
56 Commutation Relations, Normal Ordering, and Stirling Numbers
Theorem 3.13 (Schl¨omilch) The Stirling numbers of the first kind are given for n ≥ k ≥
1 by
s(n, k)=
n−k
h=0
(−1)
h
n − 1+h
n − k + h
2n − k
n − k −h
S(n − k + h, h). (3.12)
Inserting the explicit expression for S(n, k) into (3.12), one obtains a double sum with only
elementary terms.
The next result is the analog of Theorem 3.3.
Theorem 3.14 For al l k ≥ 0, the exponential generating function of the Stirling numbers
of the first kind is given by
n≥k
s(n, k)
x
n
n!
=
[log(1 + x)]
k
k!
.
Proof Let f
k
(x)=
n≥k
s(n, k)
x
n
n!
. By multiplying the recurrence relation (1) of Theorem
3.12 with
x
n−1
(n−1)!
and summing ...