
230 Commutation Relations, Normal Ordering, and Stirling Numbers
obtain a nice limit for q →−1. It is discussed – in the spirit of taking the limit q →−1–
in [1002] that the fermionic binomial coefficients are given by
n
k
−1
=
0, if n is even and k is odd,
n/2
k/2
, otherwise,
(7.9)
and that the corresponding Galois numbers (see Appendix A) are given by
G
n
(−1) = 2
n/2
.
Here we denoted by x the largest integer less than or equal to x and, similarly, by x the
smallest integer larger than or equal to x. Manin and Radul [760] introduced the coefficients
%
n
k
&
−1
in a different context as superbinomial coefficients. From recurrence (A.2) of the q-
binomial