Variations of the α-Eulerian polynomials and gamma positivity

We prove a connection formula between two multivariate generalizations of the Eulerian polynomials A^ₙ(x,y, t\,|\,α) and Aₙ(u₁,u₂,u₃,u₄, f, g, t\,|\,α, β), which enumerate permutations related to excedance and descent based statistics respectively. By exploring this connection, we derive the exponential generating function of the latter polynomials and several γ-positivity formulas for variants of Eulerian polynomials. In particular, our results generalise the main results in two recent papers by Ji {Ji23} and Ji-Lin {JL23}. Our proofs are combinatorial in nature and involve Foata's fundamental transformation and a cyclic analogue of valley-hopping.