
The q-Deformed Weyl Algebra and the Meromorphic Weyl Algebra 255
Starting directly from the recurrence relations, the following result was obtained in [973].
Proposition 7.37 The fermionic Stirling numbers of the first kind are given for n ≥ k ≥ 1
by
s
f
(n, k)=(−1)
k−
n+1
2
n
2
k −
n+1
2
.
The fermionic Stirling numbers of the second kind are given for n ≥ k ≥ 1 by
S
f
(n, k)=(−1)
k
2
n −
k
2
−1
k−1
2
.
In [1124], one can also find algebraic proofs for the above expressions. See also [1015] for a
different perspective. It is a straightforward calculation to check (do it!) that the following
orthogonality relations hold true,
n
k=m
S
f
(n, k)s
f
(k, m)=
n
k=0
s
f
(n,