The Boson normal ordering problem and generalized Bell numbers

Blasiak, P.; Penson, K.A. and Solomon, A.I. (2003). The Boson normal ordering problem and generalized Bell numbers. Annals of Combinatorics, 7(2) pp. 127–139.



For any function F(x) having a Taylor expansion we solve the boson normal ordering problem for $F[(a^\dag)^ra^s]$, with r, s positive integers, $[(a, a^\dag]=1$, i.e., we provide exact and explicit expressions for its normal form $\mathcal{N} \{F[(a^\dag)^ra^s]\}$ = $F[(a^\dag)^ra^s]$, 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.

Viewing alternatives


Public Attention

Altmetrics from Altmetric

Number of Citations

Citations from Dimensions
No digital document available to download for this item

Item Actions