Closed formula from product of sine

Question

Someone knows a formula for expression below?

$$\prod_{k=2}^{n}\sin(\pi/k)$$

Answer 1

Partial Solution: As $k\to+\infty$, $\frac{\pi}k\to0$ so we can use the small-angle approximation: $\sin\left(\frac{\pi}k\right)\approx\frac{\pi}k$. We can also ignore $k=2$ and start from $k=3$, since $\sin(\pi/2)=1$.

$$\begin{aligned} P(n)&=\prod_{k=3}^n\sin\left(\frac\pi k\right)\\ &=\prod_{k=3}^n\left[\left(\frac\pi k\right)+\frac1{3!}\left(\frac\pi k\right)^3+\frac1{5!}\left(\frac\pi k\right)^5+\ldots\right]\\ &\approx\frac{\pi^{n-2}}{n!/2}\\ \end{aligned}$$

This fits reasonably well and we get a maximum absolute error of $0.2101$ (at $x=4$) and the approximation seems to have a maximum relative error of around $1.0$ (although I've only managed to check this up to $n=224$, due to the factorial term). The approximation (black) and true value (green) of $P(n)$ are shown in the figure below.

You could also consider using the fact that:

$$\begin{aligned} \ln P&=\ln\left[\prod_{k=2}^n\sin\left(\frac\pi k\right)\right]\\ &=\sum_{k=2}^n\ln\left[\sin\left(\frac\pi k\right)\right]\\ \end{aligned}$$

So if you can find a closed form for $\ln\sin(\pi/k)$, you'd have an solution.

Answer 2

In the same spirit as Jam's answer, consider $$\log(P_n)=\sum_{k=3}^n\log\left[\sin\left(\frac\pi k\right)\right]$$ and, using Taylor $$\log\left[\sin\left(\frac\pi k\right)\right]=\log \left(\frac{\pi}{k}\right)-\frac{\pi ^2}{6 k^2}-\frac{\pi ^4}{180 k^4}-\frac{\pi ^6}{2835 k^6}-\frac{\pi ^8}{37800 k^8}+O\left(\frac{1}{k^{10}}\right)$$ Now, computing the summation, $$\log(P_n)=\log \left(\frac{2}{n!}\right)-\left(\frac{\pi ^2 H_n^{(2)}}{6}+\frac{\pi ^4 H_n^{(4)}}{180}+\frac{\pi ^6 H_n^{(6)}}{2835}+\frac{\pi ^8 H_n^{(8)}}{37800}\right)+(n-2) \log (\pi )+K$$ where appear generalized harmonic numbers and $$K=\frac{5 \pi ^2}{24}+\frac{17 \pi ^4}{2880}+\frac{13 \pi ^6}{36288}+\frac{257 \pi ^8}{9676800}$$ which seems to be quite good $$\left( \begin{array}{cccc} n & \log(P_n) \text{ approx} & P_n \text{ approx} & P_n \text{ exact}\\ 3 & -0.1438373 & 0.86602864 & 0.86602540 \\ 4 & -0.4904107 & 0.61237485 & 0.61237244 \\ 5 & -1.0218043 & 0.35994491 & 0.35994349 \\ 6 & -1.7149515 & 0.17997246 & 0.17997174 \\ 7 & -2.5499301 & 0.07808712 & 0.07808681 \\ 8 & -3.5104773 & 0.02988265 & 0.02988253 \\ 9 & -4.5833629 & 0.01022047 & 0.01022043 \\ 10 & -5.7577220 & 0.00315830 & 0.00315829 \end{array} \right)$$