
A Generalization of the Weyl Algebra 287
Example 8.28 (Weyl algebra) Let δ(V )=h, that is, UV −VU = h. Iterating δ(V
m
)=
mhV
m−1
, one finds δ
r
(V
m
)=
m!
(m−r)!
h
r
V
m−r
. Inserting this into (8.14),oneobtains
U
n
V
m
=
n
k=0
n
k
m
n − k
(n − k)!h
n−k
V
m−(n−k)
U
k
.
Switching to l = n − k, one recovers (6.2).
Example 8.29 (Meromorphic Weyl algebra) Let δ(V )=−V
2
, that is, UV − VU =
−V
2
. A simple induction shows that
δ
r
(V
m
)=(−1)
r
(m + r − 1)!
(m − 1)!
V
m+r
, (8.15)
implying that (8.14) reduces for the case at hand to (7.71).
The following analog of Proposition 8.25 was given in [87, Equation (1.3)] and had
appeared earlier in [577, Equation (4)].
Proposition 8.30 (Irving; Benkart, ...