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本帖最后由 青青子衿 于 2025-1-15 21:20 编辑
\begin{align*}
\frac{\Gamma (z)}{\Gamma \left(\frac{1}{2}-z\right)}=\frac{\cos (\pi z) \Gamma (4 z-1)}{2^{6 z-3} \Gamma \left(2 z-\frac{1}{2}\right)}
\end{align*}
\begin{align*}
\frac{\Gamma \left(\frac{15}{52}\right)}{\Gamma \left(\frac{11}{52}\right)}&=2\cdot2^{7/26} \sin \left(\tfrac{11 \pi }{52}\right)\cdot\frac{ \Gamma \left(\frac{2}{13}\right)}{\Gamma \left(\frac{1}{13}\right)}\\
\frac{\Gamma \left(\frac{17}{52}\right)}{\Gamma \left(\frac{9}{52}\right)}&=2\cdot2^{1/26} \sin \left(\tfrac{9 \pi }{52}\right)\cdot\frac{ \Gamma \left(\frac{4}{13}\right)}{\Gamma \left(\frac{2}{13}\right)}\\
\frac{\Gamma \left(\frac{19}{52}\right)}{\Gamma \left(\frac{7}{52}\right)}&=2^{21/26} \sin \left(\tfrac{7 \pi }{52}\right) \cdot\frac{\Gamma \left(\frac{6}{13}\right)}{\Gamma \left(\frac{3}{13}\right)}\\
\frac{\Gamma \left(\frac{21}{52}\right)}{\Gamma \left(\frac{5}{52}\right)}&=\frac{2^{15/26} \pi \sin \left(\frac{5 \pi }{52}\right)}{\sin \left(\frac{5 \pi }{13}\right)}\cdot\frac{1}{\Gamma \left(\frac{4}{13}\right) \Gamma\left(\frac{5}{13}\right)}\\
\frac{\Gamma \left(\frac{23}{52}\right)}{\Gamma \left(\frac{3}{52}\right)}&=\frac{2^{9/26}\pi\sin\left(\frac{3 \pi }{52}\right)}{\sin \left(\frac{3 \pi }{13}\right)}\cdot\frac{1}{\Gamma \left(\frac{3}{13}\right) \Gamma \left(\frac{5}{13}\right)}\\
\frac{\Gamma \left(\frac{25}{52}\right)}{\Gamma \left(\frac{1}{52}\right)}&=\frac{2^{3/26}\pi \sin \left(\frac{\pi }{52}\right)}{\sin \left(\frac{\pi }{13}\right)}\cdot\frac{1}{\Gamma \left(\frac{6}{13}\right) \Gamma \left(\frac{1}{13}\right)}
\end{align*}
- N[ Gamma[15/52]/Gamma[11/52], 20]
- N[2* 2^(7/26) Sin[(11 \[Pi])/52 ] Gamma[2/13]/Gamma[1/13], 20]
- N[ Gamma[17/52]/Gamma[9/52], 20]
- N[2* 2^(1/26) Sin[(9 \[Pi])/52] Gamma[4/13]/Gamma[2/13], 20]
- N[Gamma[19/52]/Gamma[7/52], 20]
- N[2^(21/26) Sin[(7 \[Pi])/52 ] Gamma[6/13]/Gamma[3/13], 20]
- N[Gamma[21/52]/Gamma[5/52], 20]
- N[2^(15/26) Sin[(5 \[Pi])/52]/Sin[(5 \[Pi])/13] \[Pi]/(
- Gamma[4/13] Gamma[5/13]), 20]
- N[ Gamma[23/52]/Gamma[3/52], 20]
- N[2^(9/26) Sin[3/52 \[Pi]]/Sin[3/13 \[Pi]] \[Pi]/(
- Gamma[3/13] Gamma[5/13]), 20]
- N[ Gamma[25/52]/Gamma[1/52], 20]
- N[2^(3/26) Sin[1/52 \[Pi]]/Sin[1/13 \[Pi]] \[Pi]/(
- Gamma[6/13] Gamma[1/13]), 20]
\begin{align*}
\frac{11+3\sqrt{13}}{2}&=\frac{\Gamma \left(\frac{1}{52}\right) \Gamma \left(\frac{3}{52}\right) \Gamma \left(\frac{9}{52}\right) \Gamma \left(\frac{17}{52}\right) \Gamma \left(\frac{23}{52}\right) \Gamma \left(\frac{25}{52}\right) \Gamma \left(\frac{27}{52}\right) \Gamma \left(\frac{29}{52}\right) \Gamma \left(\frac{35}{52}\right) \Gamma \left(\frac{43}{52}\right) \Gamma \left(\frac{49}{52}\right) \Gamma \left(\frac{51}{52}\right)}{\Gamma \left(\frac{5}{52}\right) \Gamma \left(\frac{7}{52}\right) \Gamma \left(\frac{11}{52}\right) \Gamma \left(\frac{15}{52}\right) \Gamma \left(\frac{19}{52}\right) \Gamma \left(\frac{21}{52}\right) \Gamma \left(\frac{31}{52}\right) \Gamma \left(\frac{33}{52}\right) \Gamma \left(\frac{37}{52}\right) \Gamma \left(\frac{41}{52}\right) \Gamma \left(\frac{45}{52}\right) \Gamma \left(\frac{47}{52}\right)}\\
&=\frac{\sin \left(\frac{5 \pi }{52}\right) \sin \left(\frac{7 \pi }{52}\right) \sin \left(\frac{11 \pi }{52}\right) \sin \left(\frac{15 \pi }{52}\right) \sin \left(\frac{19 \pi }{52}\right) \sin \left(\frac{21 \pi }{52}\right)}{\sin \left(\frac{\pi }{52}\right) \sin \left(\frac{3 \pi }{52}\right) \sin \left(\frac{9 \pi }{52}\right) \sin \left(\frac{17 \pi }{52}\right) \sin \left(\frac{23 \pi }{52}\right) \sin \left(\frac{25 \pi }{52}\right)}
\end{align*}
mathoverflow.net/questions/318917/a-tantalizing-gamma-quotient-t ... lich-lang-conjecture
mathoverflow.net/questions/7616/when-are-some-products-of-gamma-functions-algebraic-numbers |
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