  We prove that the generalized moment-cumulant relations introduced in
[arXiv:1711.00219] are given by the action of the Eulerian idempotents on the
Solomon-Tits algebras, whose direct sum builds up the Hopf algebra of Word
Quasi-Symmetric Functions $\WQSym$. We prove $t$-analogues of these identities
(in which the coefficient of $t$ gives back the original version), and a
similar $t$-analogue of Goldberg's formula for the coefficients of the
Hausdorff series. This amounts to the determination of the action of all the
Eulerian idempotents on a product of exponentials.
