Combinatorics of Boson Normal Ordering: the Dobiński Formula Revisited

Abstract. We derive explicit formulas for the normal ordering of powers of arbitrary monomials of boson operators. These formulas lead to generalisations of conventional Bell and Stirling numbers and to appropriate generalisations of the Dobiński relations. These new combinatorial numbers are shown to be coherent state matrix elements of powers of the monomials in question. It is further demonstrated that such Bell-type numbers, when considered as power moments, give rise to positive measures on the positive half-axis, which in many cases can be written in terms of known functions.