감마곱 (Gamma Products)

수학노트
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개요

  • 자연수 n에 대한 잉여계의 부분집합 A에 대하여, 다음과 같은 감마함수의 곱이 언제 닫힌 형태로 표현되는가의 문제\[\prod_{k\in A}\Gamma(\frac{k}{n})\]

 

 

\(\Gamma \left(\frac{1}{6}\right) \Gamma \left(\frac{5}{6}\right)=2\sqrt{\pi }\)

\(\Gamma \left(\frac{1}{10}\right) \Gamma \left(\frac{3}{10}\right) \Gamma \left(\frac{7}{10}\right) \Gamma \left(\frac{9}{10}\right)=4 \pi ^2\)

\(\Gamma \left(\frac{1}{14}\right) \Gamma \left(\frac{9}{14}\right) \Gamma \left(\frac{11}{14}\right)=4{\pi ^{3/2}}\)

\(\Gamma \left(\frac{3}{14}\right) \Gamma \left(\frac{5}{14}\right) \Gamma \left(\frac{13}{14}\right)=2\pi ^{3/2}\)

\(\Gamma \left(\frac{1}{18}\right) \Gamma \left(\frac{5}{18}\right) \Gamma \left(\frac{7}{18}\right) \Gamma \left(\frac{11}{18}\right) \Gamma \left(\frac{13}{18}\right) \Gamma \left(\frac{17}{18}\right)=8 \pi ^3\)

\(\Gamma \left(\frac{1}{22}\right) \Gamma \left(\frac{3}{22}\right) \Gamma \left(\frac{5}{22}\right) \Gamma \left(\frac{7}{22}\right) \Gamma \left(\frac{9}{22}\right) \Gamma \left(\frac{13}{22}\right) \Gamma \left(\frac{15}{22}\right) \Gamma \left(\frac{17}{22}\right) \Gamma \left(\frac{19}{22}\right) \Gamma \left(\frac{21}{22}\right)=32 \pi ^5\)

\(\Gamma \left(\frac{1}{26}\right) \Gamma \left(\frac{3}{26}\right) \Gamma \left(\frac{5}{26}\right) \Gamma \left(\frac{7}{26}\right) \Gamma \left(\frac{9}{26}\right) \Gamma \left(\frac{11}{26}\right) \Gamma \left(\frac{15}{26}\right) \Gamma \left(\frac{17}{26}\right) \Gamma \left(\frac{19}{26}\right) \Gamma \left(\frac{21}{26}\right) \Gamma \left(\frac{23}{26}\right) \Gamma \left(\frac{25}{26}\right)=64 \pi ^6\)

\(\Gamma \left(\frac{1}{30}\right) \Gamma \left(\frac{17}{30}\right) \Gamma \left(\frac{19}{30}\right) \Gamma \left(\frac{23}{30}\right)=8 \pi ^2\)

\(\Gamma \left(\frac{1}{34}\right) \Gamma \left(\frac{9}{34}\right) \Gamma \left(\frac{13}{34}\right) \Gamma \left(\frac{15}{34}\right) \Gamma \left(\frac{19}{34}\right) \Gamma \left(\frac{21}{34}\right) \Gamma \left(\frac{25}{34}\right) \Gamma \left(\frac{33}{34}\right)=16 \pi ^4\)

 

 

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