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.


Let $\theta:=\theta_x$, $f(x):=x\cos(x)$ and $g(x)=x^2\sin x$. Then
$$\theta(f)=f-g, $$
$$\theta^2(f)=\theta(f)-\theta(g)=f-g-\theta(g), $$
$$\theta^3(f)=\theta(f)-\theta(g)-\theta^2(g)=f-g-\theta(g)-\theta^2(g); $$
in summary
$$\theta^n(f)=f-g-\theta(g)-\dots-\theta^{n-1}(g).$$
