"로저스-라마누잔 항등식"의 두 판 사이의 차이

수학노트
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잔글 (찾아 바꾸기 – “<h5 (.*)">” 문자열을 “==” 문자열로)
 
(같은 사용자의 중간 판 19개는 보이지 않습니다)
1번째 줄: 1번째 줄:
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==이 항목의 스프링노트 원문주소==
 
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* [[로저스-라마누잔 항등식|로저스-라마누잔 연분수와 항등식]]
 
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==개요==
 
==개요==
  
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* 모듈라 성질을 갖는 [[q-초기하급수(q-hypergeometric series) (통합됨)|q-초기하급수(q-hypergeometric series)]] 의 중요한 예
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* 모듈라 성질을 갖는 [[Q-초기하급수(q-hypergeometric series)와 양자미적분학(q-calculus)|q-초기하급수(q-hypergeometric series)]] 의 중요한 예
  
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==로저스-라마누잔 항등식==
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==로저스-라마누잔 항등식==
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* 다음의 두 항등식을 로저스-라마누잔 항등식이라 부른다
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<math>G(q) = \sum_{n=0}^\infty \frac {q^{n^2}} {(q;q)_n} =  
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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}
 
   =1+ q +q^2 +q^3 +2q^4+2q^5 +3q^6+\cdots</math>
 
   =1+ q +q^2 +q^3 +2q^4+2q^5 +3q^6+\cdots</math>
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:<math>H(q) =\sum_{n=0}^\infty \frac {q^{n^2+n}} {(q;q)_n} =  
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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}
 
  =1+q^2 +q^3 +q^4+q^5 +2q^6+\cdots</math>
 
  =1+q^2 +q^3 +q^4+q^5 +2q^6+\cdots</math>
  
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* [[Pochhammer 기호와 캐츠(Kac) 기호|Pochhammer 기호]] 참조 
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* [[Pochhammer 기호와 캐츠(Kac) 기호|Pochhammer 기호]] 참조
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:<math>(a;q)_n = \prod_{k=0}^{n-1} (1-aq^k)=(1-a)(1-aq)(1-aq^2)\cdots(1-aq^{n-1})</math>
  
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<math>(a;q)_n = \prod_{k=0}^{n-1} (1-aq^k)=(1-a)(1-aq)(1-aq^2)\cdots(1-aq^{n-1})</math>
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* [[Q-초기하급수(q-hypergeometric series)와 양자미적분학(q-calculus)|q-초기하급수(q-hypergeometric series)]] 의 틀에서 이해할 수 있다
  
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* [[q-초기하급수(q-hypergeometric series) (통합됨)|q-초기하급수(q-hypergeometric series)]] 의 틀에서 이해할 수 있다<br>
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==세타함수 표현과 모듈라 성질==
 
==세타함수 표현과 모듈라 성질==
  
 
* 세타함수를 통한 표현
 
* 세타함수를 통한 표현
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* <math>G(q)=\frac{1}{(q)_{\infty}}\sum_{n\in \mathbb{Z}}(-1)^n q^{(5n^2+n)/2}</math><br><math>H(q)=\frac{1}{(q)_{\infty}}\sum_{n\in \mathbb{Z}}(-1)^n q^{(5n^2+3n)/2}</math><br>
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:<math>G(q)=\frac{1}{(q)_{\infty}}\sum_{n\in \mathbb{Z}}(-1)^n q^{(5n^2+n)/2}</math>
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*  로저스-라마누잔 함수는 약간의 수정을 통해 modularity를 가짐<br><math>q^{-1/60}G(q) = q^{-1/60}\sum_{n=0}^\infty \frac {q^{n^2}} {(q;q)_n} = \frac {q^{-1/60}}{(q;q^5)_\infty (q^4; q^5)_\infty}</math><br><math>q^{11/60}H(q) =q^{11/60}\sum_{n=0}^\infty \frac {q^{n^2+n}} {(q;q)_n} = q^{11/60}\frac {1}{(q^2;q^5)_\infty (q^3; q^5)_\infty} </math><br>
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:<math>H(q)=\frac{1}{(q)_{\infty}}\sum_{n\in \mathbb{Z}}(-1)^n q^{(5n^2+3n)/2}</math>
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* [[데데킨트 에타함수]]가 갖는 modularity와의 유사성<br><math>\eta(\tau) = q^{1/24} \prod_{n=1}^{\infty} (1-q^{n})</math><br>
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*  로저스-라마누잔 함수는 약간의 수정을 통해 모듈라 성질을 갖게 됨
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:<math>q^{-1/60}G(q) = q^{-1/60}\sum_{n=0}^\infty \frac {q^{n^2}} {(q;q)_n} = \frac {q^{-1/60}}{(q;q^5)_\infty (q^4; q^5)_\infty}</math>
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:<math>q^{11/60}H(q) =q^{11/60}\sum_{n=0}^\infty \frac {q^{n^2+n}} {(q;q)_n} = \frac {q^{11/60}}{(q^2;q^5)_\infty (q^3; q^5)_\infty} </math>
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* 모듈라 변환
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:<math>f(\tau)=\left(
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\begin{array}{c}
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q^{-1/60}G(q) \\
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q^{11/60} H(q) \\
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\end{array}
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\right)
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</math>
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로 두면, 다음이 성립한다
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:<math>
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f(\tau+1)=
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\left(
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\begin{array}{cc}
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\zeta_{60}^{-1} & 0 \\
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0 & \zeta_{60}^{11} \\
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\end{array}
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\right)f(\tau)  
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</math>
  
