  Since Voiculescu introduced his bi-free probability theory in 2013, the major
development of the theory has been on its combinatorial side; in particular, on
the combinatorics of bi-free cumulants and its application to the bi-free
R-transform. In this article we propose a harmonic analysis approach to the
bi-free R-transform, which is solely based on integral transforms of two
variables. To accommodate the harmonic analysis tools, we confine ourselves in
the simplest situation of bi-freeness with commuting faces. Our method allows
us to treat measures with unbounded support, and we show that the classical
limit theory of infinitely divisible laws, due to Levy and Khintchine, has a
perfect bi-free analogue.
