  We follow the guiding line offered by canonical operators on the full Fock
space, in order to identify what kind of cumulant functionals should be
considered for the concept of bi-free independence introduced in the recent
work of Voiculescu. By following this guiding line we arrive to consider, for a
general noncommutative probability space (A, phi), a family of "(l,r)-cumulant
functionals" which enlarges the family of free cumulant functionals of the
space. In the motivating case of canonical operators on the full Fock space we
find a simple formula for a relevant family of (l,r)-cumulants of a (2d)-tuple
(A_1, ..., A_d, B_1, ..., B_d), with A_1, ... , A_d canonical operators on the
left and B_1, ... , B_d canonical operators on the right. This extends a known
one-sided formula for free cumulants of A_1, ..., A_d, which establishes a
basic operator model for the R-transform of free probability.
