The Boson Normal Ordering Problem and Generalized Bell Numbers |
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Authors: | P Blasiak K A Penson A I Solomon |
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Institution: | (1) Laboratoire de Physique Théoretique des Liquides, CNRS URA 7600, Université Pierre et Marie Curie, Tour 16, 5ième étage, 75252 Paris Cedex 05, France;(2) H. Niewodniczaski, Institute of Nuclear Physics, ul. Eliasza Radzikowskiego 152, 31-342 Kraków, Poland |
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Abstract: | For any function F(x) having a Taylor expansion we solve the boson
normal ordering problem for $F (a^\dag)^r a^s]$, 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.AMS Subject Classification: 81R05, 81R15, 81R30, 47N50. |
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Keywords: | boson normal order Bell numbers Stirling numbers coherent states |
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