  We consider two extensions of free probability that have been studied in the
research literature, and are based on the notions of c-freeness and
respectively of infinitesimal freeness for noncommutative random variables. In
a 2012 paper, Belinschi and Shlyakhtenko pointed out a connection between these
two frameworks, at the level of their operations of 1-dimensional free additive
convolution. Motivated by that, we propose a construction which produces a
multi-variate version of the Belinschi-Shlyakhtenko result, together with a
result concerning free products of multi-variate noncommutative distributions.
Our arguments are based on the combinatorics of the specific types of cumulants
used in c-free and in infinitesimal free probability. They work in a rather
general setting, where the initial data consists of a vector space $V$ given
together with a linear map $\Delta : V \to V \otimes V$. In this setting, all
the needed brands of cumulants live in the guise of families of multilinear
functionals on $V$, and our main result concerns a certain transformation
$\Delta^{*}$ on such families of multilinear functionals.
