R-Transforms of Free Joint Distributions and Non-crossing Partitions

We introduce an extension of the notion ofR-transform, defined by D. Voiculescu, to (free)joint distributions, i.e., normalized linear functionals on algebras of non-commutative polynomials in several indeterminates. We point out that theR-transform has good behavior with respect to the operations of free product and free convolution of joint distributions. We prove that the explicit computation of the inverse-R-transform of a joint distribution is done via a formula of summation over the lattice of non-crossing partitions, which shows that theR-transform is the operator-theoretic counterpart of the “free cumulants” considered in the combinatorial approach of R. Speicher. Moreover, in connection to the equivariance of the multidimensionalR-transform under rotations, we show that a natural free analogue of a classical result about rotations of independent random variables is holding.