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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 (Digital Object Identifier) Link: http://dx.doi.org/10.1007/s00026-003-0177-z
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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.

Item Type: Journal Article
Copyright Holders: 2003 Birkhauser-Verlag
ISSN: 0219-3094
Extra Information: Some of the symbols may not have transferred correctly into this bibliographic record.
Keywords: boson normal order; Bell numbers; Stirling numbers; coherent states
Academic Unit/Department: Science > Physical Sciences
Item ID: 4891
Depositing User: Users 6041 not found.
Date Deposited: 11 Aug 2006
Last Modified: 28 Mar 2014 12:39
URI: http://oro.open.ac.uk/id/eprint/4891
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