Conditionally bi-free independence for pairs of faces

In this paper, the notion of conditionally bi-free independence for pairs of faces is introduced. The notion of conditional (ℓ,r)-cumulants is introduced and it is demonstrated that conditionally bi-free independence is equivalent to the vanishing of mixed cumulants. Furthermore, limit theorems for the additive conditionally bi-free convolution are studied using both combinatorial and analytic techniques. In particular, a conditionally bi-free partial R-transform is constructed and a conditionally bi-free analogue of the Lévy-Hinčin formula for planar Borel probability measures is derived.