감마곱 (Gamma Products)

개요

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

예

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

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

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

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

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

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

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

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

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

메모

• http://mathoverflow.net/questions/9878/a-product-of-gamma-values-over-the-numbers-coprime-to-n

관련된 항목들

• Chowla-셀베르그 공식
• 1부터 n까지의 최소공배수

수학용어번역

• 단어사전 http://www.google.com/dictionary?langpair=en%7Cko&q=
• 발음사전 http://www.forvo.com/search/
• 대한수학회 수학 학술 용어집
  • http://mathnet.kaist.ac.kr/mathnet/math_list.php?mode=list&ftype=eng_term&fstr=
• 남·북한수학용어비교
• 대한수학회 수학용어한글화 게시판

사전 형태의 자료

• http://ko.wikipedia.org/wiki/
• http://en.wikipedia.org/wiki/
• The Online Encyclopaedia of Mathematics
• NIST Digital Library of Mathematical Functions
• The World of Mathematical Equations

관련논문

• Luschny, Peter, and Stefan Wehmeier. 2009. “The lcm(1,2,...,n) as a product of sine values sampled over the points in Farey sequences.” 0909.1838 (September 10). http://arxiv.org/abs/0909.1838 .
• Martin, Greg. 2009. “A product of Gamma function values at fractions with the same denominator.” 0907.4384 (July 24). http://arxiv.org/abs/0907.4384 .
• Nijenhuis, Albert. 2009. “Small Gamma Products with Simple Values.” 0907.1689 (July 9). http://arxiv.org/abs/0907.1689 .
• Problem 11426, M. L. Glasser, The American Mathematical Monthly, Vol. 116, No. 4 (Apr., 2009), p. 365 http://www.jstor.org/stable/40391099
  • solution http://math.la.asu.edu/~checkman/AMM/11426_Heckman.pdf
• http://www.jstor.org/action/doBasicSearch?Query=
• http://www.ams.org/mathscinet
• http://dx.doi.org/