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Formulae and Asymptotics for Coefficients of Algebraic Functions

Published online by Cambridge University Press:  11 December 2014

CYRIL BANDERIER  and
MICHAEL DRMOTA
CYRIL BANDERIER
Affiliation:
CNRS – Université Paris 13, 93430 Villetaneuse, France (e-mail: Cyril.Banderier@lipn.univ-paris13.fr, http://lipn.univ-paris13.fr/~banderier)
MICHAEL DRMOTA
Affiliation:
Institut für Diskrete Mathematik und Geometrie, TU Wien, A1040 Wien, Austria (e-mail: michael.drmota@tuwien.ac.at, http://dmg.tuwien.ac.at/drmota/)
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Abstract

We study the coefficients of algebraic functions ∑n≥0fnzn. First, we recall the too-little-known fact that these coefficients fn always admit a closed form. Then we study their asymptotics, known to be of the type fn ~ CAnnα. When the function is a power series associated to a context-free grammar, we solve a folklore conjecture: the critical exponents α cannot be 1/3 or −5/2; they in fact belong to a proper subset of the dyadic numbers. We initiate the study of the set of possible values for A. We extend what Philippe Flajolet called the Drmota–Lalley–Woods theorem (which states that α=−3/2 when the dependency graph associated to the algebraic system defining the function is strongly connected). We fully characterize the possible singular behaviours in the non-strongly connected case. As a corollary, the generating functions of certain lattice paths and planar maps are not determined by a context-free grammar (i.e., their generating functions are not ℕ-algebraic). We give examples of Gaussian limit laws (beyond the case of the Drmota–Lalley–Woods theorem), and examples of non-Gaussian limit laws. We then extend our work to systems involving non-polynomial entire functions (non-strongly connected systems, fixed points of entire functions with positive coefficients). We give several closure properties for ℕ-algebraic functions. We end by discussing a few extensions of our results (infinite systems of equations, algorithmic aspects).

Keywords

Primary 05A1505A16Secondary 68Q4568R15
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 24 , Special Issue 1: Honouring the Memory of Philippe Flajolet - Part 3 , January 2015 , pp. 1 - 53
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Copyright © Cambridge University Press 2014 

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