Closed form for repeated theta operator applied to $x\cos(x)$ let $\theta_{x}$ be the operator :
$$\theta_{x}=x\frac{d}{dx}$$

What is the closed form for :
$$\theta_{x}^{n}\left[x\cos(x)\right]$$
$n$ being an positive integer.


You can have the closed form

$$ \sum_{k=0}^{n} {n\brace k} x^k \left(x\cos\left(x+\frac{k\pi}{2}\right)+k\sin\left(x+\frac{k\pi}{2}\right)\right),$$

where ${n\brace k}$ is the Stirling numbers of the second kind. For the $n$th derivative of $x\cos(x)$, see here.
Note: We used the identity

$$ (xD)^n = \sum_{k=0}^{n}{n\brace k} x^k D^k. $$

