
The q-Deformed Weyl Algebra and the Meromorphic Weyl Algebra 235
By linearity, this can be extended to arbitrary polynomials (or formal series),
g(U)f(V )=
k≥0
h
k
[k]
q
!
f
(k)
(V )
k
q
g
(k)
(U),
see [264, Equation (37)]. This is the q-analog of (6.4). Any polynomial (or formal series) f
in these variables U and V canbewritteninauniquenormalorderedfashionf(U, V )=
f
(n)
(U, V )=
r,s
f
(n)
r,s
(q)V
r
U
s
, and, similarly, in a unique antinormal ordered expression
f(U, V )=f
(a)
(U, V )=
r,s
f
(a)
r,s
(q)U
r
V
s
.
In the undeformed case (q = 1), an explicit relation between the sets of coefficients was given
in Proposition 5.34 (where h = 1). Luo [746, Equation (2)] derived the followi ...