Asymptotics of multivariate sequences II: Multiple points of the singular variety

Abstract

Let F(z) = Σr arzr be a multivariate generating function that is meromorphic in some neighbourhood of the origin of ℂd, and let V be its set of singularities. Effective asymptotic expansions for the coefficients can be obtained by complex contour integration near points of V. In the first article in this series, we treated the case of smooth points of V. In this article we deal with multiple points of V. Our results show that the central limit (Ornstein-Zernike) behaviour typical of the smooth case does not hold in the multiple point case. For example, when V has a multiple point singularity at (1,....,1), rather than ar decaying as |r|-1/2 as |r|, ar is very nearly polynomial in a cone of directions.