power series in ${{\mathbb{R}}[[x]]^{\alpha}_{\beta}}$ are a subset of *Gevrey-1* sequences [@hsieh2012basic Ch XI-2]. Being *Gevrey-1* is not sufficient for a power series to be in ${{\mathbb{R}}[[x]]^{\alpha}_{\beta}}$. For instance, a sequence which behaves for large $n$ as $a_n \sim n! (1+ \frac{1}{\sqrt{n}}+ {\mathcal{O}}(\frac{1}{n}))$ is *Gevrey-1*, but not in ${{\mathbb{R}}[[x]]^{\alpha}_{\beta}}$.

In resurgence theory further restrictions on the allowed power series are imposed, which ensure that the Borel transformations of the sequences have proper analytic continuations or are ‘endless continuable’ [@mitschi2016divergent II.6]. These restrictions are analogous to the requirement that, apart from $f_n$, also $c_k$ has to have a well-behaved asymptotic expansion. The coefficients of this asymptotic expansion are also required to have a well-behaved asymptotic expansion and so on. These kind of restrictions will not be necessary for the presented algebraic considerations, which are aimed at combinatorial applications.

The central theme of this article is to *interpret the coefficients $c_k^f$ of the asymptotic expansion as another power series*. In fact, Definition \[def:Fpowerseries\] immediately suggests to define the following map:

\[def:basic\_asymp\_definition\] Let ${\mathcal{A}}^\alpha_\beta: {{\mathbb{R}}[[x]]^{\alpha}_{\beta}} \rightarrow {\mathbb{R}}[[x]]$ be the map that associates a power series ${\mathcal{A}}^\alpha_\beta f \in {\mathbb{R}}[[x]]$ to every power series $f \in {{\mathbb{R}}[[x]]^{\alpha}_{\beta}}$ such that, $$\begin{aligned}
 \label{eqn:basic_asymp} f_n &= \sum _{k=0}^{R-1} 