A rooted-trees $q$-series lifting a one-parameter family of Lie idempotents

Abstract

We define and study a series indexed by rooted trees and with coefficients in $\mathbb{Q}\left(q\right)$. 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.