WIAS Preprint No. 2945, (2022)

Shifted substitution in non-commutative multivariate power series with a view towards free probability


  • Ebrahimi-Fard, Kurusch
  • Patras, Frédéric
  • Tapia, Nikolas
    ORCID: 0000-0003-0018-2492
  • Zambotti, Lorenzo

2020 Mathematics Subject Classification

  • 16T05 17A30 46L53


  • Non-commutative probability theory, non-commutative power series, moments and cumulants, combinatorial Hopf algebra, pre-Lie algebra




We 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.

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