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Alexander Raichev

Publications and source records attributed to Alexander Raichev.

5 recordsLinked to original sources
arXiv2012

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

Let $F(x)= \sum_{ν\in\NN^d} F_νx^ν$ 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 $F_{rα}$ as $r \to \infty$ with $rα\in \NN^d$ for $α$ 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 $F_{rα}$ 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.

math.CO↗
arXiv2008

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

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.

math.CO↗