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This question arose from my initial attempts at answering this question. I later found a way to transform the desired sum into a sum of squares of tangents, but before I did, I found numerically that apparently

$$ \sum_{l=1}^n\tan\frac{jl\pi}{2n+1}\tan\frac{kl\pi}{2n+1}=m_{jkn}(2n+1) $$

with integer factors $m_{jkn}$, for which I haven't been able to find an explanation. If $j$ or $k$ is coprime to $2n+1$, we can sum over $jl$ or $kl$ instead, so most cases (in particular all for $2n+1$ prime) can be reduced to the case $j=1$. Here are the numerically determined factors $m_{1kn}$ for $n\le18$ (with $n$ increasing downward and $k$ increasing to the right):

$$ \begin{array}{r|rr} &1&2&3&4&5&6&7&8&9&10&11&12&13&14&15&16&17&18\\\hline1&1\\ 2&2&0\\ 3&3&-1&1\\ 4&4&0&1&0\\ 5&5&-1&1&1&1\\ 6&6&0&2&-2&0&0\\ 7&7&-1&2&-1&1&0&1\\ 8&8&0&2&0&2&2&2&0\\ 9&9&-1&3&1&1&-3&1&-1&1\\ 10&10&0&3&-2&2&-1&1&0&1&0\\ 11&11&-1&3&-1&1&-1&1&3&-1&1&1\\ 12&12&0&4&0&2&0&0&-4&0&0&0&0\\ 13&13&-1&4&1&3&0&1&-1&1&1&3&0&1\\ 14&14&0&4&-2&2&2&2&0&2&4&0&0&2&0\\ 15&15&-1&5&-1&3&-3&3&-1&3&-5&1&1&1&-1&1\\ 16&16&0&5&0&2&-1&2&0&1&-2&1&1&-2&-2&1&0\\ 17&17&-1&5&1&3&-1&2&-1&1&-1&1&5&1&0&1&1&1\\ 18&18&0&6&-2&4&0&2&0&2&-2&2&-6&0&0&4&2&0&0\\ \end{array} $$

(See also the table in this answer to the other question, which shows the case $j=k+1$; in that case the rows of the table sum to $0$ because of the identity that's the subject of the other question.)

The values $m_{11n}=n$ reflect the sum of squares of tangents that I determined in my answer to the other question. I have no explanation for the remaining values. I've tried using the product formula for the tangent; multiplying by a third tangent to use the triple tangent product formula; and finding a polynomial whose roots are the products being summed; but none of that worked out.

This vaguely reminds me of character theory; the values $\tan\frac{kl\pi}{2n+1}$ for fixed $k$ are like characters, and their dot products are integer multiples of the "group order" $2n+1$; though if they were characters the dot products couldn't be negative.

I'd appreciate any insight into this phenomenon, and of course ideally a way to calculate the $m_{jkn}$.

[Update:]

I've verified the periodicities that Brian observed in comments up to $n=250$:

$$m_{1,k,n+k} = m_{1kn}+[k \text{ odd}]\;,$$

$$m_{1,k+4d+2,k+4d+2+d}=m_{1,k,k+d}\;,$$

where the bracket is the Iverson bracket.

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Apparently when $k$ is even, $m_{1kn}$ is periodic with period $k$, and when $k$ is odd, $m_{1kn}$ is the sum Apparently when $k$ is even, $m_{1kn}$ is periodic with period $k$, and when $k$ is odd, $m_{1kn}$ is the sum of a $k$-periodic sequence and $\langle\underbrace{0,\dots,0}_k,\underbrace{1,\dots,1}_k,\underbrace{2,\dots,2}‌​_k, \dots\rangle$. –  Brian M. Scott Jul 21 '12 at 21:47
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It also appears that the diagonals may be periodic, with periods $2,6,10,14,\dots$. –  Brian M. Scott Jul 21 '12 at 21:57
    
I am very glad if you can help me on this. Thank you. math.stackexchange.com/questions/423297/… –  Maizon Jun 18 '13 at 2:56

2 Answers 2

warning: I suppose that $p=2n+1$ is prime and only prove that $m_{jkn}$ is an integer, without actually computing it (the assumption can perhaps be removed, but the method is non-explicit).

We have $a_k:=i\tan\frac{k\pi}{p}=\frac{\alpha^k-1}{\alpha^k+1}$, where $\alpha=\exp \frac{2\pi i}{p}$. Your sum is in $K:=\mathbb{Q}(\alpha)$ and is invariant under the Galois group, so it is a rational number.

