Some identities involving q-Stirling numbers of the second kind in type B

The recent interest in type Bq -Stirling numbers of the second kind prompted us to give a type B analogue of a classical identity connecting the q-Stirling numbers of the second kind and Carlitz's major q-Eulerian numbers, which turns out to be a q-analogue of an identity due to Bagno, Biagioli and Garber. We provide a combinatorial proof of this identity and an algebraic proof of a more general identity for colored permutations. In addition, we prove some q-identities about the q-Stirling numbers of the second kind in types A, B and D.