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:<math>
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f(-\frac{1}{\tau})
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=
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\frac{2}{\sqrt{5}}
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\left(
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\begin{array}{cc}
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\sin \left(\frac{2 \pi }{5}\right) & \sin \left(\frac{\pi }{5}\right) \\
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\sin \left(\frac{\pi }{5}\right) & -\sin \left(\frac{2 \pi }{5}\right) \\
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\end{array}
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\right)f(\tau)
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=
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\left(
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\begin{array}{cc}
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\sqrt{\frac{2}{5-\sqrt{5}}} & \sqrt{\frac{2}{5+\sqrt{5}}} \\
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\sqrt{\frac{2}{5+\sqrt{5}}} & -\sqrt{\frac{2}{5-\sqrt{5}}} \\
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\end{array}
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\right)f(\tau)
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</math>
  
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* [[데데킨트 에타함수]]가 갖는 modularity와의 유사성:<math>\eta(\tau) = q^{1/24} \prod_{n=1}^{\infty} (1-q^{n})</math>
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==cusp에서의 변화==
 
==cusp에서의 변화==
  
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* <math>q=e^{-t}</math> 으로 두면 <math>t\sim 0</math> 일 때,<br><math>H(q)=\sum_{n=0}^\infty \frac {q^{n^2+n}} {(q;q)_n} \sim  \sqrt\frac{2}{5+\sqrt{5}}\exp(\frac{\pi^2}{15t}+\frac{11t}{60})+o(1)</math><br><math>G(q)=\sum_{n=0}^\infty \frac {q^{n^2}} {(q;q)_n} \sim  \sqrt\frac{2}{5-\sqrt{5}}\exp(\frac{\pi^2}{15t}-\frac{t}{60})+o(1)</math><br>
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* <math>q=e^{-t}</math> 으로 두면 <math>t\sim 0</math> 일 때,
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* '''[McIntosh1995]''' 참조<br>
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:<math>H(q)=\sum_{n=0}^\infty \frac {q^{n^2+n}} {(q;q)_n} \sim  \sqrt\frac{2}{5+\sqrt{5}}\exp(\frac{\pi^2}{15t}+\frac{11t}{60})+o(1)</math>
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* 이로부터 다음을 알 수 있다<br><math>t\to 0</math> 일 때, <math>q=e^{-t}\to 1</math> 으로 두면<br><math>\frac{H(1)}{G(1)} = \sqrt{\frac{5-\sqrt{5}}{5+\sqrt{5}}}=\varphi-1=0.618\cdots</math><br>
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:<math>G(q)=\sum_{n=0}^\infty \frac {q^{n^2}} {(q;q)_n} \sim  \sqrt\frac{2}{5-\sqrt{5}}\exp(\frac{\pi^2}{15t}-\frac{t}{60})+o(1)</math>
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* '''[McIntosh1995]''' 참조
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* 이로부터 <math>t\to 0</math> 일 때, <math>q=e^{-t}\to 1</math> 으로 다음이 성립함을 알 수 있다
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:<math>\frac{H(1)}{G(1)} = \sqrt{\frac{5-\sqrt{5}}{5+\sqrt{5}}}=\varphi-1=0.618\cdots</math>
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==로저스-라마누잔 연분수==
 
