
Chapter 10
A Generalization of Touchard Polynomials
The Touchard polynomials – also called exponential polynomials – may be defined in an
operational fashion for n ∈ N by
T
n
(x)=e
−x
x
d
dx
n
e
x
, (10.1)
see Theorem 3.30. Using (1.27), one obtains from the above definition of the Touchard
polynomials the relation T
n
(x)=
n
k=0
S(n, k)x
k
= B
n
(x), where the second equation
corresponds to the definition of the conventional Bell polynomials. In the present chapter
we introduce Touchard polynomials of higher order. They are defined for any order m ∈ Z
(and n ∈ N)by
T
(m)
n
(x)=e
−x
x
m
d
dx
n
e
x
,
and reduce for m = 1 to the conventional Touchard polynomials, that is, T
(1)
n
= T
n