by $C' \Gamma(m+\beta)$ for all $m\in {\mathbb{N}}_0$. Corollary \[crll:centersum\] ensures that $\sum_{m=R}^{n-R} C^m P(m) \Gamma(n-m+\beta) \in {\mathcal{O}}(\Gamma(n-R+\beta))$. The remainder $\sum_{m=n-R+1}^{n} C^m P(m) \Gamma(n-m+\beta)= \sum_{m=0}^{R-1} C^{n-m} P(n-m) \Gamma(m+\beta)$ is obviously in ${\mathcal{O}}(\Gamma(n-R+\beta))$.

\[crll:iteratedgammasum\] If $\beta \in {\mathbb{R}}_{>0}$, then there exists a $C\in {\mathbb{R}}$ such that $$\begin{aligned}
 \sum_{\substack{\textbf{m} \in {\mathbb{N}}_0^L\\ |\textbf{m}| = n}} \prod_{l=1}^L \Gamma(m_l+\beta) \leq C^{L} \Gamma(n+\beta) & & \forall n,L \in {\mathbb{N}}_0 \text{ with } L \geq 1, \end{aligned}$$ where $\substack{\textbf{m} \in {\mathbb{N}}_0^L\\ |\textbf{m}| = n}$ denotes the simplex $\{(m_1,\ldots, m_L) \in {\mathbb{N}}_0^L | \sum_{l=1}^L m_l = n\}$.

This inequality is merely an iterated version of Lemma \[lmm:gammasum\]. It can be proven by induction in $L$. The case $L=1$ is trivial. Lemma \[lmm:gammasum\] guarantees the existence of a $C \in {\mathbb{R}}$ such that $$\begin{aligned}
 C^{L}\sum_{m_{L+1}=0}^n \Gamma(m_{L+1}+\beta) \Gamma(n-m_{L+1}+\beta) &\leq C^{L+1} \Gamma(n+\beta) && \forall n,L \in {\mathbb{N}}_0. \end{aligned}$$ Suppose the statement holds for $L$. Using the statement on $C^{L} \Gamma(n-m_{L+1}+\beta)$ on the left hand side results in $$\begin{aligned}
 &\sum_{m_{L+1}=0}^n \sum_{\substack{\textbf{m} \in {\mathbb{N}}_0^L\\ |\textbf{m}| = n-m_{L+1}}} \prod_{l=1}^{L+1} \Gamma(m_l+\beta) \leq C^{L+1} \Gamma(n+\beta) && \forall n \in {\mathbb{N}}_0, \end{aligned}$$ which is the statement for $L+1$.

An immediate consequence of Corollary \[crll:centersum\] is

\[prop:derivation\] ${{\mathbb{R}}[[x]]^{\alpha}_{\beta}}$ forms a subring of ${\mathbb{R}}[[x]]$: If $f,g\in {{\mathbb{R}}[[x]]^{\alpha}_{\beta}}$, then $f(x)g(x) \in 