  The normal ordering formulae for powers of the boson number operator
$\hat{n}$ are extended to deformed bosons. It is found that for the `M-type'
deformed bosons, which satisfy $a a^{\dagger} - q a^{\dagger} a = 1$, the
extension involves a set of deformed Stirling numbers which replace the
Stirling numbers occurring in the conventional case. On the other hand, the
deformed Stirling numbers which have to be introduced in the case of the
`P-type' deformed bosons, which satisfy $a a^{\dagger} - q a^{\dagger} a =
q^{-\hat{n}}$, are found to depend on the operator $\hat{n}$. This distinction
between the two types of deformed bosons is in harmony with earlier
observations made in the context of a study of the extended
Campbell-Baker-Hausdorff formula.
