The Boson Normal Ordering Problem and Generalized Bell Numbers

Details The Boson Normal Ordering Problem and Generalized Bell Numbers The Boson Normal Ordering Problem and Generalized Bell Numbers

2002-12-11

arxiv.org/abs/quant-ph/0212072v1

Annals of Combinatorics 7: 127-139, (2003)

Physics

Quantum Physics

10 pages

Scientific paper

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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