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.
Highlights
This article1 is concerned with real sequences fn, which admit an asymptotic expansion for large n of the form, fn = αn+βΓ(n + β) α(n c1 +β c2 1)(n for some α ∈ R>0, β ∈ R and ck ∈ R
Bender’s results are extended into a complete algebraic framework. This is achieved by making heavy use of generating functions in the spirit of the ‘analytic combinatorics’ or ‘symbolic method’ approach [20, 10, 28]
The key step is to interpret the coefficients of the asymptotic expansion as another power series
Summary
The electronic journal of combinatorics 25(4) (2018), #P4.1 there are countless examples where perturbative expansions of physical quantities admit asymptotic expansions of this kind [5, 24, 17] The restriction to this specific class of power series is inspired by the work of Bender. Resurgence provides a promising approach to cope with divergent perturbative expansions in physics Its application to these problems is an active field of research [2, 17, 3]. The formalism can be seen as a toy model of resurgence’s calcul differentieletranger [18, Vol 1] called alien calculus [25, II.6] This toy model is unable to fully reconstruct functions from asymptotic expansions, but does not rely on analytic properties of Borel transformed functions and offers itself for combinatorial applications. A detailed and illuminating account on resurgence theory is given in Sauzin’s review [25, Part II]
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