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Sometimes it is possible to express a value of the $\Gamma$-function at a rational point through values of the $\Gamma$-function at rational points with smaller denominators, e.g. $$\Gamma\!\left(\tfrac{1}{10}\right)=\frac{\sqrt{5+\sqrt{5}}}{2^{7/10}\sqrt{\pi}}\Gamma\!\left(\tfrac{1}{5}\right)\Gamma\!\left(\tfrac{2}{5}\right).$$ Is it possible to do that with $\Gamma\!\left(\tfrac{1}{50}\right)$?

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By the duplication formula, $$\Gamma\left(\frac{2}{50}\right) = C\cdot \Gamma\left(\frac{1}{50}\right)\Gamma\left(\frac{13}{25}\right)$$ hence: $$\Gamma\left(\frac{1}{50}\right) = \frac{\Gamma\left(\frac{1}{25}\right)}{C\cdot \Gamma\left(\frac{13}{25}\right)}.$$

Specifically, the constant $C$ is,

$$\Gamma\left(\frac{1}{50}\right) = 2^{24/25}\sqrt{\pi}\, \frac{\Gamma\left(\frac{1}{25}\right)}{\Gamma\left(\frac{13}{25}\right)} = 49.44221\dots$$

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  • $\begingroup$ I took the liberty of adding $C$. $\endgroup$ – Tito Piezas III Jun 22 '15 at 9:46
  • $\begingroup$ @TitoPiezasIII: well done. $\endgroup$ – Jack D'Aurizio Jun 22 '15 at 11:11

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