==로저스-라마누잔 연분수==
  
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*  두 함수의 비는 아래와 같은 연분수 표현을 가진다<br><math>\frac{H(q)}{G(q)} = \cfrac{1}{1+\cfrac{q}{1+\cfrac{q^2}{1+\cfrac{q^3}{1+\cdots}}}}</math><br>
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*  두 함수의 비는 아래와 같은 연분수 표현을 가진다
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* [[로저스-라마누잔 연분수]]  항목에서 다루기로 함
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:<math>\frac{H(q)}{G(q)} = \cfrac{1}{1+\cfrac{q}{1+\cfrac{q^2}{1+\cfrac{q^3}{1+\cdots}}}}</math>
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* [[로저스-라마누잔 연분수]] 항목에서 다루기로 함
  
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==재미있는 사실==
 
==재미있는 사실==
72번째 줄: 106번째 줄:
 
* http://mathoverflow.net/questions/29117/what-is-the-relationship-between-modular-forms-and-the-rogers-ramanujan-identitie
 
* http://mathoverflow.net/questions/29117/what-is-the-relationship-between-modular-forms-and-the-rogers-ramanujan-identitie
  
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==관련된 항목들==
 
==관련된 항목들==
81번째 줄: 115번째 줄:
 
* [[5차방정식과 정이십면체|오차방정식과 정이십면체]]
 
* [[5차방정식과 정이십면체|오차방정식과 정이십면체]]
 
* [[초기하급수(Hypergeometric series)]]
 
* [[초기하급수(Hypergeometric series)]]
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* [[q-초기하급수(q-hypergeometric series) (통합됨)|q-초기하급수(q-hypergeometric series)]]
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* [[Q-초기하급수(q-hypergeometric series)와 양자미적분학(q-calculus)]]
 
* [[다이로그 함수(dilogarithm)|Dilogarithm 함수]]
 
* [[다이로그 함수(dilogarithm)|Dilogarithm 함수]]
 
* [[연분수와 유리수 근사|연분수]]
 
* [[연분수와 유리수 근사|연분수]]
  
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==매스매티카 파일 및 계산 리소스==
 
==매스매티카 파일 및 계산 리소스==
  
 
* https://docs.google.com/leaf?id=0B8XXo8Tve1cxNmQ3NGMzZWMtZTg4OC00NjBlLTljNmUtOGExYjkyYjA3NDkx&sort=name&layout=list&num=50
 
* https://docs.google.com/leaf?id=0B8XXo8Tve1cxNmQ3NGMzZWMtZTg4OC00NjBlLTljNmUtOGExYjkyYjA3NDkx&sort=name&layout=list&num=50
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* http://www.wolframalpha.com/input/?i=
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* [http://www.research.att.com/%7Enjas/sequences/index.html The On-Line Encyclopedia of Integer Sequences]
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* http://functions.wolfram.com/
 
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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>
 
 
** [http://oeis.org/A003114 A003114] Number of partitions of n into parts 5k+1 or 5k+4
 
** [http://oeis.org/A003114 A003114] Number of partitions of n into parts 5k+1 or 5k+4
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** [http://oeis.org/A003106 A003106]         Number of partitions of n into parts 5k+2 or 5k+3.
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** [http://oeis.org/A003106 A003106]         Number of partitions of n into parts 5k+2 or 5k+3.
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* [http://numbers.computation.free.fr/Constants/constants.html Numbers, constants and computation]
 
  
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* [[매스매티카 파일 목록]]
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==사전형태의 자료==
 
==사전형태의 자료==
115번째 줄: 140번째 줄:
 
* [http://en.wikipedia.org/wiki/Gauss%27s_continued_fraction http://en.wikipedia.org/wiki/Gauss's_continued_fraction]
 
* [http://en.wikipedia.org/wiki/Gauss%27s_continued_fraction http://en.wikipedia.org/wiki/Gauss's_continued_fraction]
  
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==관련도서 및 추천도서==
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==관련도서==
  
