On the multiplication of free N -tuples of noncommutative random variables

Let a 1,. . . , a n, b 1,. . . , b n be random variables in a noncommutative probability space, such that a 1,. . . , a n is free from b 1,. . . , b n. We show how the joint distribution of the n -tuple (a 1 b 1,. . . , a n b n) can be described in terms of the joint distributions of (a 1,. . . , a n) and (b 1,. . . , 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 noncommutative random variables. A different approach to two of these applications is presented by Dan Voiculescu in an Appendix to the paper.