
104 Commutation Relations, Normal Ordering, and Stirling Numbers
In particular, the generalized Bell numbers satisfy
B
n
(α, β, r)=
1
e
1/β
k≥0
(kβ + r|α)
n
(1/β)
k
k!
. (4.58)
Proof We follow the proof given in [568]. By (4.56), we find
n≥0
B
n
(x; α, β, r)
t
n
n!
=
1
e
x/β
(1 + αt)
r/α
k≥0
(1 + αt)
kβ/α
(x/β)
k
k!
=
1
e
x/β
k≥0
(x/β)
k
k!
j,l≥
r/α
j
kβ/α
l
(αt)
j+l
,
where we used the binomial formula twice in the last equation. Equating the coefficient of
t
n
/n! on both sides and using Vandermonde’s formula, one obtains
n≥0
B
n
(x; α, β, r)
t
n
n!
=
1
e
x/β
k≥0
(x/β)
k
k!
r/α + kβ/α
n
α
n
n!.
By the identity
r/α+kβ/α
n
α
n
n!=(kβ + r|α)
n
, the assertion is shown.
From (4.56), one may easily obtain ...