"앤드류스-고든 항등식(Andrews-Gordon identity)"의 두 판 사이의 차이

수학노트
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(같은 사용자의 중간 판 21개는 보이지 않습니다)
1번째 줄: 1번째 줄:
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==이 항목의 스프링노트 원문주소==
 
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* [[앤드류스-고든 항등식(Andrews-Gordon identity)]]<br>
 
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==개요==
 
==개요==
  
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* [[로저스-라마누잔 항등식|로저스-라마누잔 연분수와 항등식]]의 일반화<br>
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* [[로저스-라마누잔 항등식|로저스-라마누잔 연분수와 항등식]]의 일반화
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*  모듈라 성질을 가지며([[모듈라 형식(modular forms)]] 참조), [[q-초기하급수(q-hypergeometric series) (통합됨)|q-초기하급수(q-hypergeometric series)]] 형태로 표현 가능<br>
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*  모듈라 성질을 가지며([[모듈라 형식(modular forms)]] 참조), [[Q-초기하급수(q-hypergeometric series)와 양자미적분학(q-calculus)|q-초기하급수(q-hypergeometric series)]] 형태로 표현 가능
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*  등각장론에서 c(2, 2k+1) minimal models 의 캐릭터<br><math>\chi_j(\tau)=q^{h_j-c/24}\prod_{n\neq 0,\pm(j+1)}(1-q^n)^{-1}</math><br>
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*  등각장론에서 c(2, 2k+1) minimal 모형에 의해 주어지는 표현의 캐릭터:<math>\chi_j(\tau)=q^{h_j-c/24}\prod_{n\neq 0,\pm(j+1)}(1-q^n)^{-1}</math>
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* [[베일리 사슬(Bailey chain)]],  [[베일리 격자(Bailey lattice)]] 의 방법으로 증명할 수 있다
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==항등식==
 
==항등식==
  
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*  자연수 <math>k\geq 2</math> , <math>1\leq i \leq k</math>에 대하여, 다음이 성립한다<br><math>\sum_{n_1,\cdots,n_{k-1}\geq0}\frac{x^{N_1^2+\cdots+N_{k-1}^2+N_i+\cdots+N_{k-1}}}{(x)_{n_1}...(x)_{n_{k-1}}}=\prod_{r\neq 0,\pm i \pmod {2k+1}}\frac{1}{1-x^r} </math><br> 여기서 <math>j\leq k-1</math>이면 <math>N_j=n_j+\cdots+n_{k-1}</math> , <math>j=k</math>이면 <math>N_j=0</math><br>
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*  자연수 <math>k\geq 2</math> , <math>1\leq i \leq k</math>에 대하여, 다음이 성립한다
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*  여러 문헌에서 다음과 같이 표현되기도 한다<br><math>\sum_{n_1\geq\cdots\geq n_{k-1}\geq0}\frac{q^{n_1^2+\cdots+n_{k-1}^2+n_i+\cdots+n_{k-1}}}{(q)_{n_{1}-n_{2}}\cdots (q)_{n_{k-2}-n_{k-1}}(q)_{n_{k-1}}}=\prod_{n\neq 0,\pm i\pmod {2k+1}}(1-q^n)^{-1}</math><br>
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:<math>\sum_{n_1,\cdots,n_{k-1}\geq0}\frac{q^{N_1^2+\cdots+N_{k-1}^2+N_i+\cdots+N_{k-1}}}{(q)_{n_1}...(q)_{n_{k-1}}}=\prod_{r\neq 0,\pm i \pmod {2k+1}}\frac{1}{1-q^r} </math>
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여기서 <math>j\leq k-1</math>이면 <math>N_j=n_j+\cdots+n_{k-1}</math> , <math>j=k</math>이면 <math>N_j=0</math>
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*  여러 문헌에서 다음과 같이 표현되기도 한다
 +
:<math>\sum_{n_1\geq\cdots\geq n_{k-1}\geq0}\frac{q^{n_1^2+\cdots+n_{k-1}^2+n_i+\cdots+n_{k-1}}}{(q)_{n_{1}-n_{2}}\cdots (q)_{n_{k-2}-n_{k-1}}(q)_{n_{k-1}}}=\prod_{n\neq 0,\pm i\pmod {2k+1}}(1-q^n)^{-1}</math>
  
