
292 Commutation Relations, Normal Ordering, and Stirling Numbers
right-hand side powers of the operator D appear and not those of E
s;1
(which corresponds
to U ). For this, we need k-Stirling numbers of the first kind s(k; m, l), defined by Lang [710]
as
X
m
D
m
=
m
l=1
s(k; m, l)X
−l(k−1)
(X
k
D)
l
, (8.29)
see (4.13). Now, we can formulate the corresponding result in the context of noncommuting
variables satisfying UV = VU + hV
s
[765].
Proposition 8.43 Let U and V be variables satisfying (8.1). Then one has the normal
ordering result (VU)
n
=
n
m=1
S
s;h
(n, m)V
s(n−m)+m
U
m
,wherethecoefficientsS
s;h
(n, m)
are given by
S
s;h
(n, m)=h
n−m
n
l=m
S(s +1;n, l)s(s; l, m).
Proof Let us first ...