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We define and study a series indexed by rooted trees and with coefficients inQ(q). We show that it is related to a family of Lie idempotents. We prove thatthis series is a q-deformation of a more classical series and that some of itscoefficients are Carlitz q-Bernoulli numbers.

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