mentioned, this intuition turns out to be false in general.

Composition {#sec:comp}
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Composition by analytic functions
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\[thm:chain\_analytic\] If $g^{1}, \ldots, g^{L} \in {{\mathbb{R}}[[x]]^{\alpha}_{\beta}}$ with each $g_0^{l}=0$ and $f \in {\mathbb{R}}\{y_1,\ldots,y_L\}$, a function in $L$ variables, which is analytic at the origin, then $h(x) = f(g^{1}(x), \ldots, g^{L}(x)) \in {{\mathbb{R}}[[x]]^{\alpha}_{\beta}}$ and $$\begin{aligned}
 \label{eqn:chain_analytic} ({\mathcal{A}}^\alpha_\beta h)(x) = ({\mathcal{A}}^\alpha_\beta (f(g^{1}, \ldots, g^{L})))(x) = \sum_{l=1}^L \frac{\partial f}{\partial g^l} (g^{1}, \ldots, g^{L}) ({\mathcal{A}}^\alpha_\beta g^{l})(x). \end{aligned}$$

In [@bender1975asymptotic] Edward Bender established this theorem for the case $L=1$ in a less ‘generatingfunctionology’ biased notation. If for example $g\in {{\mathbb{R}}[[x]]^{\alpha}_{\beta}}$ and $f \in {\mathbb{R}}\{x,y\}$, then his Theorem 1 allows us to calculate the asymptotics of the power series $f(g(x),x)$. In fact, Bender analyzed more general power series including sequences with even more rapid than factorial growth.

The following proof of Theorem \[eqn:chain\_analytic\] is a straightforward generalization of Bender’s Lemma 2 and Theorem 1 in [@bender1975asymptotic] to the multivariate case $f \in {\mathbb{R}}\{y_1,\ldots,y_L\}$.

\[lmm:prod\_hard\_estimate\] If $g^{1}, \ldots, g^{L} \in {{\mathbb{R}}[[x]]^{\alpha}_{\beta}}$, then there exists a constant $C \in {\mathbb{R}}$ such that $$\begin{aligned}
 \left| [x^n] \prod_{l=1}^L \left( g^l(x) \right)^{t_l} \right| &\leq C^{|\textbf{t}|} \alpha^n \Gamma(n+\beta) && \forall \textbf{t} \in {\mathbb{N}}_0^L, |\textbf{t}| \geq 1 \text{ and } n \in {\mathbb{N}}_0. \end{aligned}$$

