#### Vol. 3, No. 6, 2009

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

### Frédéric Chapoton

Vol. 3 (2009), No. 6, 611–636
##### Abstract

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

##### Keywords
tree series, operads, Lie idempotents, Bernoulli–Carlitz numbers
##### Mathematical Subject Classification 2000
Primary: 18D50
Secondary: 17D25, 05C05