
Generalizations of Stirling Numbers 99
to Graves. Using (4.46), one finds that
e
e
x
D
u(x)=
n≥0
e
nx
n!
n
k=0
|s(n, k)|u
(k)
(x). (4.48)
Setting u(x)=e
x
,wehaveu
(k)
(x)=e
x
. With aid of the fact
n
k=0
|s(n, k)| = n!, we
find the amusing formula e
e
x
D
e
x
=
e
x
1−e
x
. Writing the right-hand side as e
x−ln(1−e
x
)
,we
obtain e
e
x
D
e
x
= e
x−ln(1−e
x
)
, in which we recognize a particular instance of a result due to
Crofton [310, Equation (26)] from 1881.
4.2 Stirling Numbers of Hsu and Shiue: A Grand Unification
In this section we describe a generalization of Stirling numbers introduced by Hsu and
Shiue [568] in 1998. These numbers depend on three parameters. By specializing these pa-
rameters ...