This document may be downloaded, read, stored and printed for your own use within the limits of § 53 UrhG but it may not be distributed via the internet or passed on to external parties.Dieses Dokument darf im Rahmen von § 53 UrhG zum eigenen Gebrauch kostenfrei heruntergeladen, gelesen, gespeichert und ausgedruckt, aber nicht im Internet bereitgestellt oder an Außenstehende weitergegeben werden.Ebrahimi-Fard, KuruschPatras, FrédéricTapia, NikolasZambotti, Lorenzo2022-07-082022-07-082022https://oa.tib.eu/renate/handle/123456789/9703https://doi.org/10.34657/8741We study a particular group law on formal power series in non-commuting parameters induced by their interpretation as linear forms on a suitable non-commutative and non- cocommutative graded connected word Hopf algebra. This group law is left-linear and is therefore associated to a pre-Lie structure on formal power series. We study these structures and show how they can be used to recast in a group theoretic form various identities and transformations on formal power series that have been central in the context of non-commutative probability theory, in particular in Voiculescu?s theory of free probability.eng510Non-commutative probability theorynon-commutative power seriesmoments and cumulantscombinatorial Hopf algebrapre-Lie algebraReport18 S.