322 Commutation Relations, Normal Ordering, and Stirling Numbers
and Proposition 8.106 were also generalized in [997, Theorem 2.3]. The last property we
mention is the generalization of (8.91). Using generalized factorials, we can write [997,
Theorem 2.2] as
(x +(a + b)r|−a)
n
=
n
k=0
G
a;b
(n, k; r)(x|b)
k
.
8.5.6 Interpretation in Terms of Rooks
In addition to the above combinatorial interpretation, we can relate the generalized
Stirling numbers S
s;1
(n, k)fors ∈ N
0
to the s-rook numbers introduced by Goldman and
Haglund [485]; see Section 2.4.4.3. This connection generalizes the well-known interpretation
of the Stirling numbers of the second kind as particular rook numbers; see Corollary 2.56.
For s = 0 – corresponding to the Weyl algebra generated by variables ...