
The q-Deformed Weyl Algebra and the Meromorphic Weyl Algebra 245
this situation was considered by letting
ˆ
f
+
ˆ
f
−
= ψ(ˆn),
ˆ
f
−
ˆ
f
+
= ψ(ˆn +1),[ˆn,
ˆ
f
±
]=±
ˆ
f
±
.
This can be written, equivalently, in a form analogous to (7.46),
{
ˆ
f
−
,
ˆ
f
+
} = ψ(ˆn +1)+ψ(ˆn) ≡ Ψ(ˆn), [ˆn,
ˆ
f
±
]=±
ˆ
f
±
.
Choosing Ψ(ˆn)=1+2κˆn, one obtains the algebra treated in [261]. The authors observed
that one can obtain a representation of {
ˆ
f
−
,
ˆ
f
+
} =1+2κˆn and [ˆn,
ˆ
f
±
]=±
ˆ
f
±
by (see
Exercise 7.6)
ˆ
f
+
→ z,
ˆ
f
−
→ κ
d
dz
+(1− κ)D
−1
, ˆn → z
d
dz
, (7.47)
where D
−1
is the fermionic derivative given in (7.10). By normal ordering, they introduced
κ-deformed Stirling numbers of the second kind,
(
ˆ
f
+
ˆ
f
−
)
r
=
r
k=1
(
ˆ
f
+