Does the following infinite series have a closed form?
$$
\sum\_{n=1}^{\infty} {(-1)^n \frac{\Gamma(\frac{1}{3}+\frac{n}{3})}{\Gamma(1+\frac{n}{3})} \sin(\frac{2\pi n}{3})}
$$



Closed form of an infinite series
**Q:** Does the following infinite series have a closed form?


It does, according to Mathematica:
$$\sum\_{n=1}^{\infty} {(-1)^n \frac{\Gamma(\frac{1}{3}+\frac{n}{3})}{\Gamma(1+\frac{n}{3})} \sin\left(\frac{2\pi n}{3}\right)}=-2^{-10/3} \Gamma \left(\tfrac{1}{3}\right)\Big[2 \sqrt{3}+9\pi^{-1} \, \_2F\_1\left(\tfrac{2}{3},\tfrac{2}{3};\tfrac{5}{3};-1\right)\hspace{-1mm}\Big].$$


