
76 Commutation Relations, Normal Ordering, and Stirling Numbers
Note the dependence on x! Thus, we should have better indicated this dependence by
writing S
∗
p,q
(n, k, x) in (3.64). The recurrence relation (3.65) is closely related to (3.63).
In fact, writing
ˆ
S
p,q
(n, k)=Ψ
p,q
(n, k)
n−k
p
, one obtains from (3.63) the relation Ψ
p,q
(n +
1,k)=q
k−1
Ψ
p,q
(n, k − 1) + p
−k
[k]
p,q
Ψ
p,q
(n, k). The numbers S
∗
p,q
(n, k) correspond to the
case α =1,β =0,r = 0 of the type II generalized (p, q)-Stirling numbers
˜
S
1,p,q
n,k
(α, β, r)
introduced by Remmel and Wachs [928], which will be considered in Chapter 4.
It is possible to define a (p, q)-exponential function by
e
p,q
(x)=