{{\mathbb{R}}[[x]]^{\alpha}_{{\beta+1}}} \subset {{\mathbb{R}}[[x]]^{\alpha}_{{\beta+2}}} \subset \ldots \subset {{\mathbb{R}}[[x]]^{\alpha}_{{\beta+m}}} \subset \ldots \end{aligned}$$

The requirement $\beta > 0$ is therefore only a spurious restriction. The ideal $x^m {{\mathbb{R}}[[x]]^{\alpha}_{\beta}}$ can be associated with ${{\mathbb{R}}[[x]]^{\alpha}_{{\beta-m}}}$. An alternative way to think about ${{\mathbb{R}}[[x]]^{\alpha}_{\beta}}$ with $\beta \leq 0$ is to use the field of (formal) Laurent series ${\mathbb{R}}((x))$ as the target space for ${\mathcal{A}}^\alpha_\beta$ and demand that negative powers of $x$ commute with the ${\mathcal{A}}^\alpha_\beta$ operator.

A derivation for asymptotics
============================

The following lemma forms the foundation for most conclusions in this article. It provides an estimate for sums over $\Gamma$ functions. Moreover, it ensures that the subspace ${{\mathbb{R}}[[x]]^{\alpha}_{\beta}}$ of formal power series falls into a large class of sequences studied by Bender [@bender1975asymptotic]. From another perspective the lemma can be seen as an entry point to resurgence, which bypasses the necessity for analytic continuations and Borel transformations.

\[lmm:gammasum\] If $\beta \in {\mathbb{R}}_{>0}$, then there exists a $C\in {\mathbb{R}}$ such that $$\begin{aligned}
 \label{eqn:centersumexpl2} \sum_{m=0}^{n} \Gamma(m+\beta) \Gamma(n-m+\beta) &\leq C \Gamma(n+\beta) && \forall n \in {\mathbb{N}}_0. \end{aligned}$$

Recall that $\Gamma$ is a *log-convex* function in ${\mathbb{R}}_{>0}$. The functions $\Gamma(m+\beta)$ and $\Gamma(n-m+\beta)$ are also log-convex functions in $m$ on the interval $[0,n]$, as log-convexity is preserved under shifts and reflections. 