Combinatorial Hopf algebras in noncommutative probabilility
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.
- Publication:
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arXiv e-prints
- Pub Date:
- June 2020
- DOI:
- 10.48550/arXiv.2006.02089
- arXiv:
- arXiv:2006.02089
- Bibcode:
- 2020arXiv200602089L
- Keywords:
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- Mathematics - Combinatorics;
- Mathematics - Probability
- E-Print:
- 28 pages