Powers of the operator $u(z)\frac{d}{dz}$ and their connection with some combinatorial numbers

In this paper the operator $A=u(z)\frac{d}{dz}$ is considered, where $u$ is an entire or meromorphic function in the complex plane. The expansion of $A^k$ ($k\ge 1$) with the help of the powers of the differential operator $D=\frac{d}{dz}$ is obtained, and it is shown that this expansion depends on special numbers. Connections between these numbers and known combinatorial numbers are given. Some special cases of the operator $A$, corresponding to $u(z)=z$, $u(z)=e^z$, $u(z)=\frac{1}{z}$, are considered.