Coefficients of algebraic functions: formulae and asymptotics

Banderier, Cyril ; Drmota, Michael

Discrete mathematics & theoretical computer science, DMTCS Proceedings vol. AS, 25th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2013), DMTCS Proceedings vol. AS, 25th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2013) (2013).

Résumé

This paper studies the coefficients of algebraic functions. First, we recall the too-little-known fact that these coefficients $f_n$ have a closed form. Then, we study their asymptotics, known to be of the type $f_n \sim C A^n n^{\alpha}$. When the function is a power series associated to a context-free grammar, we solve a folklore conjecture: the appearing critical exponents $\alpha$ 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 $\alpha=^{-3}/_2$ 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).

Détail

DOI : 10.46298/dmtcs.2366