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I wonder how to prove ? $$\prod_{k=1}^{n}\left(1+2\cos\frac{2\pi 3^k}{3^n+1} \right)=1$$ give me a tip

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Source?${}{}{}$ –  Gerry Myerson Jan 23 '13 at 12:20
    
@GerryMyerson - do you suspect this question shouldn't be answered? –  nbubis Jan 23 '13 at 12:32
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@nbubis, I just like to know where questions come from before I put a lot of time into them. If, for example, the statement is false, I don't want to waste a lot of time trying to prove it --- so I'd like some reason for believing it's true, some source, some reference. –  Gerry Myerson Jan 23 '13 at 12:42

2 Answers 2

Hint: First prove that: $$\prod_{k=0}^n\left(1+2\cos(3^k x)\right) = 1 + 2\sum_{k=1}^{m}\cos(kx)$$ Where: $$m=\sum_{k=0}^n{3^k} = \frac{3^{1+n}-1}{2}$$

Then show that you are essentially sampling the $\cos$ function between $0$ and $2\pi$ at equal intervals, and since the average is equal to zero, the sum is therefore $1$, as required. To prove the first identity, you can use the Chebyshev relation: $$\cos(nx)=2\cos(x)\cos((n-1)x)-\cos((n-2)x)$$

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thanks!! First prove ??(very difficult) $$\prod_{k=0}^n\left(1+2\cos(3^k x)\right) = 1 + 2\sum_{k=1}^{m}\cos(kx)$$ –  Young Jan 23 '13 at 13:49
    
@user59343 - you asked for a tip.. as I said, you can get this using the Chebyshev relation and starting from the left hand side. –  nbubis Jan 23 '13 at 13:52
    
@user59343 - is that a question? please try to use english as well. –  nbubis Jan 23 '13 at 16:27
    
is it true? $$\sum_{k=1}^{ \frac{3^{1+n}-1}{2}} \cos kx=0$$ where $x=\frac{2\pi}{3^n+1}$ –  Young Jan 23 '13 at 16:31
    
Dear nbubis. I'm sorry. The essence of the previous question is that how I can finish the proof from the first Identity –  Young Jan 23 '13 at 16:49

Let $S_n = \sum_{k=0}^n 3^k = \frac{3^{n+1}-1}{2}$. Then

$$3^{n}- S_{n-1} = 3^{n} - \frac{3^{n}-1}{2} = \frac{3^{n}+1}{2} = S_{n-1}+1. $$

Now by induction we have the following product identity for $n \geq 0$:

$$ \begin{eqnarray} \prod_{k=0}^{n}\left(z^{3^k}+1+z^{-3^k}\right) &=& \left(z^{3^{n}}+1+z^{-3^{n}}\right)\prod_{k=0}^{n-1}\left(z^{3^k}+1+z^{-3^k}\right) \\ &=& \left(z^{3^{n}}+1+z^{-3^{n}}\right) \left(\sum_{k=-S_{n-1}}^{S_{n-1}} z^k\right) \\ &=&\sum_{k=S_{n-1}+1}^{S_n}z^k + \sum_{k=-S_{n-1}}^{S_{n-1}}z^k+\sum_{k=-S_n}^{-S_{n-1}-1} z^k \\ &=& \sum_{k=-S_n}^{S_n} z^k \end{eqnarray} $$

Now take $z = \exp\left(\frac{\pi \, i}{3^n + 1}\right)$ and use that $z^{3^n+1}=-1$ to get

$$\begin{eqnarray} \prod_{k=0}^n\left(1 + 2 \cos \left(\frac{2 \pi \,3^k}{3^n+1}\right)\right) &=& \sum_{k=-S_n}^{S_n}z^{2k} = \frac{z^{2S_n+1}-z^{-2S_n-1}}{z-z^{-1}} = \frac{z^{3^{n+1}}-z^{-3^{n+1}}}{z-z^{-1}} \\ &=& \frac{z^{3(3^n+1)-3} - z^{-3(3^n+1)+3}}{z-z^{-1}} = \frac{z^3-z^{-3}}{z-z^{-1}} = z^2 + 1 + z^{-2} \\ &=& 1 + 2\cos\left(\frac{2\pi}{3^n+1}\right) \end{eqnarray}$$

and your identity follows.

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But the identity in the OP says that the big product is equal to $1$, while your identity says it is equal to $1 + 2\cos\left(\frac{2\pi}{3^n+1}\right)$ ? –  Ewan Delanoy Jan 26 '13 at 15:51
    
@EwanDelanoy look at the range of the index $k$. –  WimC Jan 26 '13 at 15:52
    
indeed. One slightly different index changes everything ... –  Ewan Delanoy Jan 26 '13 at 15:55

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