How to evaluate $\sum\limits_{k=0}^{n-1} \sin^t(\pi k/2n)$?

Question

How to evaluate $\displaystyle\sum_{k=0}^{n-1} \sin^t\left(\frac{\pi k}{2n}\right)$?

$t,n$ are constants $\in \Bbb{Z}$.

My try:

$$\begin{align} \zeta:=e^{i\pi/2n} \implies & \sum_{k=0}^{n-1}\sin^t\left(\frac{\pi k}{2n}\right) \\ = &\frac{1}{2^t}\sum_{k=0}^{n-1}\left(\zeta^k- \zeta^{-k}\right)^{t} \\ = &\frac{1}{2^t}\sum_{k=0}^{n-1}\left(\zeta^{-kt}\right)\left(\zeta^{2k}-1\right)^{t}\\ \end{align}$$

How to proceed from here?

If a closed-form solution is not possible, can we still work out cases such as $t = n, t = 2n, $ etc?

EDIT: Sangchul Lee provided an answer for even t. Still looking for a solution for odd t as well.

Answer 1

Here is an answer when $t = 2s$ is even.

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Let $n, s$ be non-negative integers. Then with $\zeta = \mathrm{e}^{i\pi/n}$ and $\omega = \zeta^2 = \mathrm{e}^{2i\pi/n}$, we have

\begin{align*} 2^{2s} \sum_{k=0}^{n-1} \sin^{2s} \left(\frac{\pi k}{n} \right) &= \sum_{k=0}^{n-1} \left( \frac{\zeta^k + \zeta^{n-k}}{i} \right)^{2s} \\ &= (-1)^{-s} \sum_{k=0}^{n-1}\sum_{l=0}^{2s} \binom{2s}{l} \zeta^{(2s-l)k+l(n-k)} \\ &= (-1)^{-s} \sum_{k=0}^{n-1}\sum_{l=0}^{2s} \binom{2s}{l} (-1)^l \omega^{(s-l)k}. \end{align*}

Interchanging the order of summation, we have

\begin{align*} 2^{2s} \sum_{k=0}^{n-1} \sin^{2s} \left(\frac{\pi k}{n} \right) &= \sum_{l=0}^{2s} (-1)^{l-s} \binom{2s}{l} \left( \sum_{k=0}^{n-1} (\omega^{s-l})^k \right) \\ &= \sum_{l=0}^{2s} (-1)^{l-s} \binom{2s}{l} \left( n \cdot \mathbf{1}_{\{ l \equiv s \ (\mathrm{mod} \ n)\}} \right) \\ &= n \sum_{j} (-1)^{nj} \binom{2s}{s+nj}, \end{align*}

where the last summation runs over all integers $j$ such that $-\frac{s}{n} \leq j \leq \frac{s}{n}$.

Examples. As special cases, plugging $s = n$ yields

$$ \sum_{k=0}^{n-1} \sin^{2n} \left(\frac{\pi k}{n} \right) = \frac{n}{2^{2n}} \left[ \binom{2n}{n} + (-1)^n 2 \right], $$

and similarly, replacing $n$ by $2n$ and plugging $s = n$ yields

$$ \sum_{k=0}^{n-1} \sin^{2n} \left(\frac{\pi k}{2n} \right) = \frac{n}{2^{2n}} \binom{2n}{n} - \frac{1}{2}. $$

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Addendum. Here is an intuition on why we expect the sum to be simplified. If $t = 2s$ is even, then we can expand $\sin^{2s} x$ into a linear combination of $1, \cos 2x, \cos 4x, \cdots, \cos 2sx$. So in our case, we may write

$$ \sin^{2s} \left(\frac{\pi k}{n}\right) = \sum_{j=0}^{s} a_j \cos \left(\frac{2\pi j k}{n}\right). $$

Now, as you sum this over $k = 0, \cdots, n-1$, all the cosine terms will cancel out except when $j$ is a multiple of $n$. This means that

$$ \sum_{k=0}^{n-1} \sin^{2s} \left(\frac{\pi k}{n}\right) = n (a_0 + a_n + a_{2n} + \cdots). $$

So it is enough to identify $a_j$ for $j$'s multiple of $n$. This can be done by expanding $\sin^{2s} x$ using complex exponential and the binomial theorem. This is essentially what we did in the computation above.

