  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
$A^{k}$ ($k\geq1$) 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.
