I was wondering if anyone could show me how to prove q-binomial identities? I do not have a single example in my notes, and I can't seem to find any online.
For example, consider: ${a + 1 + b \brack b}_q = \sum\limits_{j=0}^{b} q^{(a+1)(b-j)}{a+j \brack j}_q$
I haven't made much progress on this one, but here's one that I have managed to get something out of: ${2n \brack n}_q = \sum\limits_{k=0}^{n} q^{k^{2}}{n \brack k}_q$
Using the q-binomial theorem from my notes, which is as follows: $(1+qx)(1+q^{2}x)...(1+q^{n}x) = \sum\limits_{k=0}^{n} q^{k(k+1)/2}{n \brack k}_q x^{k}$, I have managed to show that the coefficient of $x^{n}$ is equal to: $\sum\limits_{k=0}^{n} q^{(2k^{2} - 2nk + n^{2} + n)/2} {n \brack k}_q {n \brack n-k}_q$, which is when I was working on the right hand side of the identity. In order to get here, I considered the product of $(1+qx)...(1+q^{n}x)(1+qx)...(1+q^{n}x)$, then tried obtaining the coefficient of $x^n$, as one would in the ordinary binomial proof. I've been trying to mimic the proofs of the regular binomial counterparts of these identities but without much luck.
Help would be appreciated, as I have a midterm exam coming up soon. Thanks :)