366 Commutation Relations, Normal Ordering, and Stirling Numbers
Corollary 9.56 (Corcino, Celeste, Gonzales) Let s ∈ N
0
and h ∈ C \{0}.Theq-
deformed generalized Stirling numbers associated to a word ω = V
r
n
U
t
n
···V
r
1
U
t
1
in A
s;h|q
are given by
S
r,t
s;h|q
(k)=r
(s;h)
|t|−k
(B
ω
,q).
In particular, if ω =(VU)
n
,thenduetoS
1,1
s;h|q
(k)=S
s;h|q
(n, k) one has that
S
s;h|q
(n, k)=r
(s;h)
n−k
(J
n,1
,q). (9.49)
Using the rook theoretic interpretation from above, the following theorem was shown
in [289]. The second equation is a q-analog of Theorem 8.93.
Theorem 9.57 (Corcino, Celeste, Gonzales) Let s ∈ N
0
and let n, k ∈ N. Then the
q-deformed generalized Stirling numbers satisfy
S
s;h|q
(n, k)=
n−1
r=k−1
h
n−r−1
q
r
n − 1
r
q
s
S
s;h|q
(r, k − 1)
n−r−2
j=0
[1 + js]
q
.
Furthermore, the q-deforme ...