As the title says, given a general $q$-binomial $\binom{n}{k}\_q$,
is there some general result regarding its value at a root of unity, $q = \exp(2\pi i r/N)$? 



Q-binomials at roots of unity
So, the canonical answer is the [q-Lucas theorem,](https://www.math.upenn.edu/%7Epeal/polynomials/q-analogues.htm#prelimQanaloguesQLucas) as pointed out in the comments.


This was proved in
*Olive, Gloria*, [**Generalized powers**](http://dx.doi.org/10.2307/2313851), Am. Math. Mon. 72, 619-627 (1965). [ZBL0215.07003](https://zbmath.org/?q=an:0215.07003).


