Laguerre-type derivatives: Dobinski relations and combinatorial identities

TL;DR

This paper introduces generalized Laguerre-type derivatives involving bosonic operators, deriving explicit formulas for their normal ordering and revealing combinatorial structures through coherent state expectations.

Contribution

It provides explicit formulas for the normal ordering of generalized Laguerre-type derivatives and connects their expectation values to combinatorial numbers.

Findings

Explicit formulas for normal ordering of D(r,M)

Identification of combinatorial structures in expectation values

Generalization of Dobinski relations

Abstract

We consider properties of the operators D(r,M)=a^r(a^\dag a)^M (which we call generalized Laguerre-type derivatives), with r=1,2,..., M=0,1,..., where a and a^\dag are boson annihilation and creation operators respectively, satisfying [a,a^\dag]=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 Dobinski 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.