# Why does $\sum_{p=1}^n \exp\left(\frac{i\pi p l}{2m}\right)/\prod_{k=1,k\neq p}^n\sin\left(\frac{\pi (k-p)}{2m}\right)$ vanish?

While trying to prove some identities for generating functions, I ended up needing to show that

$$\sum_{p=1}^n \exp\left(\frac{i\pi p l}{2m}\right)\prod_{\substack{k=1\\k\neq p}}^n\frac{1}{\sin\left(\frac{\pi (k-p)}{2m}\right)} \stackrel{?}{=} 0$$

for integers $m \geq 1$, $2\leq n\leq 2m$, and $l = -n+2,-n+4,\ldots n-2$. Is this identity known? I have checked it to be valid for small values, but so far I have been unable to prove the general case.

Let us consider the case when $n$ is odd. Let $$P(x):=\exp\left(\frac{ixl}{2}\right)\left(\exp\left(\frac{ix}{2}\right)-\exp\left(\frac{-ix}{2}\right)\right),$$ and notice that the degree of $P$ as an exponential polynomial equals $$\max\left(\left|\frac{l+1}{2}\right|,\left|\frac{l-1}{2}\right|\right)\leq\frac{n-1}{2}.$$ Therefore this is the unique trigonometric polynomial of degree at most $\frac{n-1}{2}$ passing through any given $n$ distinct points. Let these given points be $x_p=\frac{\pi p}{m}$ for $p\in \{1,\dots,n\}$. Then, the trigonometric version of Lagrange's interpolation formula tells us that $P(x)$ can be written as $$P(x)=\sum _{p=1}^n P(x_p)\prod_{\substack{k=1\\k\neq p}}^n\frac{\sin\frac{1}{2}(x_k-x)}{\sin\frac{1}{2}(x_k-x_p)}.$$ By plugging $x=0$ we obtain $$0=\sum _{p=1}^n \exp\left(\frac{i\pi p l}{2m}\right)2i\prod_{k=1}^n\sin\left(\frac{\pi k}{2m}\right)\prod_{\substack{k=1\\k\neq p}}^n\frac{1}{\sin\left(\frac{\pi (k-p)}{2m}\right)},$$ and by dividing by the middle factors the OP's identity follows.
The case of even $n$ can be done similarly.
• Can you provide more detail? The OP's expression also depends on $n$, so I don't see immediately how such a uniform choice of $p(x)$ can solve the problem. Oct 14, 2015 at 21:46