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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
292 Commutation Relations, Normal Ordering, and Stirling Numbers
right-hand side powers of the operator D appear and not those of E
s;1
(which corresponds
to U ). For this, we need k-Stirling numbers of the first kind s(k; m, l), defined by Lang [710]
as
X
m
D
m
=
m
l=1
s(k; m, l)X
l(k1)
(X
k
D)
l
, (8.29)
see (4.13). Now, we can formulate the corresponding result in the context of noncommuting
variables satisfying UV = VU + hV
s
[765].
Proposition 8.43 Let U and V be variables satisfying (8.1). Then one has the normal
ordering result (VU)
n
=
n
m=1
S
s;h
(n, m)V
s(nm)+m
U
m
,wherethecoefficientsS
s;h
(n, m)
are given by
S
s;h
(n, m)=h
nm
n
l=m
S(s +1;n, l)s(s; l, m).
Proof Let us first ...
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Publisher Resources

ISBN: 9781466579897