ABSTRACT
The Touchard polynomials – also called exponential polynomials – may be defined in an operational fashion for n ∈ N by
Tn(x) = e −x ( x d
dx
)n ex, (10.1)
see Theorem 3.30. Using (1.27), one obtains from the above definition of the Touchard polynomials the relation Tn(x) =
∑n k=0 S(n, k)x
k = Bn(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 ( xm
d
dx
)n ex,
and reduce for m = 1 to the conventional Touchard polynomials, that is, T (1) n = Tn. We
discuss for T (m) n the recurrence relation, the exponential generating function, and other
properties. Since normal ordering ( x ddx
)n involves generalized Stirling numbers, one ex-
pects a close connection to the generalized Stirling numbers Ss;h(n, k) or Bell polynomials Bs;h|n(x) considered in Chapter 8. This will be confirmed, and relations between Touchard polynomials of low order and some well-known families of polynomials are given.
