"로저스-라마누잔 항등식"의 두 판 사이의 차이
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Pythagoras0 (토론 | 기여) |
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| (사용자 2명의 중간 판 41개는 보이지 않습니다) | |||
| 1번째 줄: | 1번째 줄: | ||
| − | + | ==개요== | |
| − | * [[ | + | * 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== | |
| − | + | * 다음의 두 항등식을 로저스-라마누잔 항등식이라 부른다 | |
| − | + | :<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> | ||
| − | + | :<math>H(q) =\sum_{n=0}^\infty \frac {q^{n^2+n}} {(q;q)_n} = | |
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\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> | ||
| − | * [[Pochhammer 기호와 캐츠(Kac) 기호| | + | * [[Pochhammer 기호와 캐츠(Kac) 기호|Pochhammer 기호]] 참조 |
| − | + | :<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> | |
| − | <math>(a;q)_n = \prod_{k=0}^{n-1} (1-aq^k)=(1-a)(1-aq)(1-aq^2)\cdots(1-aq^{n-1 | ||
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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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| − | + | ==세타함수 표현과 모듈라 성질== | |
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* 세타함수를 통한 표현 | * 세타함수를 통한 표현 | ||
| − | + | :<math>G(q)=\frac{1}{(q)_{\infty}}\sum_{n\in \mathbb{Z}}(-1)^n q^{(5n^2+n)/2}</math> | |
| − | * 로저스-라마누잔 | + | :<math>H(q)=\frac{1}{(q)_{\infty}}\sum_{n\in \mathbb{Z}}(-1)^n q^{(5n^2+3n)/2}</math> |
| − | * | + | * 로저스-라마누잔 함수는 약간의 수정을 통해 모듈라 성질을 갖게 됨 |
| − | + | :<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> | |
| − | + | :<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> | |
| − | + | * 모듈라 변환 | |
| − | + | :<math>f(\tau)=\left( | |
| − | + | \begin{array}{c} | |
| − | + | q^{-1/60}G(q) \\ | |
| − | + | q^{11/60} H(q) \\ | |
| − | + | \end{array} | |
| − | + | \right) | |
| − | + | </math> | |
| − | + | 로 두면, 다음이 성립한다 | |
| − | + | :<math> | |
| − | + | f(\tau+1)= | |
| − | + | \left( | |
| − | + | \begin{array}{cc} | |
| − | + | \zeta_{60}^{-1} & 0 \\ | |
| − | + | 0 & \zeta_{60}^{11} \\ | |
| − | + | \end{array} | |
| − | + | \right)f(\tau) | |
| − | + | </math> | |
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| − | < | + | :<math> |
| + | 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) | ||
| + | </math> | ||
| − | ( | + | * [[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== | |
| − | ( | + | * <math>q=e^{-t}</math> 으로 두면 <math>t\sim 0</math> 일 때, |
| + | :<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> | ||
| + | :<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> | ||
| + | * '''[McIntosh1995]''' 참조 | ||
| + | * 이로부터 <math>t\to 0</math> 일 때, <math>q=e^{-t}\to 1</math> 으로 다음이 성립함을 알 수 있다 | ||
| + | :<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== | |
| − | <math> | + | * 두 함수의 비는 아래와 같은 연분수 표현을 가진다 |
| + | :<math>\frac{H(q)}{G(q)} = \cfrac{1}{1+\cfrac{q}{1+\cfrac{q^2}{1+\cfrac{q^3}{1+\cdots}}}}</math> | ||
| + | * [[로저스-라마누잔 연분수]] 항목에서 다루기로 함 | ||
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| − | + | ==재미있는 사실== | |
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* 이 항등식은 통계물리의 Lee-Yang 모델과 밀접하게 관련되어 있음 | * 이 항등식은 통계물리의 Lee-Yang 모델과 밀접하게 관련되어 있음 | ||
| + | * http://mathoverflow.net/questions/29117/what-is-the-relationship-between-modular-forms-and-the-rogers-ramanujan-identitie | ||
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| − | + | ==관련된 항목들== | |
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* [[모듈라 군, j-invariant and the singular moduli|The modular group, j-invariant and the singular moduli]] | * [[모듈라 군, j-invariant and the singular moduli|The modular group, j-invariant and the singular moduli]] | ||
* [[5차방정식과 정이십면체|오차방정식과 정이십면체]] | * [[5차방정식과 정이십면체|오차방정식과 정이십면체]] | ||
