Evaluate the series $\sum_{n=1}^{\infty}\frac{\sin(\frac{\pi a}{a+b}n)}{n^3}+\frac{\sin(\frac{\pi b}{a+b}n)}{n^3}$

Question

I have to evaluate the series:

$$\sum_{n=1}^{\infty}\frac{\sin(\frac{\pi a}{a+b}n)}{n^3}+\frac{\sin(\frac{\pi b}{a+b}n)}{n^3}$$

Where $a$ and $b$ are real numbers.

Since I'm not very good with series I tried brute force by using a complex analysis method which involves multiplying by the cotangent and calculating the residue at $n=0$ (I don't know the name of this method, sorry!), but I didn't get the result my teacher gave me.

I figured this is the Fourier series expansion of some odd function, but I don't know how to guess said function.

So I was wondering if someone could give me some advice.

Answer 1

Let's rewrite the series as

$$ S(a,b)=\Im\sum_{n=1}^{\infty}\left[\frac{(e^{\pi i a/(a+b)})^n}{n^3}+\frac{(e^{\pi i b/(a+b)})^n}{n^3}\right] $$

Now we can apply the definition of the polylogarithmic functions

$$ \text{Li}_m(z)=\sum_{n=1}^{\infty}\frac{z^n}{n^m} $$

we therefore obtain

$$ S(a,b)=\Im\left[\text{Li}_3(e^{\pi i a/(a+b)})+\text{Li}_3(e^{\pi i b/(a+b)})\right] \quad (*) $$

which is i fear the best one can do for arbritary $a,b$. For example if $a,b $ are natural numbers we may rewrite this as a finite sum over Hurwitz Zeta values.

An alternative representation is terms of Clausen functions $\text{Si}_m(z)$

$$ S(a,b)=\text{Si}_3(e^{\pi i a/(a+b)})+\text{Si}_3(e^{\pi i b/(a+b)}) $$

For the special values $a=b$, $a=0$ we might find the particulary nice results

$$ S(a,a)=\frac{\pi^3}{16} \quad \\ S(0,b)=0 \quad $$

All defintions i used and much more can be found here