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* [http://www.amazon.com/Number-Theory-Spirit-Ramanujan-Berndt/dp/0821841785 Number Theory in the Spirit of Ramanujan]<br>
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* [http://www.amazon.com/Number-Theory-Spirit-Ramanujan-Berndt/dp/0821841785 Number Theory in the Spirit of Ramanujan]
 
** Bruce C. Berndt
 
** Bruce C. Berndt
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*  도서내검색<br>
 
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** http://books.google.com/books?q=
 
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** http://book.daum.net/search/contentSearch.do?query=
 
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*  도서검색<br>
 
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** http://www.amazon.com/s/ref=nb_ss_gw?url=search-alias%3Dstripbooks&field-keywords=
 
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** http://book.daum.net/search/mainSearch.do?query=
 
  
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==리뷰, 에세이, 강의노트==
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* Andrews, George E., and R. J. Baxter. “A Motivated Proof of the Rogers-Ramanujan Identities.” The American Mathematical Monthly 96, no. 5 (May 1, 1989): 401–9. doi:[http://dx.doi.org/10.2307/2325145 10.2307/2325145].
  
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==관련논문==
 
==관련논문==
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* Goodwin, Simon M., Tung Le, and Kay Magaard. “The Generic Character Table of a Sylow <math>p</math>-Subgroup of a Finite Chevalley Group of Type <math>D_4</math>.” arXiv:1508.06937 [math], August 27, 2015. http://arxiv.org/abs/1508.06937.
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* [http://dx.doi.org/10.1155/2009/941920 Probabilities as Values of Modular Forms and Continued Fractions]<br>
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* Berndt, Bruce C. Ramanujan's forty identities for the Rogers-Ramanujan functions. Vol. 181. American Mathematical Soc., 2007. http://personal.psu.edu/auy2/articles/fortyidentity.pdf
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** Riad Masri and Ken Ono, 2009
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* Gugg, Chadwick. “Modular Identities for the Rogers-Ramanujan Functions and Analogues.” University of Illinois at Urbana-Champaign, 2011. https://www.ideals.illinois.edu/handle/2142/18485.
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* [http://www.ams.org/bull/2005-42-02/S0273-0979-05-01047-5/home.html#References Continued fractions and modular functions]<br>
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* Masri, Riad, and Ken Ono. “Probabilities as Values of Modular Forms and Continued Fractions.” International Journal of Mathematics and Mathematical Sciences 2009 (September 15, 2009): e941920. doi:[http://dx.doi.org/10.1155/2009/941920 10.1155/2009/941920].
−
** W. Duke, Bull. Amer. Math. Soc. 42 (2005), 137-162
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* [http://www.ams.org/bull/2005-42-02/S0273-0979-05-01047-5/home.html#References Continued fractions and modular functions]
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* [http://arxiv.org/abs/math/0309201 Ramanujan's "Lost Notebook" and the Virasoro Algebra]<br>
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** W. Duke, Bull. Amer. Math. Soc. 42 (2005), 137-162
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** Antun Milas, Commun.Math.Phys. 251 (2004) 567-588
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* Milas, Antun. “Ramanujan’s ‘Lost Notebook’ and the Virasoro Algebra.” Communications in Mathematical Physics 251, no. 3 (November 2004): 567–88. doi:10.1007/s00220-004-1179-3. http://arxiv.org/abs/math/0309201
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* [http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.31.5875 Ramanujan’s formulas for the explicit evaluation of the Rogers–Ramanujan continued fraction and theta-functions]<br>
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* [http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.31.5875 Ramanujan’s formulas for the explicit evaluation of the Rogers–Ramanujan continued fraction and theta-functions]
−
** Soon-Yi Kang, ACTA ARITHMETICA XC.1 (1999)
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** Soon-Yi Kang, ACTA ARITHMETICA XC.1 (1999)
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* [http://citeseerx.ist.psu.edu/viewdoc/download;jsessionid=AFEE1CBFE5553E6717E8292B3F080D00?doi=10.1.1.39.4015&rep=rep1&type=pdf Ramanujan's Class Invariants With Applications To The Values Of q-Continued Fractions And Theta-Functions]<br>
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* [http://citeseerx.ist.psu.edu/viewdoc/download;jsessionid=AFEE1CBFE5553E6717E8292B3F080D00?doi=10.1.1.39.4015&rep=rep1&type=pdf Ramanujan's Class Invariants With Applications To The Values Of q-Continued Fractions And Theta-Functions]
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** Bruce C. Berndt ,  Heng Huat Chan ,  Liang-Cheng Zhang, 1997
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** Bruce C. Berndt , Heng Huat Chan , Liang-Cheng Zhang, 1997
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* [http://www.digizeitschriften.de/index.php?id=loader&tx_jkDigiTools_pi1%5BIDDOC%5D=503543 Explicit evaluations of the Rogers-Ramanujan continued fraction.]<br>
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* [http://www.digizeitschriften.de/index.php?id=loader&tx_jkDigiTools_pi1%5BIDDOC%5D=503543 Explicit evaluations of the Rogers-Ramanujan continued fraction.]
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** Berndt, B.C,Chan, H.H.,Zhang, L.-C., Journal für die reine und angewandte Mathematik 480, 1996
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** Berndt, B.C,Chan, H.H.,Zhang, L.-C., Journal für die reine und angewandte Mathematik 480, 1996
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* '''[McIntosh1995]'''[http://jlms.oxfordjournals.org/cgi/content/abstract/51/1/120 Some Asymptotic Formulae for q-Hypergeometric Series]<br>
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* '''[McIntosh1995]'''[http://jlms.oxfordjournals.org/cgi/content/abstract/51/1/120 Some Asymptotic Formulae for q-Hypergeometric Series]
 
