6
$\begingroup$

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

$\endgroup$
9
  • 2
    $\begingroup$ Cyclic sieving gives some values. See Theorem 1.1. $\endgroup$ Sep 9 '17 at 17:09
  • 1
    $\begingroup$ 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
  • 2
    $\begingroup$ 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
  • 2
    $\begingroup$ 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
  • 4
    $\begingroup$ 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
1
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.