* [[초기하급수(Hypergeometric series)]] | * [[초기하급수(Hypergeometric series)]] | ||
| − | * [[ | + | * [[Q-초기하급수(q-hypergeometric series)와 양자미적분학(q-calculus)]] |
* [[다이로그 함수(dilogarithm)|Dilogarithm 함수]] | * [[다이로그 함수(dilogarithm)|Dilogarithm 함수]] | ||
| + | * [[연분수와 유리수 근사|연분수]] | ||
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| − | + | ==매스매티카 파일 및 계산 리소스== | |
| − | + | * https://docs.google.com/leaf?id=0B8XXo8Tve1cxNmQ3NGMzZWMtZTg4OC00NjBlLTljNmUtOGExYjkyYjA3NDkx&sort=name&layout=list&num=50 | |
| + | * [http://www.research.att.com/%7Enjas/sequences/index.html The On-Line Encyclopedia of Integer Sequences] | ||
| + | ** [http://oeis.org/A003114 A003114] Number of partitions of n into parts 5k+1 or 5k+4 | ||
| + | ** [http://oeis.org/A003106 A003106] Number of partitions of n into parts 5k+2 or 5k+3. | ||
| − | + | ||
| − | + | ==사전형태의 자료== | |
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| − | + | * [http://ko.wikipedia.org/wiki/%EC%97%B0%EB%B6%84%EC%88%98 http://ko.wikipedia.org/wiki/연분수] | |
| + | * http://en.wikipedia.org/wiki/Rogers-Ramanujan_identities | ||
| + | * [http://en.wikipedia.org/wiki/Rogers%E2%80%93Ramanujan_continued_fraction http://en.wikipedia.org/wiki/Rogers–Ramanujan_continued_fraction] | ||
| + | * http://en.wikipedia.org/wiki/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== | |
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| − | * [http://www. | + | * [http://www.amazon.com/Number-Theory-Spirit-Ramanujan-Berndt/dp/0821841785 Number Theory in the Spirit of Ramanujan] |
| − | ** | + | ** Bruce C. Berndt |
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| − | + | ==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:[http://dx.doi.org/10.2307/2325145 10.2307/2325145]. | ||
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| − | < | + | ==관련논문== |
| − | + | * 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. | |
| − | * | + | * 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 |
| − | + | * 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. | |
| − | ** [http:// | + | * 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]. |
| − | ** http:// | + | * [http://www.ams.org/bull/2005-42-02/S0273-0979-05-01047-5/home.html#References Continued fractions and modular functions] |
| − | ** http:// | + | ** W. Duke, Bull. Amer. Math. Soc. 42 (2005), 137-162 |
| − | ** http:// | + | * 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 |
| − | + | * [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) | |
| + | * [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] | ||
| + | ** Bruce C. Berndt , Heng Huat Chan , Liang-Cheng Zhang, 1997 | ||
| + | * [http://www.digizeitschriften.de/index.php?id=loader&tx_jkDigiTools_pi1%5BIDDOC%5D=503543 Explicit evaluations of the Rogers-Ramanujan continued fraction.] | ||
| + | ** Berndt, B.C,Chan, H.H.,Zhang, L.-C., Journal für die reine und angewandte Mathematik 480, 1996 | ||
| + | * '''[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 | ||
| + | * Watson, G. N. | ||
| + | ** [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 | ||
| − | + | ==블로그== | |
| − | + | * [http://bomber0.byus.net/index.php/2008/06/24/673 수학과 대학원생이 되면 좋은점 - 라마누잔 이야기] | |
| + | ** 피타고라스의 창, 2008-6-24 | ||
| + | [[분류:q-급수]] | ||
| − | * [ | + | ==메타데이터== |
| − | + | ===위키데이터=== | |
| − | * | + | * ID : [https://www.wikidata.org/wiki/Q7359380 Q7359380] |
| − | * | + | ===Spacy 패턴 목록=== |
| + | * [{'LOWER': 'rogers'}, {'OP': '*'}, {'LOWER': 'ramanujan'}, {'LEMMA': 'identity'}] | ||
2021년 2월 17일 (수) 05:40 기준 최신판
개요
- 모듈라 성질을 갖는 q-초기하급수(q-hypergeometric series) 의 중요한 예
로저스-라마누잔 항등식
