
188 Commutation Relations, Normal Ordering, and Stirling Numbers
We will give a proof of the q-deformed version of this identity later in Section 7.2.4. The
same method of proof shows that one also has (VUV)
n
= V
n
U
n
V
n
. Equation (6.18) can
be generalized to the product having m factors of VU inside the parentheses, that is, one
has that
(UVU···VU
! "
m times VU
)
n
= U
n
V
n
U
n
···V
n
U
n
! "
m times V
n
U
n
. (6.19)
In the representation U → D, V → X ≡ ˆx ≡ x, identity (6.18) becomes
(DxD)
n
= D
n
x
n
D
n
, (6.20)
which is the result (6.17) of Tait.
Let us turn to an application. Carlitz [189] considered the expansion of (DxD)
n
and
found that
(DxD)
n
=
n
k=0
n
k
n!
k!
x
k
D
n+k
, (6.21) ...