Asymptotics of multivariate sequences, part I: smooth points of the singular variety

Robin Pemantle and Mark C. Wilson, Journal of Combinatorial Theory Series A 97 (2002), 129-161.

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Abstract: Given a multivariate generating function $latex F(z_1 , ldots , z_d) = sum a_{r_1 , ldots , r_d} z_1^{r_1} cdots z_d^{r_d}$, we determine asymptotics for the coefficients. Our approach is to use Cauchy’s integral formula near singular points of $latex F$, resulting in a tractable oscillating integral. This paper treats the case where the singular point of $latex F$ is a smooth point of a surface of poles. Companion papers will treat singular points of $latex F$ where the local geometry is more complicated, and for which other methods of analysis are not known.