- 다음의 두 항등식을 로저스-라마누잔 항등식이라 부른다
\[G(q) = \sum_{n=0}^\infty \frac {q^{n^2}} {(q;q)_n} = \frac {1}{(q;q^5)_\infty (q^4; q^5)_\infty} =1+ q +q^2 +q^3 +2q^4+2q^5 +3q^6+\cdots\] \[H(q) =\sum_{n=0}^\infty \frac {q^{n^2+n}} {(q;q)_n} = \frac {1}{(q^2;q^5)_\infty (q^3; q^5)_\infty} =1+q^2 +q^3 +q^4+q^5 +2q^6+\cdots\]
\[(a;q)_n = \prod_{k=0}^{n-1} (1-aq^k)=(1-a)(1-aq)(1-aq^2)\cdots(1-aq^{n-1})\]
- q-초기하급수(q-hypergeometric series) 의 틀에서 이해할 수 있다
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 \[ f(\tau+1)= \left( \begin{array}{cc} \zeta_{60}^{-1} & 0 \\ 0 & \zeta_{60}^{11} \\ \end{array} \right)f(\tau) \]
\[ f(-\frac{1}{\tau}) = \frac{2}{\sqrt{5}} \left( \begin{array}{cc} \sin \left(\frac{2 \pi }{5}\right) & \sin \left(\frac{\pi }{5}\right) \\ \sin \left(\frac{\pi }{5}\right) & -\sin \left(\frac{2 \pi }{5}\right) \\ \end{array} \right)f(\tau) = \left( \begin{array}{cc} \sqrt{\frac{2}{5-\sqrt{5}}} & \sqrt{\frac{2}{5+\sqrt{5}}} \\ \sqrt{\frac{2}{5+\sqrt{5}}} & -\sqrt{\frac{2}{5-\sqrt{5}}} \\ \end{array} \right)f(\tau) \]
- 데데킨트 에타함수가 갖는 modularity와의 유사성\[\eta(\tau) = q^{1/24} \prod_{n=1}^{\infty} (1-q^{n})\]
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
\[\frac{H(q)}{G(q)} = \cfrac{1}{1+\cfrac{q}{1+\cfrac{q^2}{1+\cfrac{q^3}{1+\cdots}}}}\]
- 로저스-라마누잔 연분수 항목에서 다루기로 함
재미있는 사실
- 이 항등식은 통계물리의 Lee-Yang 모델과 밀접하게 관련되어 있음
- http://mathoverflow.net/questions/29117/what-is-the-relationship-between-modular-forms-and-the-rogers-ramanujan-identitie
관련된 항목들
- The modular group, j-invariant and the singular moduli
- 오차방정식과 정이십면체
- 초기하급수(Hypergeometric series)
- Q-초기하급수(q-hypergeometric series)와 양자미적분학(q-calculus)
- Dilogarithm 함수
- 연분수
매스매티카 파일 및 계산 리소스
- https://docs.google.com/leaf?id=0B8XXo8Tve1cxNmQ3NGMzZWMtZTg4OC00NjBlLTljNmUtOGExYjkyYjA3NDkx&sort=name&layout=list&num=50
- The On-Line Encyclopedia of Integer Sequences
사전형태의 자료
- http://ko.wikipedia.org/wiki/연분수
- http://en.wikipedia.org/wiki/Rogers-Ramanujan_identities
- http://en.wikipedia.org/wiki/Rogers–Ramanujan_continued_fraction
- http://en.wikipedia.org/wiki/Continued_fraction
- http://en.wikipedia.org/wiki/Gauss's_continued_fraction
관련도서
- Number Theory in the Spirit of Ramanujan
- Bruce C. Berndt
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.
관련논문
- Goodwin, Simon M., Tung Le, and Kay Magaard. “The Generic Character Table of a Sylow \(p\)-Subgroup of a Finite Chevalley Group of Type \(D_4\).” arXiv:1508.06937 [math], August 27, 2015. http://arxiv.org/abs/1508.06937.
- 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
- 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.
- 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:10.1155/2009/941920.
- Continued fractions and modular functions
- W. Duke, Bull. Amer. Math. Soc. 42 (2005), 137-162
- 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
- Ramanujan’s formulas for the explicit evaluation of the Rogers–Ramanujan continued fraction and theta-functions
- Soon-Yi Kang, ACTA ARITHMETICA XC.1 (1999)
- Ramanujan's Class Invariants With Applications To The Values Of q-Continued Fractions And Theta-Functions
- Bruce C. Berndt , Heng Huat Chan , Liang-Cheng Zhang, 1997
- Explicit evaluations of the Rogers-Ramanujan continued fraction.
- Berndt, B.C,Chan, H.H.,Zhang, L.-C., Journal für die reine und angewandte Mathematik 480, 1996
- [McIntosh1995]Some Asymptotic Formulae for q-Hypergeometric Series
- Richard J. McIntosh, Journal of the London Mathematical Society 1995 51(1):120-136
- Watson, G. N.
블로그
- 수학과 대학원생이 되면 좋은점 - 라마누잔 이야기
- 피타고라스의 창, 2008-6-24
메타데이터
위키데이터
- ID : Q7359380
Spacy 패턴 목록
- [{'LOWER': 'rogers'}, {'OP': '*'}, {'LOWER': 'ramanujan'}, {'LEMMA': 'identity'}]