The Boson Normal Ordering Problem and Generalized Bell Numbers

Abstract

For any function F(x) having a Taylor expansion we solve the boson normal ordering problem for $F [(a^\dag)^r a^s]$, with r, s positive integers, $F [(a, a^\dag]=1$, i.e., we provide exact and explicit expressions for its normal form $\mathcal{N} \{F [(a^\dag)^r a^s]\} = F [(a^\dag)^r a^s]$, where in $ \mathcal{N} (F) $ all a's are to the right. The solution involves integer sequences of numbers which, for $ r, s \geq 1 $, are generalizations of the conventional Bell and Stirling numbers whose values they assume for $ r=s=1 $. A complete theory of such generalized combinatorial numbers is given including closed-form expressions (extended Dobinski-type formulas), recursion relations and generating functions. These last are special expectation values in boson coherent states.