Generalized Eulerian Polynomials with a Nonnegative Gamma Vector

We define a family of generalized Eulerian polynomials depending on three parameters. We prove that these polynomials have a nonnegative gamma vector, and we provide a combinatorial description of the corresponding gamma coefficients. By assigning suitable integer values to the parameters, we obtain a new expansion of the nth Eulerian polynomial over the symmetric group $\mathfrak{S}_{n-1}$, a new description of the associated gamma vector, and an identity relating the derangements of $\mathfrak{S}_{2n}$ to the alternating permutations of $\mathfrak{S}_{2n+1}$.