How to write $\left(x\frac{d}{dx}\right)^n $ in terms of Stirling numbers of the first kind ${n\brack m}$?
I know that we can write the above in terms of Stirling numbers of the second kind ${n\brace m}$,
as below:
$$\left(x\frac{d}{dx}\right)^n=\sum_{m=1}^n {n\brace m} x^m \left(\frac{d}{dx}\right)^m, \qquad n \ge 1.$$
Is there a similar way of writing this for the Stirling numbers of the first kind ${n\brack m}$?