앤드류스-고든 항등식(Andrews-Gordon identity)

수학노트
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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