On the other hand, this trick does not work for odd $t$. Indeed, when $t = 2s+1$ is odd, we can write

$$ \sin^{2s+1} \left(\frac{\pi k}{n}\right) = \sum_{j=0}^{s} a_j \sin \left(\frac{\pi (2j+1)k}{n}\right) $$

for some constants $a_0, \cdots, a_n$. (In fact, $a_j = (-1)^j 2^{-2s} \binom{2s+1}{s-j}$.) Now summing both sides for $k = 0, \cdots, n-1$,

\begin{align*} \sum_{k=0}^{n-1}\sin^{2s+1} \left(\frac{\pi k}{n}\right) &= \sum_{j=0}^{s} a_j \sum_{k=0}^{n-1} \sin \left(\frac{\pi (2j+1)k}{n}\right) \\ &= \frac{1}{2^{2s}} \sum_{j=0}^{s} (-1)^j \binom{2s+1}{s-j} \cot \left(\frac{\pi(2j+1)}{2n}\right). \end{align*}

It is hard for me to believe that this will ever simplify except for some nice $s$.

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Addendum 2 (Just for fun). Although no longer simpler than the original sum, one can also prove that for complex $t$ with $\Re(t) > 0$ and for positive integer $n$ the following formula holds:

$$\sum_{k=0}^{n-1} \sin^{t} \left(\frac{\pi k}{n} \right) = \frac{n}{2^{t}} \sum_{j=-\infty}^{\infty} (-1)^{nj} \binom{t}{\frac{t}{2}+nj}. $$

Here, $\binom{n}{k} = \frac{\Gamma(n+1)}{\Gamma(k+1)\Gamma(n-k+1)}$ is the extended binomial coefficient.

Answer 2

This integral remind me of Matsubara sum. Let me use contour integral to provide another perspective of the case $t = 2s$.

Define $a_n = \sum_{k = 1}^{n-1} \sin ^{t} \left( \frac{\pi k }{n} \right) $. What OP is looking for is \begin{equation} \sum_{k=1}^{n-1} \sin^{t} \left( \frac{\pi k }{2n} \right) = \frac{1}{2} \left( \sum_{k=1}^{2n-1} \sin^{t} \left( \frac{\pi k }{2n} \right) -1 \right) = \frac{1}{2} ( a_{2n } - 1 ) \end{equation} One can extend the summation to all the roots of $\zeta^{2n} = 1$ if $t = 2s$ \begin{equation} a_n = \frac{1}{2} \sum_{k=1}^{2n} \sin^{2s} \left( \frac{\pi k }{n} \right ) = \frac{1}{2} \sum_{ k = 1}^{2n} f( \zeta_k ) \qquad \zeta_k = \exp( \frac{ik\pi }{n}) \end{equation} where \begin{equation} f(z ) = \left( \frac{z - \frac{1}{z}}{2i} \right)^{2s} \end{equation}

The summation can be regarded as the sum of the residue of the follow contour integral \begin{equation} \sum_{ k = 1}^{2n} f( \zeta_k ) = \frac{2n}{2\pi i }\oint_{C} \frac{z^{2n-1}}{ z^{2n} - 1} f( z) dz \end{equation} where the integrand is specifically designed such that \begin{equation} \text{Res} ( \frac{ 2n z^{2n-1}}{z^{2n} - 1} f(z) , \zeta_k ) = 2n \frac{z^{2n-1}(z - \zeta_k)}{z^{2n} - \zeta^{2n}_k} f(\zeta_k ) \Big|_{z = \zeta_k } = f(\zeta_k ) \end{equation} Here the closed contour should enclose all $\zeta_k$ but avoid the singularities of $f$, which can be taken as two concentric circles as shown in the following figure(red spots are those roots $\xi_k$).

