Is it possible to recover $f\_n(x)$ for any $n$ based off [this question](https://math.stackexchange.com/questions/3155741/simplify-recurrence-fracddx-f-n-1x-f-nx-f-n-1x-f-1x)? The relation is:

$ \frac{d^n}{dx^n} f\_1(x)$ from $\frac{d}{dx} f\_{k-1}(x)=f\_k(x)-f\_{k-1}(x) f\_1(x)$

$f\_0(x)=1$ and I add $f\_1(x)=e^x$

Using this you can find $f\_n(x)$ for all $n$ when $n$ is a whole number.

* $f\_0(x)=1,f\_1(x)=e^x$
* $f\_2(x)=e^x+e^{2x}$
* $f\_3(x)=e^x+3e^{2x}+e^{3x}$
* $f\_4(x)=e^x+7e^{2x}+6e^{3x}+e^{4x}$, etc.

My question is can you generalize $f\_n(x)$ get $n$ to be any number. If so how would you solve $f\_{1/2} (x)$ or $f\_{i}(x)$