  In the free probability theory of Voiculescu two of the most frequently used
*-distributions are those of a Haar unitary and of a circular element. We
define an $R$-diagonal pair as a generalization of these distributions by the
requirement that their two-dimensional $R$-transform (or free cumulants) have a
special diagonal form. We show that the class of such $R$-diagonal pairs has an
absorption property under nested multiplication of free pairs. This implies
that in the polar decomposition of such an element the polar part and the
absolute value are free. Our calculations are based on combinatorial statements
about non-crossing partitions, in particular on a canonical bijection between
the set of intervals of NC(n) and the set of 2-divisible partitions in NC(2n).
In a forthcoming paper the theory of $R$-diagonal pairs will be used to solve
the problem of the free commutator.
