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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