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==k=2인 경우 : 로저스-라마누잔 항등식==
 
==k=2인 경우 : 로저스-라마누잔 항등식==
  
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*  k=2인 경우, [[로저스-라마누잔 항등식|로저스-라마누잔 연분수와 항등식]]을 얻는다<br>
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*  k=2인 경우, [[로저스-라마누잔 항등식|로저스-라마누잔 연분수와 항등식]]을 얻는다
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*  i=1인 경우<br>  <math>H(q) =\sum_{n=0}^\infty \frac {q^{n^2+n}} {(q;q)_n} =  
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*  i=1인 경우
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:<math>H(q) =\sum_{n=0}^\infty \frac {q^{n^2+n}} {(q;q)_n} =  
 
  \frac {1}{(q^2;q^5)_\infty (q^3; q^5)_\infty}
 
  \frac {1}{(q^2;q^5)_\infty (q^3; q^5)_\infty}
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  =1+q^2 +q^3 +q^4+q^5 +2q^6+\cdots</math><br>
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  =1+q^2 +q^3 +q^4+q^5 +2q^6+\cdots</math>
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*  i=2인 경우<br><math>G(q) = \sum_{n=0}^\infty \frac {q^{n^2}} {(q;q)_n} =  
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*  i=2인 경우
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:<math>G(q) = \sum_{n=0}^\infty \frac {q^{n^2}} {(q;q)_n} =  
 
  \frac {1}{(q;q^5)_\infty (q^4; q^5)_\infty}
 
  \frac {1}{(q;q^5)_\infty (q^4; q^5)_\infty}
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   =1+ q +q^2 +q^3 +2q^4+2q^5 +3q^6+\cdots</math><br>
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   =1+ q +q^2 +q^3 +2q^4+2q^5 +3q^6+\cdots</math>
  
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==k=3인 경우==
 
==k=3인 경우==
  
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*  i=1인 경우<br><math>\sum_{n_1,n_{2}\geq0}\frac{q^{n_{1}^2+2n_1n_2+2n_{2}^{2}+n_1+2n_2}}{(q)_{n_1}(q)_{n_{2}}}=\prod_{r\neq 0,\pm 1 \pmod {7}}\frac{1}{1-q^r}=\frac{(q;q^7)_\infty (q^6; q^7)_\infty(q^7;q^7)_\infty}{(q)_\infty}</math><br>
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*  i=1인 경우
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*  i=2인 경우<br><math>\sum_{n_1,n_{2}\geq0}\frac{q^{n_{1}^2+2n_1n_2+2n_{2}^{2}+n_2}}{(q)_{n_1}(q)_{n_{2}}}=\prod_{r\neq 0,\pm 2 \pmod {7}}\frac{1}{1-q^r}=\frac{(q^2;q^7)_\infty (q^5; q^7)_\infty(q^7;q^7)_\infty}{(q)_\infty}</math><br>
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:<math>\sum_{n_1,n_{2}\geq0}\frac{q^{n_{1}^2+2n_1n_2+2n_{2}^{2}+n_1+2n_2}}{(q)_{n_1}(q)_{n_{2}}}=\prod_{r\neq 0,\pm 1 \pmod {7}}\frac{1}{1-q^r}=\frac{(q;q^7)_\infty (q^6; q^7)_\infty(q^7;q^7)_\infty}{(q)_\infty}</math>
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*  i=3인 경우<br>
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*  i=2인 경우
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:<math>\sum_{n_1,n_{2}\geq0}\frac{q^{n_{1}^2+2n_1n_2+2n_{2}^{2}+n_2}}{(q)_{n_1}(q)_{n_{2}}}=\prod_{r\neq 0,\pm 2 \pmod {7}}\frac{1}{1-q^r}=\frac{(q^2;q^7)_\infty (q^5; q^7)_\infty(q^7;q^7)_\infty}{(q)_\infty}</math>
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<math>\sum_{n_1,n_{2}\geq0}\frac{q^{n_{1}^2+2n_1n_2+2n_{2}^{2}}}{(q)_{n_1}(q)_{n_{2}}}=\prod_{r\neq 0,\pm 3 \pmod {7}}\frac{1}{1-q^r}=\frac{(q^3;q^7)_\infty (q^4; q^7)_\infty(q^7;q^7)_\infty}{(q)_\infty}</math>
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*  i=3인 경우
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:<math>\sum_{n_1,n_{2}\geq0}\frac{q^{n_{1}^2+2n_1n_2+2n_{2}^{2}}}{(q)_{n_1}(q)_{n_{2}}}=\prod_{r\neq 0,\pm 3 \pmod {7}}\frac{1}{1-q^r}=\frac{(q^3;q^7)_\infty (q^4; q^7)_\infty(q^7;q^7)_\infty}{(q)_\infty}</math>
  
