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{\bf Alexander Raichev and Mark C. Wilson }
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{\bf Asymptotics of Coefficients of Multivariate Generating Functions: Improvements for Smooth Points}
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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).

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