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Commutation Relations, Normal Ordering, and Stirling Numbers
book

Commutation Relations, Normal Ordering, and Stirling Numbers

by Toufik Mansour, Matthias Schork
September 2015
Intermediate to advanced content levelIntermediate to advanced
528 pages
19h 34m
English
Chapman and Hall/CRC
Content preview from Commutation Relations, Normal Ordering, and Stirling Numbers
The q-Deformed Generalized Weyl Algebra 361
Before closing this section, let us point out that many of the above relations can also
be shown directly using the operator interpretation. For example, Corollary 9.40 can be
shown as follows. If (f )(x)=f(qx) (see Appendix A), then ({(q 1)E
2;1|q
}f)(x)=((q
1)X
2
D
q
f)(x) is given by (q 1)x
2
f(qx)f(x)
(q1)x
=((X( 1))f)(x), implying for the left-hand
side (X +(q 1)E
2;1|q
)
n
=(X X( 1))
n
=(X)
n
. On the other hand, using X = qX
as well as (q 1)
nk
E
nk
2;1|q
=(X( 1))
nk
= X
nk
(q
nk1
1) ···(q 1)( 1), one
finds for the right-hand side
n
k=0
n
k
q
q
(
k
2
)
X
k
(q 1)
nk
E
nk
2;1|q
= X
n
n
k=0
n
k
q
q
(
k
2
)
(
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Publisher Resources

ISBN: 9781466579897