I have this problem involving a differential operator that I encountered while studying an enduring problem in chemical kinetics. However, I am unable to find previous results or develop a solution myself. Here is my problem:
Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be sufficiently differentiable and define operator $T_n(x) := (x\partial_x)^n$ for $x \in \mathbb{R}$ that acts on the function $f$ an $n \in \mathbb{N}$ number of times. So explicitly,
$$ T_3(x)f(x) = x\partial_x(x\partial_x(x\partial_xf(x))) $$
I have been attempting to find some sort of recurrence relation that tells me what the operator should be in terms of $x$ and $n$ with no derivatives. Can this be done? As a preliminary result to help, if $f(x) = e^x$, then $T_n(x)$ will be polynomials, but I can't seem to develop a formula for those polynomials in terms of $n$. Can anyone provide some insight to this?
I have tried using the sequence
$$ T_n(x) = x\partial_xT_{n-1}(x) = x(n-1)T_{n-2}(x)[x\partial^2_x + \partial_x] = \cdot\cdot\cdot $$
but that seems to get me nowhere as I just get another build up of $(x\partial^2_x + \partial_x)^{some\,power}$ terms.
An explicit expression helps as this sequence describes a probability distribution. A distribution is difficult to describe and model in terms of derivatives.