<s>[INST] On the Clausen triple $8\rm{Cl}_2\left(\frac{\pi}2\right)+3\rm{Cl}_2\left(\frac{\pi}3\right)=12\,\rm{Cl}_2\left(\frac{\pi}6\right)$ While doing research on the Clausen function, I came across this nice identity,
$$8\operatorname{Cl}_2\left(\frac{\pi}2\right)+3\operatorname{Cl}_2\left(\frac{\pi}3\right)=12\operatorname{Cl}_2\left(\frac{\pi}6\right)$$
The two addends on the LHS are Catalan's constant and Gieseking's constants. It made me wonder if there were similar relations. Define,
$$\begin{aligned}
\text{Cl}_m\left(\frac{\pi}2\right) &= \sum_{k=1}^\infty\frac{\sin\left(k\,\frac{\pi}2\right)}{k^m}=\sum_{n=0}^\infty\left(\frac1{(4n+1)^m}-\frac1{(4n+3)^m}\right)\\
\text{Cl}_m\left(\frac{\pi}3\right) &= \sum_{k=1}^\infty\frac{\sin\left(k\,\frac{\pi}3\right)}{k^m}=\frac{2^{m-1}+1}{2^m}\sum_{n=0}^\infty\left(\frac{\sqrt3}{(3n+1)^m}-\frac{\sqrt3}{(3n+2)^m}\right)\\
\text{Cl}_m\left(\frac{\pi}6\right) &= \sum_{k=1}^\infty\frac{\sin\left(k\,\frac{\pi}6\right)}{k^m}
\end{aligned}$$
and,
$$\begin{aligned}
a &= 2^{m-1}(3^{m-1}+1)\\ 
b &= 3^{m-1}\\ 
c &= 2^m\,3^{m-1}
\end{aligned}$$

Q: How do we prove, with $a,b,c$ as defined above, that,
  $$a\operatorname{Cl}_m\left(\frac{\pi}2\right)+b\operatorname{Cl}_m\left(\frac{\pi}3\right)=c\operatorname{Cl}_m\left(\frac{\pi}6\right)$$
  or equivalently,
  $$a\sum_{k=1}^\infty\frac{\sin\left(k\,\frac{\pi}2\right)}{k^m}+b\sum_{k=1}^\infty\frac{\sin\left(k\,\frac{\pi}3\right)}{k^m} = c\sum_{k=1}^\infty\frac{\sin\left(k\,\frac{\pi}6\right)}{k^m}$$
  for all integer $m>1$?

 [/INST] Let $~~m,n\in\mathbb{N}~~$ and $~~0<x<1~$ .
$\displaystyle \text{E}_m(x):=\text{Li}_m(e^{i2\pi x})~ , ~~ \text{Cl}_m(2\pi x)=\Im \text{E}_m(x)~ , ~~ \overline{\text{E}_m(x)}~$ is conjugated complex to $~\text{E}_m(x)$
We have:

$$ \text{E}_m(1-x) = \overline{\text{E}_m(x)}~~ , ~~~~ \sum\limits_{k=0}^{n-1}\text{E}_m \left(\frac{x+k}{n}\right) = n^{1-m}~\text{E}_m(x)$$

It follows  
$\displaystyle (n,x):=\left(2,\frac{1}{6}\right) : \hspace{0.5cm} \text{E}_m \left(\frac{1}{12}\right) + \text{E}_m \left(\frac{7}{12}\right) = 2^{1-m}~\text{E}_m \left(\frac{1}{6}\right) \hspace{2.8cm}(\text{I})$ 
$\displaystyle (n,x):=\left(3,\frac{1}{4}\right) : \hspace{0.5cm} \text{E}_m \left(\frac{1}{12}\right) + \text{E}_m \left(\frac{5}{12} \right)+ \text{E}_m \left(\frac{3}{4}\right) = 3^{1-m}~\text{E}_m \left(\frac{1}{4}\right)$ 
$\hspace{4cm}$ and conjugated we get     
$\displaystyle \hspace{4.2cm} \overline{\text{E}_m \left(\frac{1}{12}\right)} + \text{E}_m \left(\frac{7}{12}\right) + \text{E}_m \left(\frac{1}{4}\right) = 3^{1-m}~\overline{\text{E}_m \left(\frac{1}{4}\right)} \hspace{0.6cm}(\text{II})$ 
such that the difference $~\text{(I)}-\text{(II)}~$ of the equations yields the following relationship:

$$\text{E}_m \left(\frac{1}{12}\right)  - \overline{\text{E}_m \left(\frac{1}{12}\right)} - \text{E}_m \left(\frac{1}{4}\right) = 2^{1-m}~\text{E}_m \left(\frac{1}{6}\right) - 3^{1-m}~\overline{\text{E}_m \left(\frac{1}{4}\right)}$$

We take the imaginary part of this equation, add $\displaystyle ~\Im\text{E}_m \left(\frac{1}{4}\right)~$, multiply with $~6^{m-1}~$ and receive the desired result:
$\displaystyle 2\Im\text{E}_m \left(\frac{1}{12}\right) - \Im\text{E}_m \left(\frac{1}{4}\right) = 2^{1-m} \Im\text{E}_m \left(\frac{1}{6}\right) + 3^{1-m} \Im\text{E}_m \left(\frac{1}{4}\right)$
$\displaystyle 2^{m-1} (3^{m-1}+1) \Im\text{E}_m \left(\frac{1}{4}\right) + 3^{m-1} \Im\text{E}_m \left(\frac{1}{6}\right) = 2^{m}3^{m-1} \Im\text{E}_m \left(\frac{1}{12}\right)$
$\displaystyle 2^{m-1} (3^{m-1}+1)\text{Cl}_m \left(\frac{\pi}{2}\right) + 3^{m-1}\text{Cl}_m \left(\frac{\pi}{3}\right) = 2^{m}3^{m-1}\text{Cl}_m \left(\frac{\pi}{6}\right)$
</s>