  We investigate properties of exponential operators preserving the particle
number using combinatorial methods developed in order to solve the boson normal
ordering problem. In particular, we apply generalized Dobinski relations and
methods of multivariate Bell polynomials which enable us to understand the
meaning of perturbation-like expansions of exponential operators. Such
expansions, obtained as formal power series, are everywhere divergent but the
Pade summation method is shown to give results which very well agree with exact
solutions got for simplified quantum models of the one mode bosonic systems.
