Cumulants in Noncommutative Probability Theory III. Creation- and annihilation operators on Fock spaces

Fock space constructions give rise to natural exchangeable families and are thus well suited for cumulant calculations. In this paper we develop some general formulas and compute cumulants for generalized Toeplitz operators, notably for q-Fock spaces, previously considered by M. Anshelevich and A. Nica, and Fock spaces for characters of the infinite symmetric group, which where constructed by Bozejko and Guta. An expression for cumulants in terms of the cycle-cover polynomials of certain directed graphs is obtained in this case.