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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.

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Well, the basic observation about these equations is that they are non-local in that they do not relate values of $f$ and its derivates at the same point $x$ but instead one needs to look at the point $f^{[n-1]}(x)$. This makes the problem similar (although much harder) to that of shift-differential equations. –  Marek Jun 20 '11 at 1:30
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