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==k=4인 경우==
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* k=4, i=3인 경우 [http://oeis.org/A000726 A000726]
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:<math>\frac{q^{n_1^2+2 n_2 n_1+2 n_3 n_1+2 n_2^2+3 n_3^2+4 n_2 n_3+n_3}}{(q)_{n_1} (q)_{n_2} (q)_{n_3}}=\prod_{r\neq 0,\pm 3 \pmod {9}}\frac{1}{1-q^r}=1+q+2 q^2+2 q^3+4 q^4+5 q^5+7 q^6+9 q^7+13 q^8+16 q^9+22 q^{10}+O(q^11)</math>
  
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==얻어지는 이차형식==
 
==얻어지는 이차형식==
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* <math>k=2</math>, <math>n_{1}^{2}</math>
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* <math>k=3</math>, <math>(n_{1}+n_{2})^{2}+n_{2}^{2}</math>
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* <math>k=4</math>, <math>(n_{1}+n_{2}+n_{3})^{2}+(n_{2}+n_{3})^{2}+n_{3}^{2}</math>
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* <math>k=5</math>, <math>(n_{1}+n_{2}+n_{3}+n_{4})^{2}+(n_{2}+n_{3}+n_{4})^{2}+(n_{3}+n_{4})^{2}+n_{4}^{2}</math>
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** 이차형식에 대응되는 행렬은 다음과 같이 주어진다
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:<math>
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\left(
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\begin{array}{cccc}
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1 & 1 & 1 & 1 \\
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1 & 2 & 2 & 2 \\
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1 & 2 & 3 & 3 \\
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1 & 2 & 3 & 4 \\
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\end{array}
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\right)
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</math>
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* 이는 [[양의 정부호 행렬(positive definite matrix)]]의 예이다
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<math>n_{1}^{2}</math>
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<math>(n_{1}+n_{2})^{2}+n_{2}^{2}</math>
 
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<math>(n_{1}+n_{2}+n_{3})^{2}+(n_{2}+n_{3})^{2}+n_{3}^{2}</math>
 
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<math>(n_{1}+n_{2}+n_{3}+n_{4})^{2}+(n_{2}+n_{3}+n_{4})^{2}+(n_{3}+n_{4})^{2}+n_{4}^{2}</math>
 
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행렬은
 
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<math>\text{A=}\left( \begin{array}{ccccc} 2 & 2 & 2 & 2 & 2 \\  2 & 4 & 4 & 4 & 4 \\  2 & 4 & 6 & 6 & 6 \\  2 & 4 & 6 & 8 & 8 \\  2 & 4 & 6 & 8 & 10 \end{array} \right)</math>
 
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==역사==
 
==역사==
  
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*  1961 고든<br>
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*  1961 고든
 
* http://www.google.com/search?hl=en&tbs=tl:1&q=ramanujan+gordon+andrews
 
* http://www.google.com/search?hl=en&tbs=tl:1&q=ramanujan+gordon+andrews
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* [[수학사연표 (역사)|수학사연표]]
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* [[수학사 연표]]
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*  
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*
  
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==메모==
 
==메모==
  
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==관련된 항목들==
 
==관련된 항목들==
  
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* [[다이로그 항등식 (dilogarithm identities)]]<br>
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* [[다이로그 항등식 (dilogarithm identities)]]
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* [[정다각형의 대각선의 길이]]<br>
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* [[정다각형의 대각선의 길이]]
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* [[베일리 격자(Bailey lattice)]]
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* [[베일리 사슬(Bailey chain)]]
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==수학용어번역==
 
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* 단어사전 http://www.google.com/dictionary?langpair=en|ko&q=
 
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* 발음사전 http://www.forvo.com/search/
 
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* [http://mathnet.kaist.ac.kr/mathnet/math_list.php?mode=list&ftype=&fstr= 대한수학회 수학 학술 용어집]<br>
 
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** http://mathnet.kaist.ac.kr/mathnet/math_list.php?mode=list&ftype=eng_term&fstr=
 
