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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≥0f nzn. First, we recall the too-little-known fact that these coefficientsf n always admit a closed form. Then we study their asymptotics, known to beof the type f n ~ CA nn α. When the function is a power seriesassociated to a context-free grammar, we solve a folklore conjecture: thecritical exponents α cannot be 1/3 or −5/2; they in factbelong to a proper subset of the dyadic numbers. We initiate the study of theset of possible values for A. We extend what Philippe Flajoletcalled the Drmota–Lalley–Woods theorem (which states thatα=−3/2 when the dependency graph associated to thealgebraic system defining the function is strongly connected). We fullycharacterize the possible singular behaviours in the non-strongly connectedcase. As a corollary, the generating functions of certain lattice paths andplanar maps are not determined by a context-free grammar (i.e.,their generating functions are not ℕ-algebraic). We give examples ofGaussian limit laws (beyond the case of theDrmota–Lalley–Woods theorem), and examples of non-Gaussianlimit laws. We then extend our work to systems involving non-polynomial entirefunctions (non-strongly connected systems, fixed points of entire functions withpositive coefficients). We give several closure properties forℕ-algebraic functions. We end by discussing a few extensions of ourresults (infinite systems of equations, algorithmic aspects).

Keywords

Primary 05A1505A16Secondary 68Q4568R15

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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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