résumé

Let $\sum_{\beta\in{\Bbb N}^d} F_\beta x^\beta$ be a multivariate power series. For example $\sum F_\beta x^\beta$ could be a generating function for a combinatorial class. Assume that in a neighbourhood of the origin this series represents a nonentire function $F=G/H^p$ where $G$ and $H$ are holomorphic and $p$ is a positive integer. Given a direction $\alpha\in{\Bbb N}_+^d$ for which the asymptotics are controlled by a smooth point of the singular variety $H = 0$, we compute the asymptotics of $F_{n \alpha}$ as $n\to\infty$. 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).

Détail

DOI : 10.37236/813

Classification : 05A15, 05A16
Mots-clés : higher-order terms, multivariate singularity analysis, multivariate power series, full asymptotic expansion