Cyril Banderier ; Michael Drmota - Coefficients of algebraic functions: formulae and asymptotics

dmtcs:2366 - Discrete Mathematics & Theoretical Computer Science, January 1, 2013, DMTCS Proceedings vol. AS, 25th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2013) - https://doi.org/10.46298/dmtcs.2366
Coefficients of algebraic functions: formulae and asymptoticsConference paper

Authors: Cyril Banderier ORCID1; Michael Drmota 2

  • 1 Laboratoire d'Informatique de Paris-Nord
  • 2 Institut für Diskrete Mathematik und Geometrie [Wien]

This paper studies the coefficients of algebraic functions. First, we recall the too-little-known fact that these coefficients fn have a closed form. Then, we study their asymptotics, known to be of the type fnCAnnα. When the function is a power series associated to a context-free grammar, we solve a folklore conjecture: the appearing critical exponents α can not be 1/3 or 5/2, they in fact belong to a subset of dyadic numbers. We extend what Philippe Flajolet called the Drmota-Lalley-Woods theorem (which is assuring α=3/2 as soon as a "dependency graph" associated to the algebraic system defining the function is strongly connected): We fully characterize the possible critical exponents in the non-strongly connected case. As a corollary, it shows that certain lattice paths and planar maps can not be generated by a context-free grammar (i.e., their generating function is not $\mathbb{N}-algebraic). We end by discussing some extensions of this work (limit laws, systems involving non-polynomial entire functions, algorithmic aspects).


Volume: DMTCS Proceedings vol. AS, 25th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2013)
Section: Proceedings
Published on: January 1, 2013
Imported on: November 21, 2016
Keywords: analytic combinatorics,generating function,algebraic function,singularity analysis,context-free grammars,critical exponent,non-strongly connected positive systems,Gaussian limit laws,N-algebraic function,[INFO.INFO-DM]Computer Science [cs]/Discrete Mathematics [cs.DM]

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