The general boson normal ordering problem

Abstract

We solve the boson normal ordering problem for F[(a*)^r a^s], with r,s positive integers, where a* and a are boson creation and annihilation operators satisfying [a,a*]=1. That is, we provide exact and explicit expressions for the normal form wherein all a's are to the right. The solution involves integer sequences of numbers which are generalizations of the conventional Bell and Stirling numbers whose values they assume for r=s=1. A comprehensive theory of such generalized combinatorial numbers is given including closed-form expressions (extended Dobinski-type formulas)and generating functions. These last are special expectation values in boson coherent states.