  In this paper we give an analytic interpretation of free convolution of type
B, introduced by Biane, Goodman and Nica, and provide a new formula for its
computation. This formula allows us to show that free additive convolution of
type B is essentially a re-casting of conditionally free convolution. We put in
evidence several aspects of this operation, the most significant being its
apparition as an 'intertwiner' between derivation and free convolution of type
A. We also show connections between several limit theorems in type A and type B
free probability. Moreover, we show that the analytical picture fits very well
with the idea of considering type B random variables as infinitesimal
deformations to ordinary non-commutative random variables.