** Richard J. McIntosh, Journal of the London Mathematical Society 1995 51(1):120-136
 
** Richard J. McIntosh, Journal of the London Mathematical Society 1995 51(1):120-136
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*  Watson, G. N.
−
* [http://www.jstor.org/stable/2325145 A Motivated Proof of the Rogers-Ramanujan Identities]<br>
 
−
** George E. Andrews and R. J. Baxter, <cite style="line-height: 2em;">The American Mathematical Monthly</cite>, Vol. 96, No. 5 (May, 1989), pp. 401-409
 
−
 
 
−
*  Watson, G. N.<br>
 
 
** [http://www.google.com/url?sa=t&ct=res&cd=1&url=http%3A%2F%2Fjlms.oxfordjournals.org%2Fcgi%2Freprint%2Fs1-4%2F3%2F231&ei=JY1hSLWRLpSY8gSI7JSiBQ&usg=AFQjCNElhd9FwCl3m3Qcb3hW7j87K1P5FQ&sig2=4OhMIB56amm8h4EOGNSk6g Theorems Stated by Ramanujan (IX): Two Continued Fractions.], 1929
 
** [http://www.google.com/url?sa=t&ct=res&cd=1&url=http%3A%2F%2Fjlms.oxfordjournals.org%2Fcgi%2Freprint%2Fs1-4%2F3%2F231&ei=JY1hSLWRLpSY8gSI7JSiBQ&usg=AFQjCNElhd9FwCl3m3Qcb3hW7j87K1P5FQ&sig2=4OhMIB56amm8h4EOGNSk6g Theorems Stated by Ramanujan (IX): Two Continued Fractions.], 1929
 
** [http://www.google.com/url?sa=t&ct=res&cd=1&url=http%3A%2F%2Fjlms.oxfordjournals.org%2Fcgi%2Freprint%2Fs1-4%2F13%2F39&ei=HY5hSNa6E5ym8ASu_biqBQ&usg=AFQjCNGfZ9Hu3vXz6bawkdnRZ2UU6jDUPA&sig2=dEC2KNSntm2J6L5GwTii3A Theorems Stated by Ramanujan (VII): Theorems on a Continued Fraction.], 1929
 
** [http://www.google.com/url?sa=t&ct=res&cd=1&url=http%3A%2F%2Fjlms.oxfordjournals.org%2Fcgi%2Freprint%2Fs1-4%2F13%2F39&ei=HY5hSNa6E5ym8ASu_biqBQ&usg=AFQjCNGfZ9Hu3vXz6bawkdnRZ2UU6jDUPA&sig2=dEC2KNSntm2J6L5GwTii3A Theorems Stated by Ramanujan (VII): Theorems on a Continued Fraction.], 1929
  
