On the multiplication of free $n$-tuples of non-commutative random variables

Let $a_1,\mathrm{.}\mathrm{.}\mathrm{.},a_n,b_1,\mathrm{.}\mathrm{.}\mathrm{.},b_n$ be random variables in some (non-commutative) probability space, such that $\{a_1,\mathrm{.}\mathrm{.}\mathrm{.},a_n\}$ is free from $\{b_1,\mathrm{.}\mathrm{.}\mathrm{.},b_n\}$. We show how the joint distribution of the $n$-tuple $(a_1{b}_1,\mathrm{.}\mathrm{.}\mathrm{.},a_n{b}_n)$ can be described in terms of the joint distributions of $(a_1,\mathrm{.}\mathrm{.}\mathrm{.},a_n)$ and $(b_1,>\mathrm{.}\mathrm{.}\mathrm{.},b_n)$, by using the combinatorics of the $n$-dimensional $R$-transform. We point out a few applications that can be easily derived from our result, concerning the left-and-right translation with a semicircular element (see Sections 1.6-1.10) and the compression with a projection (see Sections 1.11-1.14) of an $n$-tuple of non-commutative random variables. A different approach to two of these applications is presented by Dan Voiculescu in an Appendix to the paper.