
A Generalization of Touchard Polynomials 385
m ∈{2, 3,...}. Since the case s(m) = 0 (that is, m = 1) is the conventional case, we restrict
to the case m ∈{2, 3,...}. Using Theorem 8.97, one obtains that
T
(m)
n+1
(x)=x
(m−1)(n+1)
B
m−1
m
;m|n+1
(x)
= x
m−1
x
(m−1)n
x
n
k=0
n
k
Γ(n − k +
m
m−1
)
Γ(
m
m−1
)
(m − 1)
n−k
B
m−1
m
;m|k
(x)
= x
m
n
k=0
n
k
Γ(n − k +
m
m−1
)
Γ(
m
m−1
)
7
(m − 1)x
(m−1)
8
n−k
T
(m)
k
(x).
The last equation is the sought-for recurrence relation in the case m ≥ 2. Let us turn to
Touchard polynomials of negative order, that is, to T
(−m
)
n
(x) with m
∈ N. Using (10.6),
we see that the parameter s(−m
) of the corresponding generalized Bell polynomial is given
by s(−m
)=
m
+1
m
∈