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2021년 2월 17일 (수) 05:40 기준 최신판

개요



로저스-라마누잔 항등식

  • 다음의 두 항등식을 로저스-라마누잔 항등식이라 부른다

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

\[(a;q)_n = \prod_{k=0}^{n-1} (1-aq^k)=(1-a)(1-aq)(1-aq^2)\cdots(1-aq^{n-1})\]



세타함수 표현과 모듈라 성질

  • 세타함수를 통한 표현

\[G(q)=\frac{1}{(q)_{\infty}}\sum_{n\in \mathbb{Z}}(-1)^n q^{(5n^2+n)/2}\] \[H(q)=\frac{1}{(q)_{\infty}}\sum_{n\in \mathbb{Z}}(-1)^n q^{(5n^2+3n)/2}\]

  • 로저스-라마누잔 함수는 약간의 수정을 통해 모듈라 성질을 갖게 됨

\[q^{-1/60}G(q) = q^{-1/60}\sum_{n=0}^\infty \frac {q^{n^2}} {(q;q)_n} = \frac {q^{-1/60}}{(q;q^5)_\infty (q^4; q^5)_\infty}\] \[q^{11/60}H(q) =q^{11/60}\sum_{n=0}^\infty \frac {q^{n^2+n}} {(q;q)_n} = \frac {q^{11/60}}{(q^2;q^5)_\infty (q^3; q^5)_\infty} \]

  • 모듈라 변환

\[f(\tau)=\left( \begin{array}{c} q^{-1/60}G(q) \\ q^{11/60} H(q) \\ \end{array} \right) \] 로 두면, 다음이 성립한다 \[ f(\tau+1)= \left( \begin{array}{cc} \zeta_{60}^{-1} & 0 \\ 0 & \zeta_{60}^{11} \\ \end{array} \right)f(\tau) \]

\[ f(-\frac{1}{\tau}) = \frac{2}{\sqrt{5}} \left( \begin{array}{cc} \sin \left(\frac{2 \pi }{5}\right) & \sin \left(\frac{\pi }{5}\right) \\ \sin \left(\frac{\pi }{5}\right) & -\sin \left(\frac{2 \pi }{5}\right) \\ \end{array} \right)f(\tau) = \left( \begin{array}{cc} \sqrt{\frac{2}{5-\sqrt{5}}} & \sqrt{\frac{2}{5+\sqrt{5}}} \\ \sqrt{\frac{2}{5+\sqrt{5}}} & -\sqrt{\frac{2}{5-\sqrt{5}}} \\ \end{array} \right)f(\tau) \]



cusp에서의 변화

  • \(q=e^{-t}\) 으로 두면 \(t\sim 0\) 일 때,

\[H(q)=\sum_{n=0}^\infty \frac {q^{n^2+n}} {(q;q)_n} \sim \sqrt\frac{2}{5+\sqrt{5}}\exp(\frac{\pi^2}{15t}+\frac{11t}{60})+o(1)\] \[G(q)=\sum_{n=0}^\infty \frac {q^{n^2}} {(q;q)_n} \sim \sqrt\frac{2}{5-\sqrt{5}}\exp(\frac{\pi^2}{15t}-\frac{t}{60})+o(1)\]

  • [McIntosh1995] 참조
  • 이로부터 \(t\to 0\) 일 때, \(q=e^{-t}\to 1\) 으로 다음이 성립함을 알 수 있다

\[\frac{H(1)}{G(1)} = \sqrt{\frac{5-\sqrt{5}}{5+\sqrt{5}}}=\varphi-1=0.618\cdots\]



로저스-라마누잔 연분수

  • 두 함수의 비는 아래와 같은 연분수 표현을 가진다

\[\frac{H(q)}{G(q)} = \cfrac{1}{1+\cfrac{q}{1+\cfrac{q^2}{1+\cfrac{q^3}{1+\cdots}}}}\]




재미있는 사실



관련된 항목들



매스매티카 파일 및 계산 리소스


사전형태의 자료



관련도서



리뷰, 에세이, 강의노트

  • Andrews, George E., and R. J. Baxter. “A Motivated Proof of the Rogers-Ramanujan Identities.” The American Mathematical Monthly 96, no. 5 (May 1, 1989): 401–9. doi:10.2307/2325145.


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