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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
48 Commutation Relations, Normal Ordering, and Stirling Numbers
where
f denotes the compositional inverse of f ,that,isf (f(t)) = f(f(t)) = t.Observethat
(2.26) is of the form (2.22). The associated raising and lowering operators are then given by
M
g,f
=
X
g
(D)
g(D)
1
f
(D)
,P
f
= f (D), (2.27)
see [940, Theorem 3.7.1]. The order of terms in M
g,f
is important; note that X enters
only linearly. M
g,f
is called Sheffer shift. In this situation, we say that u
n
(x) is Sheffer for
(g(t),f(t)) and we also denote this by u
n
(x) (g(t),f(t)). Clearly, x
n
(1,t).
Example 2.79 Let P (D)=
1
2
D, that is, f (t)=
1
2
t.Letg(t)=e
t
2
/4
. Then we find from
(2.27) that M(X, D)=2X D
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Publisher Resources

ISBN: 9781466579897