  A formula of Stembridge states that the permutation peak polynomials and
descent polynomials are connected via a quadratique transformation. The aim of
this paper is to establish the cycle analogue of Stembridge's formula by using
cycle peaks and excedances of permutations. We prove a series of new general
formulae expressing polynomials counting permutations by various excedance
statistics in terms of refined Eulerian polynomials. Our formulae are
comparable with Zhuang's generalizations [Adv. in Appl. Math. 90 (2017) 86-144]
using descent statistics of permutations. Our methods include permutation
enumeration techniques involving variations of classical bijections from
permutations to Laguerre histories, explicit continued fraction expansions of
combinatorial generating functions in Shin and Zeng [European J. Combin. 33
(2012), no. 2, 111--127] and cycle version of modified Foata-Strehl action. We
also prove similar formulae for restricted permutations such as derangements
and permutations avoiding certain patterns. Moreover, we provide new
combinatorial interpretations for the $\gamma$-coefficients of the inversion
polynomials restricted on $321$-avoiding permutations.
