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

TL;DR

This paper advances the asymptotic analysis of multivariate generating functions near smooth singular points, providing explicit formulas for asymptotics and improving numerical approximations in combinatorial contexts.

Contribution

It introduces refined multivariate singularity analysis techniques for smooth points, yielding explicit asymptotic formulas and enhanced accuracy over previous methods.

Findings

Derived explicit asymptotic formulas for coefficients in multivariate generating functions.

Improved numerical approximation accuracy demonstrated through examples.

Extended previous work by Pemantle and the second author to broader cases.

Abstract

Let $\sum_{\beta\in\nats^d} F_\beta x^\beta$ 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 $\alpha\in\pnats^d$ 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 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 examples.