The Boson Normal Ordering Problem and Generalized Bell Numbers
  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. 
P. Blasiak, K.A. Penson and A.I. Solomon (Niewodniczanski Institute of   Nuclear Physics, Krakow, Poland and University of P...