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

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    $\begingroup$ Cyclic sieving gives some values. See Theorem 1.1. $\endgroup$ Sep 9 '17 at 17:09
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    $\begingroup$ 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. $\endgroup$ Sep 9 '17 at 17:23
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    $\begingroup$ Is the $q$-Lucas theorem (see, eg. Lemma 3.1 in arxiv.org/pdf/1101.1020.pdf) of any help? $\endgroup$ Sep 9 '17 at 18:08
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    $\begingroup$ 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. $\endgroup$ Sep 9 '17 at 20:57
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    $\begingroup$ 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. $\endgroup$
    – Ira Gessel
    Sep 11 '17 at 23:40

So, the canonical answer is the q-Lucas theorem, as pointed out in the comments.

This was proved in Olive, Gloria, Generalized powers, Am. Math. Mon. 72, 619-627 (1965). ZBL0215.07003.


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