
356 Commutation Relations, Normal Ordering, and Stirling Numbers
We now generalize this. Recalling from Proposition 9.3 that a representation of variables
(U, V ) satisfying UV = qV U + hV
s
is given by (E
s;h|q
,X)whereE
s;h|q
= hE
s;1|q
= hX
s
D
q
,
we can write (9.17) equivalently as
(X + hE
s;1|q
)
n
=
n
j=0
n
j
q
H
(s)
n;j
(X, h; q)h
j
E
j
s;1|q
, (9.32)
wherewedefined
H
(s)
n;j
(X, h; q)=
n
j
−1
q
X
n−j
n−j−1
i=0
d
(s)
n
(j, i; q)h
i
X
i(s−2)
. (9.33)
To be more concrete, we record here the first few cases which may be derived explicitly
using the commutation relation E
s;1|q
X
k
= q
k
X
k
E
s;1|q
+[k]
q
X
k−1+s
.Intheq-deformed
case, these expressions quickly become rather messy:
(X + hE
s;1|q
)
1
=X + h