# Q-binomials at roots of unity

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, $q = \exp(2\pi i r/N)$?

• Cyclic sieving gives some values. See Theorem 1.1. Sep 9 '17 at 17:09
• I figure you may be aware, but perhaps the comment will be interesting and relevant to others. It's an interesting question. I am not sure what can be said about other cases. Sep 9 '17 at 17:23
• Is the $q$-Lucas theorem (see, eg. Lemma 3.1 in arxiv.org/pdf/1101.1020.pdf) of any help? Sep 9 '17 at 18:08
• You can put $q=\exp(2\pi ir/N)$ in the $q$-binomial theorem (say as stated in equation (1.87) of Enumerative Combinatorics, second ed.) and equate coefficients of $x^k$ to get an answer. A special case is Exercise 1.98. Sep 9 '17 at 20:57
• To give credit where credit is due, the $q$-Lucas theorem was first published, as far as I know, by Gloria Olive, Generalized powers, Amer. Math. Monthly 72 (1965), 619–625, equation (1.2.4). (The proof, which is not difficult, is apparently in her 1963 Ph.D. thesis, which I have not seen.) The theorem has been rediscovered many times since then, but I have not come across any earlier occurrence of it. Sep 11 '17 at 23:40