Generating asymptotics for factorially divergent sequences

Michael Borinsky¹

¹Humboldt-Universität zu Berlin

The electronic journal of combinatorics, Tome 25 (2018) no. 4

Résumé

The algebraic properties of formal power series, whose coefficients show factorial growth and admit a certain well-behaved asymptotic expansion, are discussed. It is shown that these series form a subring of $\mathbb{R}[[x]]$. This subring is also closed under composition and inversion of power series. An `asymptotic derivation' is defined which maps a power series to the asymptotic expansion of its coefficients. Product and chain rules for this derivation are deduced. With these rules asymptotic expansions of the coefficients of implicitly defined power series can be obtained. The full asymptotic expansions of the number of connected chord diagrams and the number of simple permutations are given as examples.

Détail

DOI : 10.37236/5999

Classification : 05A16, 05A05
Mots-clés : asymptotic expansions, formal power series, chord diagrams, simple permutations