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A hybrid of Darboux’s method and singularity analysis in combinatorial asymptotics. (English) Zbl 1111.05006
Summary: A “hybrid method”, dedicated to asymptotic coefficient extraction in combinatorial generating functions, is presented, which combines Darboux’s method and singularity analysis theory. This hybrid method applies to functions that remain of moderate growth near the unit circle and satisfy suitable smoothness assumptions – this, even in the case when the unit circle is a natural boundary. A prime application is to coefficients of several types of infinite product generating functions, for which full asymptotic expansions (involving periodic fluctuations at higher orders) can be derived. Examples relative to permutations, trees, and polynomials over finite fields are treated in this way.

05A16 Asymptotic enumeration
05A15 Exact enumeration problems, generating functions
30B10 Power series (including lacunary series) in one complex variable
33B30 Higher logarithm functions
40E10 Growth estimates
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