중심이항계수 (central binomial coefficient)

이 항목의 스프링노트 원문주소

• 중심이항계수(central binomial coefficient)
  

 

 

개요

• 다음과 같은 이항계수로 정의
  \({2n \choose n}=\frac{(2n)!}{(n!)^2}\)
  

 

 

급수와 중심이항계수

• 이항급수와 이항정리
  \(\frac{1}{\sqrt{1-4z}}=\sum_{n=0}^{\infty} {{2n}\choose {n}} z^n\)
  
• 역삼각함수
  \(2(\arcsin x)^2=\sum_{n=1}^{\infty}\frac{(2x)^{2n}}{n^2\binom{2n}{n}}\)
  \(\frac{2x \arcsin x}{\sqrt{1-x^2}}=\sum_{n=1}^{\infty}\frac{(2x)^{2n}}{n\binom{2n}{n}}\)
  
• 카탈란 수열(Catalan numbers) 의 생성함수
  \(G(x)= c_0 + c_1 x + c_2 x^2 + \cdots + c_n x^n + \cdots=\frac{1-\sqrt{1-4x}}{2x}\)
  \(c_n = \frac{1}{n+1}{2n\choose n}\)
  

 

 

 

리만제타함수

\(\zeta(2)=3\sum_{n=1}^{\infty}\frac{1}{n^{2}\binom{2n}{n}}\)

\(\zeta(3) = \frac{5}{2} \sum_{n=1}^\infty \frac{(-1)^{n-1}}{n^3\binom{2n}{n}}\)

\(\zeta(4) = \frac{36}{17} \sum_{n=1}^\infty \frac{1}{n^4\binom{2n}{n}}\)

 

 

재미있는 사실

 

• Math Overflow http://mathoverflow.net/search?q=
• 네이버 지식인 http://kin.search.naver.com/search.naver?where=kin_qna&query=

 

 

역사

 

• http://www.google.com/search?hl=en&tbs=tl:1&q=
• 수학사연표
•  

 

 

 

 

관련된 항목들

• 카탈란 수열(Catalan numbers)
  
• ζ(3)는 무리수이다(아페리의 정리)
  
• 폴리로그 함수(polylogarithm)
  
• 이항급수와 이항정리
  

 

 

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/Central_binomial_coefficient
• http://mathworld.wolfram.com/CentralBinomialCoefficient.html
• http://mathworld.wolfram.com/BinomialSums.html
• http://www.wolframalpha.com/input/?i=
• NIST Digital Library of Mathematical Functions
• The On-Line Encyclopedia of Integer Sequences
  
  • http://www.research.att.com/~njas/sequences/?q=

 

 

관련논문

• Experimental Determination of Apéry-like Identities for ζ(2n + 2)
  
  • David H. Bailey, Jonathan M. Borwein, and David M. Bradley
• Evaluations of binomial series
  
  • Jonathan M. Borwein1  and Roland Girgensohn, 2004
• Central Binomial Sums, Multiple Clausen Values and Zeta Values
  
  • J. M. Borwein, D. J. Broadhurst, J. Kamnitzer, 2000
• Interesting Series Involving the Central Binomial Coefficient
  
  • D. H. Lehmer, The American Mathematical Monthly, Vol. 92, No. 7 (Aug. - Sep., 1985), pp. 449-457

• On the series Σk = 1∞(k2k)−1 k−n and related sums
  
  • I. J. Zucker, Journal of Number Theory, Volume 20, Issue 1, February 1985, Pages 92-102   
• Some wonderful formulas ... an introduction to polylogarithms
  
  • A.J. Van der Poorten, Queen's papers in Pure and Applied Mathematics, 54 (1979), 269-286

• http://www.jstor.org/action/doBasicSearch?Query=
• http://www.ams.org/mathscinet
• http://dx.doi.org/

 

 

관련도서

• 도서내검색
  
  • http://books.google.com/books?q=
  • http://book.daum.net/search/contentSearch.do?query=
• 도서검색
  
  • http://books.google.com/books?q=
  • http://book.daum.net/search/mainSearch.do?query=
  • http://book.daum.net/search/mainSearch.do?query=

 

 

관련기사

• 네이버 뉴스 검색 (키워드 수정)
  
  • http://news.search.naver.com/search.naver?where=news&x=0&y=0&sm=tab_hty&query=
  • http://news.search.naver.com/search.naver?where=news&x=0&y=0&sm=tab_hty&query=
  • http://news.search.naver.com/search.naver?where=news&x=0&y=0&sm=tab_hty&query=

 

 

블로그

• 구글 블로그 검색
  
  • http://blogsearch.google.com/blogsearch?q=
• 네이버 오늘의과학
• 수학동아
• Mathematical Moments from the AMS
• BetterExplained