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Séminaire Lotharingien de Combinatoire, 78B.28 (2017), 12 pp.

# Michael Borinsky

# Generating Asymptotics for Factorially Divergent Sequences

**Abstract.**
The algebraic properties of formal power series with factorial growth
which admit a certain well-behaved asymptotic expansion are
discussed. These series form a subring of
**R**[[*x*]] which is
closed under composition. An "asymptotic derivation" is defined which
maps a power series to its asymptotic expansion. Leibniz and chain
rules for this derivation are deduced. With these rules asymptotic
expansions 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.

Received: November 14, 2016.
Accepted: February 17, 2017.
Final version: April 1, 2017.

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