Asymptotics of Multivariate Sequences, part I. Smooth points of the singular variety

Given a multivariate generating function F, 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 singular points of F where the local geometry is more complicated, and for which other methods of analysis are not known.

[1]  Roderick Wong,et al.  Asymptotic approximations of integrals , 1989, Classics in applied mathematics.

[2]  Edward A. Bender,et al.  Multivariate Asymptotics for Products of Large Powers with Applications to Lagrange Inversion , 1999, Electron. J. Comb..

[3]  R. Lyons,et al.  Coalescing Particles on an Interval , 1999 .

[4]  Bruno Salvy,et al.  Non-Commutative Elimination in Ore Algebras Proves Multivariate Identities , 1998, J. Symb. Comput..

[5]  Hsien-Kuei Hwang,et al.  On Convergence Rates in the Central Limit Theorems for Combinatorial Structures , 1998, Eur. J. Comb..

[6]  Hsien-Kuei Hwang,et al.  LARGE DEVIATIONS OF COMBINATORIAL DISTRIBUTIONS II. LOCAL LIMIT THEOREMS , 1998 .

[7]  Philippe Flajolet,et al.  The Average Case Analysis of Algorithms : Multivariate Asymptotics and Limit Distributions , 1997 .

[8]  Edward A. Bender,et al.  Admissible Functions and Asymptotics for Labelled Structures by Number of Components , 1996, Electron. J. Comb..

[9]  J. Propp,et al.  Local statistics for random domino tilings of the Aztec diamond , 1996, math/0008243.

[10]  Hsien-Kuei Hwang,et al.  Large deviations for combinatorial distributions. I. Central limit theorems , 1996 .

[11]  Hsien-Kuei Kwang,et al.  Asymptotic Expansions for the Stirling Numbers of the First Kind , 1995, J. Comb. Theory, Ser. A.

[12]  Timothy S. Murphy,et al.  Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals , 1993 .

[13]  Andrea L. Bertozzi,et al.  Multidimensional Residues, Generating Functions, and Their Application to Queueing Networks , 1993, SIAM Rev..

[14]  Philippe Flajolet,et al.  General combinatorial schemas: Gaussian limit distributions and exponential tails , 1993, Discret. Math..

[15]  Zhicheng Gao,et al.  Central and local limit theorems applied to asymptotic enumeration IV: multivariate generating functions , 1992 .

[16]  Philippe Flajolet,et al.  Singularity Analysis of Generating Functions , 1990, SIAM J. Discret. Math..

[17]  L. Lipshitz,et al.  The diagonal of a D-finite power series is D-finite , 1988 .

[18]  J. T. Chayes,et al.  Ornstein-Zernike behavior for self-avoiding walks at all noncritical temperatures , 1986 .

[19]  Edward A. Bender,et al.  Central and Local Limit Theorems Applied to Asymptotic Enumeration II: Multivariate Generating Functions , 1983, J. Comb. Theory, Ser. A.

[20]  Henry C. Thacher,et al.  Applied and Computational Complex Analysis. , 1988 .

[21]  E. Bender.  Asymptotic Methods in Enumeration , 1974 .

[22]  L. Hörmander,et al.  An introduction to complex analysis in several variables , 1973 .

[23]  Edward A. Bender,et al.  Central and Local Limit Theorems Applied to Asymptotic Enumeration , 1973, J. Comb. Theory A.

[24]  D. Klarner,et al.  The diagonal of a double power series , 1971 .

[25]  Harry Furstenberg,et al.  Algebraic functions over finite fields , 1967 .

[26]  M. Fisher.  Statistical Mechanics of Dimers on a Plane Lattice , 1961 .

[27]  G. Fischer,et al.  Plane Algebraic Curves , 1921, Nature.