A rooted-trees q-series lifting a one-parameter family of Lie   idempotents
  We define and study a series indexed by rooted trees and with coefficients in Q(q). We show that it is related to a family of Lie idempotents. We prove that this series is a q-deformation of a more classical series and that some of its coefficients are Carlitz q-Bernoulli numbers. 
Fr\'ed\'eric Chapoton (ICJ)