The inner contour gives (minus) the residue at $0$, and the outer contour gives (minus) the residue at $\infty$, hence \begin{equation} \begin{aligned} a_n &= n \left[ -\text{Res}( \frac{z^{2n-1}}{ z^{2n} - 1} f( z) , 0 ) - \text{Res}( \frac{z^{2n-1}}{ z^{2n} - 1} f( z) , \infty ) \right] \\ \end{aligned} \end{equation} Convert the residue of $\infty$ to $0$ \begin{equation} \begin{aligned} a_n =& n \left[ \text{Res}( \frac{z^{2n-1}}{ 1 - z^{2n} } f( z) , 0 ) + \text{Res}( \frac{\frac{1}{z}}{1 - z^{2n} } f( \frac{1}{z}) , 0 ) \right] \\ =& n \left[ \text{Res}( \frac{1}{z}\frac{z^{2n} + 1 }{ 1 - z^{2n} } f(z) , 0 ) \right] \\ =& n \left[ \text{Res}( \frac{1}{z} ( 1 + 2\sum_{k=1}^{\infty} z^{2nj} ) f(z) , 0 ) \right] \\ \end{aligned} \end{equation} Finally, expand $f(z)$ and extract the terms with power $0, -2nj, -4nj$ etc, \begin{equation} \begin{aligned} a_n &= n \frac{(-1)^s}{2^{2s}} \left( (-1)^s {2s \choose s } + 2 \sum_{0 < nj < s } (-1)^{s+ nj} {2s \choose s+nj } \right)\\ &= \frac{n}{2^{2s}} \sum_{ -s < nj < s } (-1)^{nj} {2s \choose s + nj } \\ \end{aligned} \end{equation} This is the same as what @Sangchul Lee obtained.

Answer 3

I found this long ago:

$$\begin{aligned} {S_t}(n) &= \sum\limits_{k = 0}^n {{{\sin }^p}} (ak)\\ {\text{For odd p}}&:\\ {S_t}(n) &= \frac{1}{{{2^p}}}\sum\limits_{k = 0}^{\frac{{p - 1}}{2}} {{{( - 1)}^k}} C(p,\frac{{p - 1}}{2} - k){\text{Long}}\\ {\text{Long}} &= \left[ {\cot \left( {\frac{a}{2}(2k + 1)} \right) - \csc \left( {\frac{a}{2}(2k + 1)} \right)\cos \left( {\frac{a}{2}(4kn + 4k + 4n + 1)} \right)} \right]\\ {\text{For even p}}&:\\ {S_t}(n) &= \frac{1}{{{2^p}}}\left[ {\left( {\frac{1}{2} + n} \right)C(p,\frac{p}{2}) + \sum\limits_{k = 1}^{\frac{p}{2}} {{{( - 1)}^k}} C(p,\frac{p}{2} - k)\csc (ak)\sin (ak(2n + 1))} \right]\\ \end{aligned}$$

$C(n,k) = \left( \begin{gathered} n \\ k \\ \end{gathered} \right)$, just as a short to input $\LaTeX$.

I try to find my proof but in vain, and I am seeking for a new elegent proof.

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So, for this question:

Let $p \to t,n \to n - 1,a \to \pi /2n$

It's:

$$\begin{aligned} {S_p}(n) &= \sum\limits_{k = 0}^{n - 1} {{{\sin }^t}} \left( {\frac{\pi }{{2n}}k} \right)\\ {\text{For odd n}}&:\\ {S_p}(n) &= {2^{ - t}}\sum\limits_{k = 0}^{\frac{{t - 1}}{2}} {{{( - 1)}^k}C(t,\frac{1}{2}( - 2k + t - 1)){\text{Long}}} \\ {\text{Long}} &= \csc \left( {\frac{{2\pi k + \pi }}{{4n}}} \right)\left( {\sin \left( {\frac{{\pi (k(4n - 2) - 1)}}{{4n}}} \right) + \cos \left( {\frac{{2\pi k + \pi }}{{4n}}} \right)} \right)\\ {\text{For even n}}&:\\ {S_p}(n) &= {2^{ - t}}\left( {\left( {n - \frac{1}{2}} \right)C(t,\frac{t}{2}) + \sum\limits_{k = 1}^{\frac{t}{2}} {{{( - 1)}^k}} C(t,\frac{t}{2} - k){\text{Long}}} \right)\\ {\text{Long}} &= \sin \left( {\frac{{\pi (k(2(n - 1) + 1))}}{{2n}}} \right)\csc \left( {\frac{{\pi k}}{{2n}}} \right)\\ \end{aligned}$$

It we want to find some nice closed-form , the $\text{Long}$ part should be simple.