
312 Commutation Relations, Normal Ordering, and Stirling Numbers
Example 8.91 Let s =2and h =1. From Corollary 8.88, we obtain for the exponential
generating function of the generalized Bell numbers that
Be
2,1
(x)=e
1−
√
1−2x
, (8.73)
and from Corollary 8.90 that the generalized Bell numbers are given by
B
2;1
(n)=e
j≥0
(−1)
n−j
j
n
j!
n−1
k=0
1 −
2k
j
.
Example 8.92 Let s = −1 and h =1. From Corollary 8.88, we obtain for the exponential
generating function of the generalized Bell numbers that
Be
−1,1
(x)=e
x+
1
2
x
2
, (8.74)
and from Corollary 8.89 that the generalized Bell numbers are given by
B
−1;1
(n)=
1
√
e
j≥0
(2j)
n
j!2
j
.
Relation (8.74) shows that B
−1;1
(n) equals the tot