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).