I was solving a probability problem and got stuck on the following situation, where each x_i is independent of others:
$$f=(x_1+x_2+..x_k)^N$$
I'm interested in the expression obtained after performing the operation $\phi$, $F$ times to the above.
$$\phi(f) = [D_{x_L}D_{x_{L-1}}\dots D_{x_1}(f)]\times (x_1 x_2\dots x_L)$$
(so we are looking at $\phi^F(f)$)
Work thus far:
I've used computer to find this but $n$ and $k$ are huge numbers, while $F\leq 1000$ and $L\times F \leq 100000$.
I observed that I get exactly $1+(F-1)L$ terms, in terms of different z $(x1+x2+..xk)^z$ which is quite computable, z decreasing from $n-L$ to $n-L-(F-1)L$.
What will these coefficients be ? I've failed to find further the first 2 terms though there seems some symmetry.
The operators $x_i D_{x_i}$ for $i \ne j$ conmute, so you should start by seeing what happens with $(x_1 D_{x_1})^F f$. I bet Leibnitz' rule helps cutting down the verbiage, and gives a clue on the terms you mention.