\prod_{l=1}^L \left(g^{l}(x)\right)^{t_l} + \sum_{\substack{\textbf{t} \in {\mathbb{N}}_0^{L} \\ |\textbf{t}| > R}} f_{t_1,\ldots,t_L} \prod_{l=1}^L \left(g^{l}(x)\right)^{t_l} \qquad \forall R\in {\mathbb{N}}_0 . \end{gathered}$$ The left sum is just the composition by a polynomial. Corollary \[crll:polynomial\_comp\] asserts that this sum is in ${{\mathbb{R}}[[x]]^{\alpha}_{\beta}}$. It has the asymptotic expansion given in eq.  which agrees with the right hand side of eq.  up to order $R-1$, because the partial derivative reduces the order of a polynomial by one and $g_l=0$. It is left to prove that the coefficients of the power series given by the right sum are in ${\mathcal{O}}(\alpha^n \Gamma(n-R+\beta))$. Corollary \[crll:power\_prod\_estimate\] and the fact that there is a constant $C$, such that $|f_{t_1,\ldots,t_L}|\leq {C}^{|\textbf{t}|}$ for all $\textbf{t} \in {\mathbb{N}}_0^L$, due to the analyticity of $f$, ensure that there is a constant $C'\in {\mathbb{R}}$ such that $$\begin{gathered}
 \sum_{\substack{\textbf{t} \in {\mathbb{N}}_0^{L} \\ n \geq |\textbf{t}| > R}} \left| f_{t_1,\ldots,t_L} [x^n] \prod_{l=1}^L \left(g^{l}(x)\right)^{t_l} \right| \leq \alpha^{n} \sum_{t=R+1}^n {C'}^{t} \Gamma(n+\beta-t+1) \sum_{\substack{\textbf{t} \in {\mathbb{N}}_0^{L} \\ |\textbf{t}| = t }} 1 , \end{gathered}$$ for all $n \geq R+1$. The result of the last sum $| \{ t_1,\ldots, t_L \in {\mathbb{N}}_0 | t_1 + \ldots + t_L = t \}| = { t + L -1 \choose L-1 }$ is a polynomial 