Suppose we have a function $f(x):\mathbb{R}\rightarrow \mathbb{R}$ which is analytic. We can write its Taylor expansion around $x=0$ as: $f(x) = a_0 + a_1x + a_2x^2 + a_3x^3 + \dots$.

The function $f^{(k)}(x) = f(x)\circ\dots\circ f(x)$, that is the composition of $f(x)$ with itself $k$ times, is also analytic as a composition of analytic functions. So we can also write its Taylor expansion around $x=0$ as $f^{(k)}(x)=b_0 + b_1x + b_2x^2 + b_3x^3 + \dots$.

My question: is there a way to get a close form of the coefficients $b_0,b_1,\dots$ in terms of $a_0,a_1,\dots$?

In particular, suppose the coefficients of $f(x)$ goes to zero as $O\left(\frac{1}{n!}\right)$ (e.g. $f(x)=\exp(x)$), how fast the coefficients of $f^{(k)}(x)$ will go to zero?


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