
6 Commutation Relations, Normal Ordering, and Stirling Numbers
FIGURE 1.2: Stirling numbers of the second kind from Stirling’s Methodus Differentialis.
Writing this in one sum and using the recurrence given in Theorem 1.17, one obtains that
z
n+1
=
n+1
k=1
S(n +1,k)(z)
k
,aswastobeshown.
We introduce Stirling numbers of the first kind in analogy to (1.3) as connection coeffi-
cients.
Definition 1.21 The Stirling numbers of the first kind s(n, k) are defined as connection
coefficients between falling polynomials and monomials,
(z)
n
=
n
k=1
s(n, k)z
k
. (1.4)
Combining (1.3) and (1.4), this gives the orthogonality relations
n
k=1
s(n, k)S(k, l)=
n
k=1
S(n, k)s(k, l)=δ
n,l
,