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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
Chapter 10
A Generalization of Touchard Polynomials
The Touchard polynomials also called exponential polynomials may be defined in an
operational fashion for n N by
T
n
(x)=e
x
x
d
dx
n
e
x
, (10.1)
see Theorem 3.30. Using (1.27), one obtains from the above definition of the Touchard
polynomials the relation T
n
(x)=
n
k=0
S(n, k)x
k
= B
n
(x), where the second equation
corresponds to the definition of the conventional Bell polynomials. In the present chapter
we introduce Touchard polynomials of higher order. They are defined for any order m Z
(and n N)by
T
(m)
n
(x)=e
x
x
m
d
dx
n
e
x
,
and reduce for m = 1 to the conventional Touchard polynomials, that is, T
(1)
n
= T
n
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Publisher Resources

ISBN: 9781466579897