Generating asymptotics for factorially divergent sequences

TL;DR

This paper explores the algebraic structure of formal power series with factorial growth coefficients, introducing an asymptotic derivation to obtain asymptotic expansions, demonstrated on combinatorial sequences.

Contribution

It establishes that factorially divergent series form a subring closed under composition and inversion, and introduces an asymptotic derivation with product and chain rules.

Findings

Full asymptotic expansions for connected chord diagrams

Asymptotic formulas for simple permutations

Method for implicit power series expansions

Abstract

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.