A Summation Formula Related to $q$-Eulerian Polynomials

Abstract

Brenti (J. Combin. Theory Ser. A $91(2000))$ considered a $q$-analogue of the Eulerian polynomials by enumerating permutations in the symmetric group $S_n$ with respect to the numbers of excedances and cycles. Here we establish a connection between these $q$-Eulerian polynomials and some infinite generating functions.