250 Commutation Relations, Normal Ordering, and Stirling Numbers
whereweusedthatforallβ ∈ F
+
β
(cF (c, c
†
)) one has :β:= (c
†
)
−1
:β
:. This formal notation
means that the degree of the creation operator has to be decreased by one. This is possible
since, by definition, there is at least one singleton of type C in every β
, namely the one
which becomes “free” after deleting the edge in step (i) from above. Let us write, as above,
:β
:= (c
†
)
a
β
c
b
β
. Assuming for the moment that
β∈F
+
β
(cF (c,c
†
))
W
q
(β)=[a
β
]
q
W
q
(β
), (7.55)
we have, therefore, shown that
β∈F
+
(cF (c,c
†
))
W
q
(β):β:=
β
∈F (F (c,c
†
))
W
q
(β
)[a
β
]
q
(c
†
)
a
β
−1
c
b
β
. (7.56)
Switching to the more convenient notation β
γ and inserting (7.54) and (7.56) into
(7.53) yields
γ∈F (F (c,c
†
))
W
q
(γ)
'
q
a
γ
(c
†
)
a
γ
c