
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
1−x
=
K
r,s,n
x
r−1
A
r,s
(x)
(1−x)
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