A generalization of the boson normal ordering

In this paper we generalize some results of Katriel [J. Opt. B: Quantum Semiclass. Opt. 4:S200--S203, 2002]. The main motivation is to shed further light onto the combinatorics arising from algebraic and Fock space properties of boson operators. Our generalization is based on the number of contractions whose vertices are next to each other in the linear representation of the boson operator function. In this way we introduce a parameter that allows to refine the set of all contractions and to specify a statistics (in a combinatorial sense). The standard normally ordered form is obtained as a special case