Let k be a positive integer and let G k denote the set of all joint distributions of k-tuples (a 1 , . . . , a k ) in a non-commutative probability space (A, ϕ) such that ϕ(a 1 ) = · · · = ϕ(a k ) = 1. G k is a group under the operation of free multiplicative convolution ⊠. We identify G k , ⊠ as the group of characters of a certain Hopf algebra Y (k) . Then, by using the log map from characters to infinitesimal characters of Y (k) , we introduce a transform LS µ for distributions µ ∈ G k . LS µ is a power series in k non-commuting indeterminates z 1 , . . . , z k ; its coefficients can be computed from the coefficients of the R-transform of µ by using summations over chains in the lattices N C(n) of non-crossing partitions. The LS-transform has the "linearizing" property thatIn the particular case k = 1 one has that Y (1) is naturally isomorphic to the Hopf algebra Sym of symmetric functions, and that the LS-transform is very closely related to the logarithm of the S-transform of Voiculescu, by the formulaIn this case the group (G 1 , ⊠) can be identified as the group of characters of Sym, in such a way that the S-transform, its reciprocal 1/S and its logarithm log S relate in a natural sense to the sequences of complete, elementary and respectively power sum symmetric functions.

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