
The q-Deformed Generalized Weyl Algebra 371
Proposition 9.65 Let s =1/2 and h =2.Theq-deformed generalized Stirling numbers
can be expressed by
√
q-deformed Lah numbers as
S
1
2
;2|q
(n, k)=
2
1+
√
q
n−k
L
√
q
(n, k). (9.62)
Proof Let T (n, k)=(
2
1+
√
q
)
n−k
L
√
q
(n, k). It follows that T (n +1,k)isgivenby
2
1+
√
q
n−(k−1)
√
q
n+k−1
L
√
q
(n, k − 1) +
2
1+
√
q
n−k+1
[n + k]
√
q
L
√
q
(n, k). Thus, one has
the recurrence relation
T (n +1,k)=
√
q
n+k−1
T (n, k −1) +
2
1+
√
q
[n + k]
√
q
T (n, k),
for all n ≥ 0andk ≥ 1. T (n, k) satisfies the same recurrence relation as S
1
2
;2|q
(n, k). Since
T (n, 0) = δ
n,0
and T (0,k)=δ
0,k
for all n, k ∈ N
0
, the initial values also coincide, completing
the proof.
Coroll ...