Abstract
In this paper, we use the multivariate analytic techniques of Pemantle and Wilson to find asymptotic for- mulae for the coefficients of a broad class of multivariate generating functions with algebraic singularities. Flajolet and Odlyzko (1990) analyzed the coefficients of a class of univariate generating functions with algebraic singularities. These results have been extended to classes of multivariate generating functions by Gao and Richmond (1992) and Hwang (1996, 1998), in both cases by immediately reducing the multivariate case to the univariate case. Pemantle and Wilson (2013) outlined new multivariate analytic techniques and used them to analyze the coefficients of rational generating functions.
Highlights
Generating functions are a powerful, convenient tool to encode an array of numbers into a single function.Given that a generating function can be computed with limited information about the corresponding array, it is often desirable to learn more about the array from the generating function itself
A useful goal is to approximate the coefficients of a generating function asymptotically as their indices grow
In [PW13], Pemantle and Wilson outline a program which greatly extends the results of previous work on multivariate generating function analysis
Summary
Generating functions are a powerful, convenient tool to encode an array of numbers into a single function. A useful goal is to approximate the coefficients of a generating function asymptotically as their indices grow. Researchers have derived asymptotic expansions for the coefficients of many classes of univariate and multivariate generating functions through singularity analysis. In 1990, Flajolet and Odlyzko found asymptotics for a large class of univariate functions with algebraic singularities by using the Cauchy integral formula and explicit contour manipulations. When researchers first extended these results to classes of bivariate functions, they relied on temporarily fixing a variable and applying univariate results, which required special restrictions on the bivariate functions. Pemantle and Wilson begin with the multivariate Cauchy integral formula and manipulate it directly.
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