로저스-라마누잔 항등식

개요

• 모듈라 성질을 갖는 q-초기하급수(q-hypergeometric series) 의 중요한 예

로저스-라마누잔 항등식

• 다음의 두 항등식을 로저스-라마누잔 항등식이라 부른다

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

• Pochhammer 기호 참조

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

• 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
  • A003114 Number of partitions of n into parts 5k+1 or 5k+4
  • A003106 Number of partitions of n into parts 5k+2 or 5k+3.

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

관련논문

• 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.
  • Theorems Stated by Ramanujan (IX): Two Continued Fractions., 1929
  • Theorems Stated by Ramanujan (VII): Theorems on a Continued Fraction., 1929

블로그

• 수학과 대학원생이 되면 좋은점 - 라마누잔 이야기
  • 피타고라스의 창, 2008-6-24