6 0. GAUSSIAN BINOMIAL COEFFICIENT S
for all n.
One defines
[n]!
for integers n 0 by
[0]! = 1 and [n]! = [1][2] [n] for n 0.
One has then
for all a n 0.
[n\ [a n\
0.2. An easy calculation shows for all a and n with n 0
"a + l~
n
= v
- n
al
n
j
+
1/a—n+1
a
n - 1
(6)
(7)
(1)
This implies (by induction on n) that all Gaussian binomial coefficients are con-
tained in Z[v,
v~1].
(For fixed n use induction on a for all a n; use 0.1(3) to deal
with a 0.)
The ring
Z[v,v_1]
has a unique involutory automorphism that maps v to
v~l
and v
- 1
to v. Denote it by x »— x. We have clearly [a] = [a] for all a G Z, hence
also
n6
for all a and n. If we apply it to (1), then we get for all a and
(2)
a + 1
+ v
—a+«—1
a
n - 1
An elementary calculation shows for all r 1 that
^(-i)V^- 1 )
2 = 0
0.
(3)
If we apply the automorphism above, then we get also
^ ( _ ! ) ^ - ^ - i )
i=0
0. (4)
0.3. If A : is a ring (with identity) and q G k a unit in k, then there is a unique
ring homomorphism
Z[t;,v-1]
k with i; i- q (and 1 i- 1). All [a],
[n]!,
and
are contained in Z[v, v
- 1
]. We denote their images by [a]v=zq etc.; by abuse of
n
notation we shall often write just [a] = [a]v=q etc., if it is clear which q we consider.
Note that [a]v=i = a, [n]\JZ=1 = n!, and
0.4.
J v=l
f
Warning: The definition of the Gaussian binomial coefficients in 0.1 is not
the traditional one that you can find (e.g.) in [Macdonald], I, §2, ex. 3.
You get that old version, if you change the definition of [a] in 0.1(1) to
(va
l)/(v l). In general, those "classical" Gaussian binomial coefficients
(also called g-binomial coefficients) differ from those here by a q power
(positive or negative).
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