Let $f^{[n]}(x)$ be the $n$-th functional iterate of $f(x)$, so that $f^{[1]}(x)=f(x)$ and $f^{[n+1]}(x)=f(f^{[n]}(x)$. And let $f^{(n)}(x) = \frac{d^{n}}{dx^{n}} \left(f(x)\right)$
Has there been any research into solving equations like:
$f^{[n]}(x)=f^{(n)}(x)$
The case $n=1$ reduces to the exponential. What about $n>1$? Note: I do not mean that there is one $f$ which solves the above equation for all values of $n$, but I am stuck on how to notate each solution, given the preponderance of superscripted $n$s.