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The Boson Normal Ordering Problem and Generalized Bell Numbers

P. Blasiak·K. A. Penson·A. I. Solomon
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Abstract

For any function F(x) having a Taylor expansion we solve the boson normal ordering problem for F[(a*)^r a^s], with r,s positive integers,[a,a*]=1, i.e. we provide exact and explicit expressions for its normal form which has all a's to the right. The solution involves integer sequences of numbers which, for r,s >=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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P. Blasiak, K. A. Penson, A. I. Solomon. 2002-12-11. The Boson Normal Ordering Problem and Generalized Bell Numbers. https://arxiv.org/abs/quant-ph/0212072

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