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We follow the guiding line offered by canonical operators on the full Fockspace, in order to identify what kind of cumulant functionals should beconsidered for the concept of bi-free independence introduced in the recentwork of Voiculescu. By following this guiding line we arrive to consider, for ageneral noncommutative probability space (A, phi), a family of "(l,r)-cumulantfunctionals" which enlarges the family of free cumulant functionals of thespace. In the motivating case of canonical operators on the full Fock space wefind 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 theleft and B_1, ... , B_d canonical operators on the right. This extends a knownone-sided formula for free cumulants of A_1, ..., A_d, which establishes abasic operator model for the R-transform of free probability.

Double-ended queues and joint moments of left-right canonical operators on full Fock space