UUUWe dedicate this article to the memory of Philippe Flajolet, who was and will remain a guide and a wonderful source of inspiration for so many of us. UUU[ This article will appear in Combinatorics, Probability, and Computing, in the special volume dedicated to Philippe Flajolet. ] Cyril Banderier, CNRS/Univ. Paris 13, Villetaneuse (France). Cyril.Banderier at lipn.univ-paris13.fr, http://lipn.univ-paris13.fr/∼banderier Michael Drmota, TU Wien (Austria). drmota at dmg.tuwien.ac.at, http://dmg.tuwien.ac.at/drmota/ Date: March 22, 2013 (revised March 22, 2014. Key words and phrases. analytic combinatorics, generating function, algebraic function, singularity analysis, context-free grammars, critical exponent, non-strongly connected positive systems, Gaussian limit laws, N-algebraic function.
2 FORMULAE AND ASYMPTOTICS FOR COEFFICIENTS OF ALGEBRAIC FUNCTIONSAbstract. We study the coefficients of algebraic functions n≥0 fnz n . 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 ∼ CA n n α . When the function is a power series associated to a contextfree 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 behaviors 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).Résumé. Cet article a pour héros les coefficients des fonctions algébriques. Après avoir rappelé le fait trop peu connu que ces coefficients fn admettent toujours une forme close, nousétudions leur asymptotique fn ∼ CA n n α . Lorsque la fonction algébrique est la série génératrice d'une grammaire noncontextuelle, nous résolvons une vieille conjecture du folklore : les exposants critiques α ne peuvent pasêtre 1/3 ou −5/2 et sont en fait restreintsà un sousensemble des nombres dyadiques. Nous amorçons aussi l'étude de l'ensemble des valeurs possibles pour A. Nousétendons ce que Philippe Flajolet appelait le théorème de Drmota-Lalley-Woods (qui affirme que α = −3/2 dès lors qu'un "graphe de dépendance" associé au sys...