
The q-Deformed Generalized Weyl Algebra 361
Before closing this section, let us point out that many of the above relations can also
be shown directly using the operator interpretation. For example, Corollary 9.40 can be
shown as follows. If (f )(x)=f(qx) (see Appendix A), then ({(q − 1)E
2;1|q
}f)(x)=((q −
1)X
2
D
q
f)(x) is given by (q −1)x
2
f(qx)−f(x)
(q−1)x
=((X( −1))f)(x), implying for the left-hand
side (X +(q − 1)E
2;1|q
)
n
=(X − X( − 1))
n
=(X)
n
. On the other hand, using X = qX
as well as (q − 1)
n−k
E
n−k
2;1|q
=(X( − 1))
n−k
= X
n−k
(q
n−k−1
− 1) ···(q − 1)( − 1), one
finds for the right-hand side
n
k=0
n
k
q
q
(
k
2
)
X
k
(q − 1)
n−k
E
n−k
2;1|q
= X
n
n
k=0
n
k
q
q
(
k
2
)
(