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* [http://www.nktech.net/science/term/term_l.jsp?l_mode=cate&s_code_cd=MA 남·북한수학용어비교]
 
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* [http://kms.or.kr/home/kor/board/bulletin_list_subject.asp?bulletinid=%7BD6048897-56F9-43D7-8BB6-50B362D1243A%7D&boardname=%BC%F6%C7%D0%BF%EB%BE%EE%C5%E4%B7%D0%B9%E6&globalmenu=7&localmenu=4 대한수학회 수학용어한글화 게시판]
 
  
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==매스매티카 파일 및 계산 리소스==
 +
* https://docs.google.com/file/d/0B8XXo8Tve1cxUFkzQXUyeXNMTHc/edit
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==사전 형태의 자료==
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==사전 형태의 자료==
  
 
* http://ko.wikipedia.org/wiki/
 
* http://ko.wikipedia.org/wiki/
119번째 줄: 111번째 줄:
 
* http://www.proofwiki.org/wiki/
 
* http://www.proofwiki.org/wiki/
 
* http://mathworld.wolfram.com/Andrews-GordonIdentity.html
 
* http://mathworld.wolfram.com/Andrews-GordonIdentity.html
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* http://www.wolframalpha.com/input/?i=
 
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* [http://dlmf.nist.gov/ NIST Digital Library of Mathematical Functions]
 
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* [http://www.research.att.com/%7Enjas/sequences/index.html The On-Line Encyclopedia of Integer Sequences]<br>
 
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** http://www.research.att.com/~njas/sequences/?q=
 
  
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==관련논문==
 
==관련논문==
 +
* Kazakov, Vladimir, Sebastien Leurent, and Dmytro Volin. “T-System on T-Hook: Grassmannian Solution and Twisted Quantum Spectral Curve.” arXiv:1510.02100 [hep-Th, Physics:math-Ph], October 7, 2015. http://arxiv.org/abs/1510.02100.
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* Larson, Hannah. “Generalized Andrews-Gordon Identities.” arXiv:1506.05063 [math], June 16, 2015. http://arxiv.org/abs/1506.05063.
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* Capparelli, S., J. Lepowsky, and A. Milas. 2006. “The Rogers-Selberg Recursions, the Gordon-Andrews Identities and Intertwining Operators.” The Ramanujan Journal. An International Journal Devoted to the Areas of Mathematics Influenced by Ramanujan 12 (3): 379–397. doi:10.1007/s11139-006-0150-7.
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* Richmond, Bruce, and George Szekeres. 1981. “Some Formulas Related to Dilogarithms, the Zeta Function and the Andrews-Gordon Identities.” Australian Mathematical Society. Journal. Series A 31 (3): 362–373. http://dx.doi.org/10.1017/S1446788700019492
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* Andrews, George E. 1974. “A General Theory of Identities of the Rogers-Ramanujan Type.” Bulletin of the American Mathematical Society 80: 1033–1052. http://projecteuclid.org/euclid.bams/1183536000
 +
* Andrews, George E. 1974. On the General Rogers-Ramanujan Theorem. Providence, R.I.: American Mathematical Society. http://www.math.psu.edu/andrews/pdf/58.pdf
 +
**  Andrews, G. E. Providence, RI: Amer. Math. Soc., 1974.
 +
* G. Andrews, An analytic generalization of the Rogers–Ramanujan identities for odd moduli, Proc. Natl. Acad. Sci. USA 71 (1974), 4082–4085. http://www.pnas.org/content/71/10/4082.short
 +
* Gordon, Basil. 1961. “A Combinatorial Generalization of the Rogers-Ramanujan Identities.” American Journal of Mathematics 83: 393–399. http://www.jstor.org/stable/2372962
  
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* [http://dx.doi.org/10.1007/s11139-006-0150-7 The Rogers–Selberg recursions, the Gordon–Andrews identities and intertwining operators]<br>
+
[[분류:q-급수]]
−
** Stefano Capparelli, James Lepowsky, Antun Milas, 2004
 
−
* [http://dx.doi.org/10.1017/S1446788700019492 Some formulas related to dilogarithms, the zeta function and the Andrews–Gordon identities]<br>
 
−
** B. Richmond and G. Szekeres, 1981
 
−
* [http://projecteuclid.org/euclid.bams/1183536000 A general theory of identities of the Rogers-Ramanujan type]<br>
 
−
** George E. Andrews, Bull. Amer. Math. Soc. Volume 80, Number 6 (1974), 1033-1052.
 
