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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
Generalizations of Stirling Numbers 99
to Graves. Using (4.46), one finds that
e
e
x
D
u(x)=
n0
e
nx
n!
n
k=0
|s(n, k)|u
(k)
(x). (4.48)
Setting u(x)=e
x
,wehaveu
(k)
(x)=e
x
. With aid of the fact
n
k=0
|s(n, k)| = n!, we
find the amusing formula e
e
x
D
e
x
=
e
x
1e
x
. Writing the right-hand side as e
xln(1e
x
)
,we
obtain e
e
x
D
e
x
= e
xln(1e
x
)
, in which we recognize a particular instance of a result due to
Crofton [310, Equation (26)] from 1881.
4.2 Stirling Numbers of Hsu and Shiue: A Grand Unification
In this section we describe a generalization of Stirling numbers introduced by Hsu and
Shiue [568] in 1998. These numbers depend on three parameters. By specializing these pa-
rameters ...
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Publisher Resources

ISBN: 9781466579897