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

The well-known Worpitzky identity $$\begin{aligned} (x+1)^n = \sum \limits _{k=0}^{n-1} A_{n,k} {{x+n-k} \atopwithdelims (){n}} \end{aligned}$$ provides a connection between two bases of $$\mathbb {Q}[x]$$ : the standard basis $$(x+1)^n$$ and binomial $${{x+n-k} (){n}}$$ , where Eulerian numbers $$A_{n,k}$$ for symmetric group serve as entries transformation matrix. Brenti has generalized this to Coxeter groups types $$B_n$$ $$D_n$$ (signed even-signed permutations groups, respectively) using generatingfunctionology. Motivated by Foata–Schützenberger’s Rawlings’ proof in group, we provide combinatorial proofs generalizations their q-analogues . Our utilize language P-partitions - -posets, introduced Chow Stembridge, respectively.