
346 Commutation Relations, Normal Ordering, and Stirling Numbers
By Lemma 9.10, we obtain that
U
m+1
V =
m
j=0
h
j
⎛
⎝
λ∈T
m−j,j
q
m+1−j−
j
i=1
(λ
i
+1)
g
j
(λ
∗
; V )
⎞
⎠
U
m+1−j
+
m
j=0
h
j+1
⎛
⎝
λ∈T
m−j,j
q
m−j−|λ|
g
j+1
(λ
∗
; V )
⎞
⎠
U
m−j
=
m
j=0
h
j
⎛
⎝
λ∈T
m−j,j
q
m+1−j−
j
i=1
(λ
i
+1)
g
j
(λ
∗
; V )
⎞
⎠
U
m+1−j
+
m+1
j=0
h
j
⎛
⎝
λ
∗
∈T
m+1−j,j
q
m+1−j−|λ
∗
|
g
j
(λ
∗
; V )
⎞
⎠
U
m+1−j
,
where the sum over λ
∗
∈T
m+1−j,j
is the sum over all Young diagrams in T
m+1−j,j
such that
the last column is empty (for convenience, we define T
k,−1
to be the empty set). Therefore,
U
m+1
V =
m+1
j=0
h
j
⎛
⎝
λ
∗
∈T
m+1−j,j
q
m+1−j−|λ
∗
|
g
j
(λ
∗
; V )
⎞
⎠
U
m+1−j
+
m+1
j=0
h
j
⎛
⎝
λ
∗
∈T
m+1−j,j
q
m+1−j−|λ
∗
|
g
j
(λ
∗
; V )
⎞
⎠
U
m+1−j
,
where the sum over λ
∗
∈T
m+1−j,j
is the sum over all