  Using Reiner's definition of Stirling numbers of the second kind in types $B$
and $D$, we generalize two well-known identities concerning the classical
Stirling numbers of the second kind. The first identity relates them with
Eulerian numbers and the second identity interprets them as entries in a
transition matrix between the elements of two standard bases of the polynomial
ring $R[x]$. Finally, we generalize these identities to the group of colored
permutations $G_{m,n}$.
