
124 Commutation Relations, Normal Ordering, and Stirling Numbers
In [1046], two kinds of p-Stirling numbers were introduced by ξ
k
= k
p
and ξ
k
=
k+p−1
p
(p ∈ N
0
) and interpreted combinatorially in terms of “matrix partitions” and “matrix
permutations”. See [1047] for more properties of these numbers. In [293], “dually weighted
Stirling numbers” were introduced by defining them in terms of symmetric functions in
analogy to (4.94) and (4.95), but where two sequences of weight functions are used.
Let us note that the reciprocal relations
a
n
=
n
k=0
s
ξ
(n, k)b
k
⇔ b
n
=
n
k=0
S
ξ
(n, k)a
k
hold true if either the sequence (a
k
)
k∈N
0
or the sequence (b
k
)
k∈N
0
is given arbitrarily ...