constant $C$ such that $|f_n| \leq C \alpha^n \Gamma(n+\beta)$ and $|g_n| \leq C \alpha^n \Gamma(n+\beta)$ for all $n \geq 0$. Hence, $$\begin{aligned}
 \sum_{m=R}^{n-R} |f_m g_{n-m}| \leq C^2 \alpha^n \sum_{m=R}^{n-R}\Gamma(m+\beta)\Gamma(n-m+\beta) \qquad \forall n \geq 2R \end{aligned}$$ shows that this sum is in ${\mathcal{O}}(\alpha^n \Gamma(n-R+\beta))$ by Corollary \[crll:centersum\].

\[crll:chain\_for\_product\] If $g^{1}, \ldots, g^{L} \in {{\mathbb{R}}[[x]]^{\alpha}_{\beta}}$, then $$\begin{aligned}
 \left( {\mathcal{A}}^\alpha_\beta \left( \prod_{l=1}^L g^{l}(x) \right)\right)(x) = \sum_{l=1}^L \left(\prod_{\substack{m=1 \\ m\neq l}}^L g^{m}(x)\right) ( {\mathcal{A}}^\alpha_\beta g^{l})(x) . \end{aligned}$$

Proof by induction on $L$ using the Leibniz rule.

\[crll:chain\_for\_product\_pow\] If $g^{1}, \ldots, g^{L} \in {{\mathbb{R}}[[x]]^{\alpha}_{\beta}}$ and $\textbf{t} = (t_1, \ldots, t_L) \in {\mathbb{N}}_0^L$, then $$\begin{aligned}
 \left( {\mathcal{A}}^\alpha_\beta \left( \prod_{l=1}^L (g^{l}(x))^{t_l} \right)\right)(x) = \sum_{l=1}^L t_l (g^l(x))^{t_l-1} \left(\prod_{\substack{m=1 \\ m\neq l}}^L (g^{m}(x))^{t_m}\right) ( {\mathcal{A}}^\alpha_\beta g^{l})(x) . \end{aligned}$$

\[crll:polynomial\_comp\] If $g^{1}, \ldots, g^{L} \in {{\mathbb{R}}[[x]]^{\alpha}_{\beta}}$ and $p\in {\mathbb{R}}[y_1,\ldots,y_L]$ is polynomial in $L$ variables, then $h(x) = p(g^{1}(x), \ldots, g^{L}(x))\in{{\mathbb{R}}[[x]]^{\alpha}_{\beta}}$ and $$\begin{aligned}
 \label{eqn:polynomial_comp} ({\mathcal{A}}^\alpha_\beta h)(x) = ({\mathcal{A}}^\alpha_\beta (p(g^{1}, \ldots, g^{L})))(x) = \sum_{l=1}^L \frac{\partial p}{\partial g^l} (g^{1}, \ldots, g^{L}) ({\mathcal{A}}^\alpha_\beta g^{l})(x). \end{aligned}$$

Although the last three statements are only basic general properties of commutative derivation rings, they suggest that ${\mathcal{A}}^\alpha_\beta$ fulfills a simple chain rule. In fact, Corollary \[crll:polynomial\_comp\] can still be generalized from polynomials to analytic functions, but, as already 