
Generalizations of Stirling Numbers 89
Equating both sides yields
n
m=1
(d
m−1
+ l)
s
m
=
s
1
+···+s
n
k=s
1
S
r,s
(k)(l)
k
. (4.36)
This implies for the corresponding polynomials
n
m=1
(d
m−1
+ x)
s
m
=
s
1
+···+s
n
k=s
1
S
r,s
(k)(x)
k
.
This gives an interpretation of the numbers S
r,s
(k) as expansion coefficients. Dividing (4.35)
by l! and summing it over l,onegets
X
r
n
D
s
n
···X
r
1
D
s
1
e
x
=
l≥s
1
n
m=1
(d
m−1
+ l)
s
m
x
l+d
n
l!
.
Combining this with (4.33), we showed the following result [807].
Theorem 4.18 (Generalized Dobi´nski formula) The generalized Bell polynomials
satisfy
B
r,s
(x)=e
−x
k≥s
1
n
m=1
(d
m−1
+ k)
s
m
x
k
k!
. (4.37)
For x = 1, one obtains a generalized Dobi´nski formula for the generalized