
Generalizations of Stirling Numbers 109
where the Gould–Hopper numbers G(n, k; r, s)aregivenby
G(n, k; r, s)=
1
k!
k
j=0
(−1)
k−j
k
j
(rj + s)
n
. (4.63)
The Gould–Hopper numbers can be expressed as
G(n, k; r, s)=
1
k!
%
Δ
k
(rt + s)
n
&
t=0
. (4.64)
Comparing (4.63) with (4.53), we find that G(n, k; r, s)=r
k
S(n, k;1,r,s). Chak [216] con-
sidered certain generalized Laguerre polynomials defined by
G
(a)
n,b
(x)=x
−a−bn
e
x
(x
b+1
D)
n
x
a
e
−x
.
He showed that G
(a)
n,b
(x)=
n
k=0
(−1)
k
A
(a)
n,k;b
x
k
,where
A
(a)
n,k;b
=
(−1)
n
k!
k
j=0
(−1)
k−j
k
j
(−j − a|b)
n
. (4.65)
Thus, we deduce that A
(a)
n,k;b
=(−1)
n
b
n
G(n, k; −1/a, −a/b). On the other hand, we can use
the expansion (x
b+1
D)
n
= x
nb
n
k=0
S
b+1,1
(n, k)x
k
D