
390 Commutation Relations, Normal Ordering, and Stirling Numbers
whereweusedthat
n
k=0
|s(n, k)| = n!; see Section 8.5.2.
The reason for calling these functions Comtet–Touchard functions is that Comtet [279]
considered expansions of
g(x)
d
dx
n
in detail; see Theorem 4.45.
Proposition 10.35 The Comtet–Touchard functions associated to g are given for any n ∈
N by T
(g)
n
(x)=
n
l=1
T
(g)
n,l
(x),whereT
(g)
n,l
(x) are given by Theorem 4.45.
Proof Inserting the expansion given in Theorem 4.45 into (10.38) shows the assertion.
Let us give the first few Comtet–Touchard functions explicitly, where we denote the
derivative with respect to x by a prime:
T
(g)
1
(x)=g(x),
T
(