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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
84 Commutation Relations, Normal Ordering, and Stirling Numbers
Example 4.7 (Narayana polynomials) Agapito [6] showed that
(XD
2
)
n
1
1 x
=
n!(n +1)!N
n
(x)
(1 x)
2n+1
, (4.19)
where N
n
(x)=
n
k=1
N(n, k)x
k
is the Narayana polynomial and where N (n, k) are
Narayana numbers (A001263 in [1019]). The Narayana numbers have many combinatorial
interpretations, for example, as the number of Dyck paths of length n with exactly k peaks.
Agapito [6, Definition 3.4] considered (X
r
D
s
)
n
1
1x
=
K
r,s,n
x
r1
A
r,s
(x)
(1x)
ns+1
and derived prop-
erties of the so defined polynomials A
r,s
(x)=
k
A
r,s
(n, k)x
k
. One has that [6, Theorem
4.1]
A
r,s
(x)=
1
K
r,s,n
nr
k=r
(n(s r)+k)!S
r,s
(n, k)x
k
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Publisher Resources

ISBN: 9781466579897