Variations of the -Eulerian polynomials and gamma positivity

We prove a connection formula between two multivariate generalizations of the Eulerian polynomials A^ₙ (x, y, t\, |\, ) and Aₙ (u₁, u₂, u₃, u₄, f, g, t\, |\, , ), 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 -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.