Asymptotics of multivariate sequences, part III: quadratic points

ec 2 00 8 Asymptotics of multivariate sequences , part III : quadratic points

We consider a number of combinatorial problems in which rational generating functions may be obtained, whose denominators have factors with certain singularities. Specifically, there exist points near which one of the factors is asymptotic to a nondegenerate quadratic. We compute the asymptotics of the coefficients of such a generating function. The computation requires some topological deforma...

Asymptotics of Multivariate Sequences , Part I : Smooth Points Ofthe Singular

Asymptotics of Multivariate Sequences, Part I: Smooth Points of the Singular Variety

Given a multivariate generating function F (z1, . . . , zd) = ∑ ar1,... ,rdz r1 1 · · · z rd d , we determine asymptotics for the coefficients. Our approach is to use Cauchy’s integral formula near singular points of F , resulting in a tractable oscillating integral. This paper treats the case where the singular point of F is a smooth point of a surface of poles. Companion papers will treat sin...

Asymptotics of Multivariate Sequences

Let F (z) = ∑ r arz r be a multivariate generating function which is meromorphic in some neighborhood of the origin of C, 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 ...

Combinatorial Methods in Complex Analysis

The theme of this thesis is combinatorics, complex analysis and algebraic geometry. The thesis consists of six articles divided into four parts. Part A: Spectral properties of the Schrödinger equation This part consists of Papers I-II, where we study a univariate Schrödinger equation with a complex polynomial potential. We prove that the set of polynomial potentials that admits solutions to the...