closed form for $\prod_{k=0}^{n}\left(1-2\alpha \cos\frac{2\pi k}{n}\right) $

Question

Does anybody know a closed form for this multiplication?

$$\prod_{k=0}^{n}\left(1-2\alpha \cos\frac{2\pi k}{n}\right)$$ where $\alpha$ is a real number

or maybe even a potential method to use to evaluate it?

Answer 1

With $\zeta=e^{2\pi\mathrm{i}/n}$, we have $z^n-1=\prod\limits_{k=0}^{n-1}(z-\zeta^k)=\prod\limits_{k=0}^{n-1}(z-\zeta^{-k})$. Multiplying these, we get $$(z^n-1)^2=\prod_{k=0}^{n-1}\left(z^2-2z\cos\frac{2\pi k}{n}+1\right)=(z^2+1)^n\prod_{k=0}^{n-1}\left(1-\frac{2z}{z^2+1}\cos\frac{2\pi k}{n}\right).$$ Thus, to get your product, you have to solve $z/(z^2+1)=\alpha$ and add an extra term (with $k=n$).