
380 Commutation Relations, Normal Ordering, and Stirling Numbers
Proposition 10.10 Let s = −1 and h =1.Thenth generalized Bell polynomial B
−1;1|n
(x)
can be expressed by Hermite polynomials, that is,
B
−1;1|n
(x)=
i
√
x
√
2
n
H
n
√
x
i
√
2
.
In particular, the nth generalized Bell number B
−1;1
(n) is given by
B
−1;1
(n)=
i
√
2
n
H
n
1
i
√
2
.
10.1.2 Exponential Generating Functions
In this section we determine the exponential generating function of the generalized
Touchard polynomials. Recall that the case m = 1 corresponds to the conventional Bell
polynomials, yielding the well-known result
n≥0
λ
n
n!
T
(1)
n
(x)=
n≥0
λ
n
n!
B
n
(x)=e
x(e
λ
−1)
,
see Theorem 3.29(3). From the definition of ...