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arXiv · 2006.02089

Combinatorial Hopf algebras in noncommutative probabilility

Franz Lehner·Jean-Christophe Novelli·Jean-Yves Thibon
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Abstract

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.

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Franz Lehner, Jean-Christophe Novelli, Jean-Yves Thibon. 2020-06-03. Combinatorial Hopf algebras in noncommutative probabilility. https://arxiv.org/abs/2006.02089

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