
134 Commutation Relations, Normal Ordering, and Stirling Numbers
Mullen, and Shiue [567]. We will follow their presentation. For any fixed a ∈ R,theDickson
polynomial of degree n is defined by
D
n
(x, a)=
n/2
k=0
n
n − k
n −k
k
(−a)
k
x
n−2k
,
where x denotes the greatest integer less than or equal to x. In particular, we define
D
0
(x, a) = 2 for all real x and a. Define the Dickson–Stirling numbers of the first and of
the second kind, denoted D
1
(n, k; a)andD
2
(n, k; a), by
(x − a)
n
=
n
k=0
D
1
(n, k; a)(D
k
(x, a) − c
k
),
D
n
(x, a) − c
n
=
n
k=0
D
2
(n, k; a)(x − a)
k
,
for n ∈ N
0
,wherec
0
=1andc
k
=0fork ≥ 1. Note that for a =0,wehave
D
n
(x, 0) = x
n
, and the Dickson–Stirling numbers ...