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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
A Generalization of Touchard Polynomials 395
Let us turn to the case m = 1. Recall that the q-deformed Bessel polynomials were
discussed in Section 9.4.3.
Proposition 10.49 The q-deformed Touchard polynomials of order m = 1 can be ex-
pressed by q-deformed Bessel polynomials as
T
(1)
n|q
(x)=x
n
q
(n1)
2
y
n1
1
x
; q
. (10.45)
Proof Theorem 10.45 implies that T
(1)
n|q
(x)=x
2n
B
2;q
1
|n|q
1 (x),whereweused
that [1]
q
= q
1
. The same argument as in Lemma 8.98 shows that B
s;h|n|q
(x)=
h
n
B
s;1|n|q
(
x
h
). Thus, T
(1)
n|q
(x)=x
2n
q
n
B
2;1|n|q
1
(qx). Inserting the expression resulting
for B
2;1|n|q
1
(qx) from (9.61) gives, after some simplifications, the assertion.
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Publisher Resources

ISBN: 9781466579897