A pr 2 00 9 Laguerre - type derivatives : Dobiński relations and combinatorial identities

We consider properties of the operators D (r , M) = a r (a † a) M (which we call generalized Laguerre-type derivatives) a and a † are boson annihilation and creation operators respectively , satisfying [ a , a † ] = 1. We obtain explicit formulas for the normally ordered form of arbitrary Taylor-expandable functions of D (r , M) with the help of an operator relation which generalizes the Dobi´nski formula. Coherent state expectation values of certain operator functions of D (r , M) turn out to be generating functions of combinatorial numbers. In many cases the corresponding combinatorial structures can be explicitly identified .