this notation. Tuples of non-negative integers will be denoted by bold letters $\textbf{t} = (t_1,\ldots,t_L) \in {\mathbb{N}}_0^L$. The notation $|\textbf{t}|$ will be used as a short form for $\sum_{l=1}^L t_l$.

Prerequisites {#sec:prereq}
=============

The first step is to establishing a suitable notion of a power series with a well-behaved asymptotic expansion.

\[def:Fpowerseries\] For given $\alpha,\beta \in {\mathbb{R}}_{>0}$ let ${{\mathbb{R}}[[x]]^{{\alpha}}_{\beta}}$ be the subset of ${\mathbb{R}}[[x]]$, such that $f \in {{\mathbb{R}}[[x]]^{{\alpha}}_{\beta}}$ if and only if there exists a sequence of real numbers $(c_k^f)_{k\in {\mathbb{N}}_0}$, which fulfills $$\begin{aligned}
 \label{eqn:basic_asymp_C} f_n &= \sum _{k=0}^{R-1} c_k^f \alpha^{n+\beta-k} \Gamma(n+\beta-k) + {\mathcal{O}}\left(\alpha^{n}\Gamma(n+\beta-R)\right) && \forall R \in {\mathbb{N}}_0. \end{aligned}$$

${{\mathbb{R}}[[x]]^{{\alpha}}_{\beta}}$ is a linear subspace of ${\mathbb{R}}[[x]]$.

The expression in eq. represents an asymptotic expansion or Poincaré expansion with the asymptotic scale $\alpha^{n}\Gamma(n+\beta)$ [@bruijn1970asymptotic Ch. 1.5].

For fixed $R$, an expansion as above with $R$ explicit summands will be called an asymptotic expansion up to order $R-1$.

The subspace ${{\mathbb{R}}[[x]]^{\alpha}_{\beta}}$ includes all (real) power series whose coefficients only grow exponentially: ${\mathbb{R}}\{x\} \subset {{\mathbb{R}}[[x]]^{\alpha}_{\beta}}$.

These with other series, which are in ${o}(\alpha^{n}\Gamma(n+\beta-R))$ for all fixed $R \geq 0$, have an asymptotic expansion of the form in eq.  with all the $c_k^f=0$.

Definition \[def:Fpowerseries\] implies that $f_n \in {\mathcal{O}}\left(\alpha^{n}\Gamma(n+\beta)\right)$. Accordingly, the 