Asymptotics of multivariate sequences, II: multiple points of the singular variety

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We consider a multivariate generating function F(z), whose coefficients are indexed by d-tuples of nonnegative integers: F(z) = sumᵣ aᵣ zʳ where zʳ denotes the product of zⱼʳʲ over j = 1,..., d. Suppose that F(z) is meromorphic in some neighborhood of the origin in complex d-space. Let V be the set where the denominator of F vanishes. 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) behavior typical of the smooth case does not hold in the multiple point case. For example, when V has a multiple point singularity at the point (1,..., 1), rather than aᵣ decaying on the order of |r|⁻¹/² as |r| goes to infinity, aᵣ is a polynomial plus a rapidly decaying term.