Formulae and Asymptotics for Coefficients of Algebraic Functions

Abstract

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...

Graph limits of random graphs from a subset of connected k‐trees

“…In the periodic case the coefficients [ z n ]B k ( z ) are positive precisely for n = 1 mod d , where . In both cases we obtain asymptotics for [ z n ]B k ( z ), see Lemma 8 and Theorem 3 in . So for the sake of simplicity, we assume that we are working in the aperiodic case .…”

“…In principle we have to distinguish between the aperiodic case, where all coefficients of B k (z) are positive and k,Ω is the only singularity on the radius of convergence |z| = k,Ω , and the periodic case, where B k (z) has several singularities on its radius |z| = k,Ω , see [9]. In the periodic case the coefficients [z n ]B k (z) are positive precisely for n = 1 mod , where = gcd(Ω out ).…”

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“…The automata serves for us as a bridge between the formal language theory and theory of computing on one side and the asymptotic theory of algebraic functions on the other. See, for instance, [4,16] and references therein for background.…”

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“…There is no known criterion for a Z-algebraic series with coefficients in N to be N-algebraic but there are some necessary conditions on the asymptotic behavior of the coefficients (see the Drmota-Lalley-Woods Theorem in [17, VII.6.1] and recent insights from Banderier and Drmota in [3,4]). …”