The Worpitzky identity for the groups of signed and even-signed permutations

The well-known Worpitzky identity provides a connection between two bases of $\mathbb{Q}[x]$: The standard basis $(x+1{)}^n$ and the binomial basis ${{x+n-i} \choose {n}}$,wheretheEuleriannumbersfortheCoxetergroupoftype$A$ (the symmetric group) serve as the entries of the transformation matrix. Brenti has generalized this identity to the Coxeter groups of types $B$ and $D$ (signed and even-signed permutations groups, respectively) using generating function techniques. Motivated by Foata-Sch"utzenberger and Rawlings’ proof for the Worpitzky identity in the symmetric group, we provide combinatorial proofs of this identity and for their $q-$analogues in the Coxeter groups of types $B$ and $D$.