
Stirling and Bell Numbers 71
see (3.21) for L. Thus, we define L
q
to be the linear functional on V
q
that satisfies L
q
(1) = 1
and
L
q
([x]
n|q
)=1, (3.47)
for all n ≥ 1. Let e
q
(x)=
n≥0
x
n
[n]
q
!
be the q-exponential function. Then,
1=
1
e
q
(1)
j≥0
1
[j]
q
!
=
1
e
q
(1)
j≥0
[j]
n|q
[j]
q
!
,
which by (3.47) implies that L
q
([x]
n|q
)=
1
e
q
(1)
j≥0
[j]
n|q
[j]
q
!
. Since [x]
n|q
is a basis of V
q
,we
obtain for any polynomial p ∈ V
q
that [824, Theorem 1.1]
L
q
(p(x)) =
1
e
q
(1)
j≥0
p(j)
[j]
q
!
. (3.48)
On the other hand, applying L
q
to (3.33), one finds
L
q
(([x]
q
)
n
)=
n
k=0
S
q
(n, k)=
n|q
.
Thus, considering p(x)=([x]
q
)
n
in (3.48), we get the following q-analog of Dobi´nski’s
formula [824, Corollary 1.1].
Theorem 3.47