On the other hand, $a_k$'s are the roots of $(1+x)^{p}-(1-x)^{p}$. Now notice that $f(x):=((1+x)^{p}-(1-x)^{p})/2x$ is a monic polynomial with integer coefficients. In particular, your sum is an algebraic integer, and as it is rational, it is an integer.

As $f(0)=p$, we thus have $N_{K/\mathbb{Q}}a_k=p$. This means that $a_k$ is a unit times $1-\alpha$, so your sum is divisible by $1-\alpha$, which means (as it is rational) that your sum is a multiple of $p$.

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Thanks for this! Though you didn't compute the values, this is exactly the sort of perspective that I was hoping to get from an answer; my acquaintance with Galois theory is very cursory. –  joriki Jul 22 '12 at 10:03
up vote 5 down vote accepted

[Just as I was finishing this I saw user8268's answer. I suspect the explanations are related.]

I wasn't intending to answer my own question in this case, but I've now found an explanation. The character theory analogy turned out to be more useful than I expected. Thinking of the values $\tan\frac{kl\pi}{2n+1}$ for fixed $k$ as vectors composed of integer multiples of mutually orthogonal vectors made me wonder what these mutually orthogonal vectors might be. A natural choice was a Fourier-style set of sines or cosines, and indeed it turns out that

$$ \sum_{l=1}^{2n}\sin\frac{2jl\pi}{2n+1}\tan\frac{kl\pi}{2n+1}= \begin{cases} \pm(2n+1)&\gcd(k,2n+1)\mid j\;,\\ 0&\text{otherwise.} \end{cases} $$

I found this surprising at first, but it's actually not too difficult to explain. We have

$$ \def\ex#1{\mathrm e^{#1}}\def\exi#1{\ex{\mathrm i#1}}\def\exm#1{\ex{-\mathrm i#1}} \ex{2n\mathrm i\phi}-\ex{-2n\mathrm i\phi}=\left(\exi{\phi}+\exm{\phi}\right)\left(\ex{(2n-1)\mathrm i\phi}-\ex{(2n-3)\mathrm i\phi}+\dotso+\ex{-(2n-3)\mathrm i\phi}-\ex{-(2n-1)\mathrm i\phi}\right)\;, $$

so

$$ \sin(2j\phi)=2\cos\phi\left(\sin((2j-1)\phi)-\sin((2j-3)\phi)+\dotso+(-1)^{j+1}\sin\phi\right)\;. $$

Thus, for $k=1$ the cosine in the denominator of the tangent is cancelled, and the remaining sine picks out the last term in the sum of alternating sines with odd frequencies, which yields

$$ \sum_{l=1}^{2n}\sin\frac{2jl\pi}{2n+1}\tan\frac{l\pi}{2n+1}=(-1)^{j+1}(2n+1)\;. $$

But since the integers $jl$ and $kl$ in the arguments of both factors only matter $\bmod2n+1$, if $k$ is coprime to $2n+1$, we can sum over $kl$ instead of $l$ and will get the result for $k^{-1}j\bmod(2n+1)$, so for $k$ coprime to $2n+1$

$$ \sum_{l=1}^{2n}\sin\frac{2jl\pi}{2n+1}\tan\frac{kl\pi}{2n+1}=(-1)^{\sigma_k(j)+1}(2n+1)\;, $$

where $\sigma_k$ is the permutation effected by multiplication with $k^{-1}\bmod(2n+1)$. If $1\lt\gcd(k,2n+1)\mid j$, the sum reduces to $\gcd(k,2n+1)$ identical copies, whereas if $\gcd(k,2n+1)\nmid j$, cancellation lets the sum vanish.

Thus, the $m_{jkn}$ are integers because the vectors $\left\{\tan\frac{kl\pi}{2n+1}\right\}_l$ are integer linear combinations of vectors $\left\{\sin\frac{2jl\pi}{2n+1}\right\}_l$ whose dot products are all either $0$ or $\pm1$, and these values can be obtained using only elementary number theory, namely permutations induced by multiplicative inverses.

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our answers might be related, but yours is much better :) –  user8268 Jul 22 '12 at 9:13
    
@user8268: I'm in a dilemma now. I doubt that there will be further answers, so I'd like to accept one of them. Usually I'd tend to accept yours rather than my own, but on the other hand that would be a bit strange when you yourself think that mine is better :-) What would you do? –  joriki Jul 22 '12 at 19:05
    
your answer is better - complete and simpler. The decision is yours (I don't want to spoil your dilemma :) –  user8268 Jul 22 '12 at 20:26
    
@user8268: Darn -- and I thought I could shift the responsibility... :-) –  joriki Jul 22 '12 at 20:41

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