302 Commutation Relations, Normal Ordering, and Stirling Numbers
As the expression for the generalized Stirling numbers S
s;h
(n, k) given in Proposi-
tion 8.43 shows, they are closely related to the generalized Stirling numbers S(s +1;n, k)=
S
s+1,1
(n, k). The generalized Stirling numbers defined by (8.50) are also very natural insofar
as many properties of the conventional Stirling numbers of the second kind have a simple
analog. For example, the interpretation of S(n, k) as rook number of a staircase Ferrers
board (see Corollary 2.56) generalizes in a beautiful way to the interpretation of S
s;h
(n, k)
as s-rook number of the staircase board (see Theorem 8.110). Although we defined S
s;h
(n, k)
above for s ∈ N
0
(to conform with the interpretation of U