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Let $N\geq 1$ and $0\leq k\leq N-1$ be fixed numbers, and $c>0$ be unknown. Suppose we have

\begin{eqnarray} \sum\limits_{i=0}^{N-1} \exp\left(-jc\sin\left(\frac{2\pi i}{N}-\frac{\pi k}{N}\right)\right)=0, \end{eqnarray} where $j:=\sqrt{-1}$. Equivalently, we have \begin{eqnarray} \sum\limits_{i=0}^{N-1} \cos\left(c\sin\left(\frac{2\pi i}{N}-\frac{\pi k}{N}\right)\right)=0,~~~~~~~~~~~~~~~\sum\limits_{i=0}^{N-1} \sin\left(c\sin\left(\frac{2\pi i}{N}-\frac{\pi k}{N}\right)\right)=0. \end{eqnarray}

Is it possible to simplify and solve the above equations to arrive at a closed form expression for $c$ in terms of $k$ and $N$?

Any help would be greatly appreciated.

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  • $\begingroup$ Is $j:=\sqrt{-1}$? $\endgroup$ – Alex R. Jun 29 '16 at 18:13
  • $\begingroup$ @AlexR. Yes. That is right. Thanks for the comment. I'll add that in the question. $\endgroup$ – PN Karthik Jun 29 '16 at 18:27
  • $\begingroup$ Can you do the case $N=1$? How about $N=2$? $\endgroup$ – GEdgar Jul 1 '16 at 12:11
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Up to $N=4$ it can be solved symbolically. But Maple finds no symbolic solution for $N=5$. When $N=5,k=0$ we get $$ Q_{5,0}(c) := 1+{{\rm e}^{-jc\sin \left( 2/5\,\pi \right) }}+{{\rm e}^{-jc\sin \left( \pi /5 \right) }}+{{\rm e}^{jc\sin \left( \pi /5 \right) }}+{ {\rm e}^{jc\sin \left( 2/5\,\pi \right) }} $$ By its symmetry, the imaginary part is always zero. $Q_{5,0}$ is almost periodic, but not periodic. $Q_{5,0}(c)=0$ has infinitely many solutions for $c$.
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