Editing Gaussian binomial coefficient (section)

===Analogs of Pascal's identity===

The analogs of [[Pascal's identity]] for the Gaussian binomial coefficients are:<ref>Mukhin, Eugene, chapter 3</ref>

:<math>{m \choose r}_q = q^r {m-1 \choose r}_q + {m-1 \choose r-1}_q</math>

and

:<math>{m \choose r}_q = {m-1 \choose r}_q + q^{m-r}{m-1 \choose r-1}_q.</math>

When <math>q=1</math>, these both give the usual binomial identity. We can see that as <math>m\to\infty</math>, both equations remain valid.

The first Pascal analog allows computation of the Gaussian binomial coefficients recursively (with respect to ''m'' ) using the initial values

:<math>{m \choose m}_q ={m \choose 0}_q=1 </math>

and also shows that the Gaussian binomial coefficients are indeed polynomials (in ''q'').

The second Pascal analog follows from the first using the substitution <math> r \rightarrow m-r </math> and the invariance of the Gaussian binomial coefficients under the reflection <math> r \rightarrow m-r </math>.

These identities have natural interpretations in terms of linear algebra. Recall that <math>\tbinom{m}{r}_q</math> counts ''r''-dimensional subspaces <math>V\subset \mathbb{F}_q^m</math>, and let <math>\pi:\mathbb{F}_q^m \to \mathbb{F}_q^{m-1} </math> be a projection with one-dimensional nullspace <math>E_1 </math>. The first identity comes from the bijection which takes <math>V\subset \mathbb{F}_q^m </math> to the subspace <math>V' = \pi(V)\subset \mathbb{F}_q^{m-1}</math>; in case <math>E_1\not\subset V</math>, the space <math>V'</math> is ''r''-dimensional, and we must also keep track of the linear function <math>\phi:V'\to E_1</math> whose graph is <math>V</math>; but in case <math>E_1\subset V</math>, the space <math>V'</math> is (''r''−1)-dimensional, and we can reconstruct <math>V=\pi^{-1}(V')</math> without any extra information. The second identity has a similar interpretation, taking <math>V</math> to <math>V' = V\cap E_{n-1}</math> for an (''m''−1)-dimensional space <math>E_{m-1}</math>, again splitting into two cases.