Combinatorics of boson normal ordering and some applications

We provide the solution to the normal ordering problem for powers and exponentials of two classes of operators. The first one consists of boson strings and more generally homogeneous polynomials, while the second one treats operators linear in one of the creation or annihilation operators. Both solutions generalize Bell and Stirling numbers arising in the number operator case. We use the advanced combinatorial analysis to provide closed form expressions, generating functions, recurrences, etc. The analysis is based on the Dobi\'nski-type relations and the umbral calculus methods. As an illustration of this framework we point out the applications to the construction of generalized coherent states, operator calculus and ordering of deformed bosons.