−
* [http://www.math.psu.edu/andrews/pdf/58.pdf On the General Rogers-Ramanujan Theorem.]<br>
 
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**  Andrews, G. E. Providence, RI: Amer. Math. Soc., 1974.<br>
 
−
* [http://www.pnas.org/content/71/10/4082.short An Analytic Generalization of the Rogers-Ramanujan Identities for Odd Moduli]<br>
 
−
**  George E. Andrews, PNAS October 1, 1974 vol. 71 no. 10 4082-4085<br>
 
−
* [http://www.jstor.org/stable/2372962 A Combinatorial Generalization of the Rogers-Ramanujan Identities]<br>
 
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**  Gordon, B. Amer. J. Math. 83, 393-399, 1961.<br>
 
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* http://www.jstor.org/action/doBasicSearch?Query=
 
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* http://www.ams.org/mathscinet
 
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* http://dx.doi.org/
 
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==관련기사==
 
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*  네이버 뉴스 검색 (키워드 수정)<br>
 
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** http://news.search.naver.com/search.naver?where=news&x=0&y=0&sm=tab_hty&query=
 
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** http://news.search.naver.com/search.naver?where=news&x=0&y=0&sm=tab_hty&query=
 
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** http://news.search.naver.com/search.naver?where=news&x=0&y=0&sm=tab_hty&query=
 
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==블로그==
 
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*  구글 블로그 검색<br>
 
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** http://blogsearch.google.com/blogsearch?q=
 
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* [http://navercast.naver.com/science/list 네이버 오늘의과학]
 
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* [http://betterexplained.com/ BetterExplained]
 

2020년 12월 28일 (월) 02:42 기준 최신판

개요



항등식

  • 자연수 \(k\geq 2\) , \(1\leq i \leq k\)에 대하여, 다음이 성립한다

\[\sum_{n_1,\cdots,n_{k-1}\geq0}\frac{q^{N_1^2+\cdots+N_{k-1}^2+N_i+\cdots+N_{k-1}}}{(q)_{n_1}...(q)_{n_{k-1}}}=\prod_{r\neq 0,\pm i \pmod {2k+1}}\frac{1}{1-q^r} \] 여기서 \(j\leq k-1\)이면 \(N_j=n_j+\cdots+n_{k-1}\) , \(j=k\)이면 \(N_j=0\)

  • 여러 문헌에서 다음과 같이 표현되기도 한다

\[\sum_{n_1\geq\cdots\geq n_{k-1}\geq0}\frac{q^{n_1^2+\cdots+n_{k-1}^2+n_i+\cdots+n_{k-1}}}{(q)_{n_{1}-n_{2}}\cdots (q)_{n_{k-2}-n_{k-1}}(q)_{n_{k-1}}}=\prod_{n\neq 0,\pm i\pmod {2k+1}}(1-q^n)^{-1}\]



k=2인 경우 : 로저스-라마누잔 항등식

\[H(q) =\sum_{n=0}^\infty \frac {q^{n^2+n}} {(q;q)_n} = \frac {1}{(q^2;q^5)_\infty (q^3; q^5)_\infty} =1+q^2 +q^3 +q^4+q^5 +2q^6+\cdots\]

  • i=2인 경우

\[G(q) = \sum_{n=0}^\infty \frac {q^{n^2}} {(q;q)_n} = \frac {1}{(q;q^5)_\infty (q^4; q^5)_\infty} =1+ q +q^2 +q^3 +2q^4+2q^5 +3q^6+\cdots\]



k=3인 경우

  • i=1인 경우

\[\sum_{n_1,n_{2}\geq0}\frac{q^{n_{1}^2+2n_1n_2+2n_{2}^{2}+n_1+2n_2}}{(q)_{n_1}(q)_{n_{2}}}=\prod_{r\neq 0,\pm 1 \pmod {7}}\frac{1}{1-q^r}=\frac{(q;q^7)_\infty (q^6; q^7)_\infty(q^7;q^7)_\infty}{(q)_\infty}\]

