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.

DOI: https://doi.org/10.1007/s00026-003-0177-z

Abstract

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.

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