The q-Deformed Generalized Weyl Algebra 351
Let us consider s = 2, and let us furthermore assume that h = q −1, that is, the variables
U and V satisfy UV = qV U +(q − 1)V
2
. Equation (9.19) with t = q − 1ands =2gives
D
(2)
n+1,n+1
(q − 1) = 1 as well as
D
(2)
n+1,j
(q − 1) = q
n−j
D
(2)
n,j
(q − 1) + q
n+1−j
D
(2)
n,j−1
(q − 1),
which, by an induction in n and j, implies that
D
(2)
n,j
(q − 1) =
n
j
q
q
(
n−j
2
)
. (9.21)
Hence, we obtain that (U + V )
n
=
n
j=0
%
n
j
&
q
q
(
j
2
)
V
j
U
n−j
, which was mentioned in (7.100).
9.3.2 Noncommutative Binomial Formula of Rida
In Section 3.2.5 the partial and complete exponential Bell polynomials were introduced
and some of their basic properties described. Rida introduced in his Ph.D. thesis [931]
and in publications with Schimming [961, 962] certain ...