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For questions that specifically ask for determining a closed form of equations, integrals etc.
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Closed form of a Hypergeometric Function ${}_2F_1$ at $z=-8$
How this can be proved?
$$
E = {}_2F_1(-\frac{1}{2}, \frac{1}{3}, \frac{4}{3},-8) = \frac{6}{5} - \frac{\chi}{2}
$$
where
$$
\chi = \frac{6\sqrt{\pi}}{5}\frac{\Gamma(\frac{1}{3})}{\Gamma(-\frac{1}{6}) …