Some identities involving second kind Stirling numbers of types B and D

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).