  • i=2인 경우

\[\sum_{n_1,n_{2}\geq0}\frac{q^{n_{1}^2+2n_1n_2+2n_{2}^{2}+n_2}}{(q)_{n_1}(q)_{n_{2}}}=\prod_{r\neq 0,\pm 2 \pmod {7}}\frac{1}{1-q^r}=\frac{(q^2;q^7)_\infty (q^5; q^7)_\infty(q^7;q^7)_\infty}{(q)_\infty}\]

  • i=3인 경우

\[\sum_{n_1,n_{2}\geq0}\frac{q^{n_{1}^2+2n_1n_2+2n_{2}^{2}}}{(q)_{n_1}(q)_{n_{2}}}=\prod_{r\neq 0,\pm 3 \pmod {7}}\frac{1}{1-q^r}=\frac{(q^3;q^7)_\infty (q^4; q^7)_\infty(q^7;q^7)_\infty}{(q)_\infty}\]


k=4인 경우

\[\frac{q^{n_1^2+2 n_2 n_1+2 n_3 n_1+2 n_2^2+3 n_3^2+4 n_2 n_3+n_3}}{(q)_{n_1} (q)_{n_2} (q)_{n_3}}=\prod_{r\neq 0,\pm 3 \pmod {9}}\frac{1}{1-q^r}=1+q+2 q^2+2 q^3+4 q^4+5 q^5+7 q^6+9 q^7+13 q^8+16 q^9+22 q^{10}+O(q^11)\]


얻어지는 이차형식

  • \(k=2\), \(n_{1}^{2}\)
  • \(k=3\), \((n_{1}+n_{2})^{2}+n_{2}^{2}\)
  • \(k=4\), \((n_{1}+n_{2}+n_{3})^{2}+(n_{2}+n_{3})^{2}+n_{3}^{2}\)
  • \(k=5\), \((n_{1}+n_{2}+n_{3}+n_{4})^{2}+(n_{2}+n_{3}+n_{4})^{2}+(n_{3}+n_{4})^{2}+n_{4}^{2}\)
    • 이차형식에 대응되는 행렬은 다음과 같이 주어진다

\[ \left( \begin{array}{cccc} 1 & 1 & 1 & 1 \\ 1 & 2 & 2 & 2 \\ 1 & 2 & 3 & 3 \\ 1 & 2 & 3 & 4 \\ \end{array} \right) \]



역사



메모

관련된 항목들


매스매티카 파일 및 계산 리소스


사전 형태의 자료



관련논문

  • Kazakov, Vladimir, Sebastien Leurent, and Dmytro Volin. “T-System on T-Hook: Grassmannian Solution and Twisted Quantum Spectral Curve.” arXiv:1510.02100 [hep-Th, Physics:math-Ph], October 7, 2015. http://arxiv.org/abs/1510.02100.
  • Larson, Hannah. “Generalized Andrews-Gordon Identities.” arXiv:1506.05063 [math], June 16, 2015. http://arxiv.org/abs/1506.05063.
  • Capparelli, S., J. Lepowsky, and A. Milas. 2006. “The Rogers-Selberg Recursions, the Gordon-Andrews Identities and Intertwining Operators.” The Ramanujan Journal. An International Journal Devoted to the Areas of Mathematics Influenced by Ramanujan 12 (3): 379–397. doi:10.1007/s11139-006-0150-7.
  • Richmond, Bruce, and George Szekeres. 1981. “Some Formulas Related to Dilogarithms, the Zeta Function and the Andrews-Gordon Identities.” Australian Mathematical Society. Journal. Series A 31 (3): 362–373. http://dx.doi.org/10.1017/S1446788700019492
  • Andrews, George E. 1974. “A General Theory of Identities of the Rogers-Ramanujan Type.” Bulletin of the American Mathematical Society 80: 1033–1052. http://projecteuclid.org/euclid.bams/1183536000
  • Andrews, George E. 1974. On the General Rogers-Ramanujan Theorem. Providence, R.I.: American Mathematical Society. http://www.math.psu.edu/andrews/pdf/58.pdf
    • Andrews, G. E. Providence, RI: Amer. Math. Soc., 1974.
  • G. Andrews, An analytic generalization of the Rogers–Ramanujan identities for odd moduli, Proc. Natl. Acad. Sci. USA 71 (1974), 4082–4085. http://www.pnas.org/content/71/10/4082.short
  • Gordon, Basil. 1961. “A Combinatorial Generalization of the Rogers-Ramanujan Identities.” American Journal of Mathematics 83: 393–399. http://www.jstor.org/stable/2372962
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