Editing Gaussian binomial coefficient (section)

==Definition==

The Gaussian binomial coefficients are defined by:<ref>Mukhin, Eugene, chapter 3</ref>

:<math>{m \choose r}_q
= 
\frac{(1-q^m)(1-q^{m-1})\cdots(1-q^{m-r+1})} {(1-q)(1-q^2)\cdots(1-q^r)} </math>

where ''m'' and ''r'' are non-negative integers.  If {{math|''r'' > ''m''}}, this evaluates to 0.  For {{math|''r'' {{=}} 0}}, the value is 1 since both the numerator and denominator are [[empty product]]s.

Although the formula at first appears to be a [[rational function]], it actually is a polynomial, because the division is exact in '''Z'''<nowiki>[</nowiki>''q''<nowiki>]</nowiki>

All of the factors in numerator and denominator are divisible by {{math|1 − ''q''}}, and the quotient is the [[Q-analog#Introductory examples|''q''-number]]:

:<math>[k]_q=\sum_{0\leq i<k}q^i=1+q+q^2+\cdots+q^{k-1}=
\begin{cases}
\frac{1-q^k}{1-q} & \text{for} & q \neq 1 \\
k & \text{for} & q = 1 
\end{cases},</math>

Dividing out these factors gives the equivalent formula

:<math>{m \choose r}_q=\frac{[m]_q[m-1]_q\cdots[m-r+1]_q}{[1]_q[2]_q\cdots[r]_q}\quad(r\leq m).</math>

In terms of the [[Q-analog#Introductory examples|''q'' factorial]] <math>[n]_q!=[1]_q[2]_q\cdots[n]_q</math>, the formula can be stated as
:<math>{m \choose r}_q=\frac{[m]_q!}{[r]_q!\,[m-r]_q!}\quad(r\leq m).</math>

Substituting {{math|''q'' {{=}} 1}} into <math>\tbinom mr_q</math> gives the ordinary binomial coefficient <math>\tbinom mr</math>.

The Gaussian binomial coefficient has finite values as <math>m\rightarrow \infty</math>:

:<math>{\infty \choose r}_q = \lim_{m\rightarrow \infty} {m \choose r}_q = \frac{1} {(1-q)(1-q^2)\cdots(1-q^r)} = \frac{1}{[r]_q!\,(1-q)^r}</math>