One q-binomial identity

Question

There is an identity on $q$-binomial coefficients: $\sum_{i=0}^{2n}(-1)^i{{2n}\choose{i}}_q = (1-q)(1-q^3)\ldots(1-q^{2n-1})$. Does anybody know where one can look for a proof or how to prove it?
I used the q-binomial theorem in order to rewrite $(1-q)(1-q^3)\ldots(1-q^{2n-1})=\sum_{k=0}^{n}q^{k^2}{{n}\choose{k}}_{q^2}$. But this is still far from the sum I would like to obtain.

Answer 1

Jonathan Azose’s 2007 Harvey Mudd College undergraduate thesis, Applications of the q-Binomial Coefficients to Counting Problems, introduces a nice combinatorial interpretation of the $q$-binomial coefficients and in Section 3.2 uses it to prove the desired result fairly easily.