앤드류스-고든 항등식(Andrews-Gordon identity)
이 항목의 스프링노트 원문주소==
개요==
- 로저스-라마누잔 연분수와 항등식의 일반화
- 모듈라 성질을 가지며(모듈라 형식(modular forms) 참조), q-초기하급수(q-hypergeometric series) 형태로 표현 가능
- 등각장론에서 c(2, 2k+1) minimal models 의 캐릭터
\(\chi_j(\tau)=q^{h_j-c/24}\prod_{n\neq 0,\pm(j+1)}(1-q^n)^{-1}\)
\(\chi_j(\tau)=q^{h_j-c/24}\prod_{n\neq 0,\pm(j+1)}(1-q^n)^{-1}\)
항등식==
- 자연수 \(k\geq 2\) , \(1\leq i \leq k\)에 대하여, 다음이 성립한다
\(\sum_{n_1,\cdots,n_{k-1}\geq0}\frac{x^{N_1^2+\cdots+N_{k-1}^2+N_i+\cdots+N_{k-1}}}{(x)_{n_1}...(x)_{n_{k-1}}}=\prod_{r\neq 0,\pm i \pmod {2k+1}}\frac{1}{1-x^r} \)
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
\(\sum_{n_1\geq\cdots\geq n_{k-1}\geq0}\frac{q^{n_1^2+\cdots+n_{k-1}^2+n_i+\cdots+n_{k-1}}}{(q)_{n_{1}-n_{2}}\cdots (q)_{n_{k-2}-n_{k-1}}(q)_{n_{k-1}}}=\prod_{n\neq 0,\pm i\pmod {2k+1}}(1-q^n)^{-1}\)
\(\sum_{n_1,\cdots,n_{k-1}\geq0}\frac{x^{N_1^2+\cdots+N_{k-1}^2+N_i+\cdots+N_{k-1}}}{(x)_{n_1}...(x)_{n_{k-1}}}=\prod_{r\neq 0,\pm i \pmod {2k+1}}\frac{1}{1-x^r} \)
여기서 \(j\leq k-1\)이면 \(N_j=n_j+\cdots+n_{k-1}\) , \(j=k\)이면 \(N_j=0\)
\(\sum_{n_1\geq\cdots\geq n_{k-1}\geq0}\frac{q^{n_1^2+\cdots+n_{k-1}^2+n_i+\cdots+n_{k-1}}}{(q)_{n_{1}-n_{2}}\cdots (q)_{n_{k-2}-n_{k-1}}(q)_{n_{k-1}}}=\prod_{n\neq 0,\pm i\pmod {2k+1}}(1-q^n)^{-1}\)
k=2인 경우 : 로저스-라마누잔 항등식==
- k=2인 경우, 로저스-라마누잔 연분수와 항등식을 얻는다
- i=1인 경우
\(H(q) =\sum_{n=0}^\infty \frac {q^{n^2+n}} {(q;q)_n} =
\frac {1}{(q^2;q^5)_\infty (q^3; q^5)_\infty}
=1+q^2 +q^3 +q^4+q^5 +2q^6+\cdots\)
- i=2인 경우
\(G(q) = \sum_{n=0}^\infty \frac {q^{n^2}} {(q;q)_n} =
\frac {1}{(q;q^5)_\infty (q^4; q^5)_\infty}
=1+ q +q^2 +q^3 +2q^4+2q^5 +3q^6+\cdots\)
\(H(q) =\sum_{n=0}^\infty \frac {q^{n^2+n}} {(q;q)_n} = \frac {1}{(q^2;q^5)_\infty (q^3; q^5)_\infty} =1+q^2 +q^3 +q^4+q^5 +2q^6+\cdots\)
\(G(q) = \sum_{n=0}^\infty \frac {q^{n^2}} {(q;q)_n} = \frac {1}{(q;q^5)_\infty (q^4; q^5)_\infty} =1+ q +q^2 +q^3 +2q^4+2q^5 +3q^6+\cdots\)
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
\(\sum_{n_1,n_{2}\geq0}\frac{q^{n_{1}^2+2n_1n_2+2n_{2}^{2}}}{(q)_{n_1}(q)_{n_{2}}}=\prod_{r\neq 0,\pm 3 \pmod {7}}\frac{1}{1-q^r}=\frac{(q^3;q^7)_\infty (q^4; q^7)_\infty(q^7;q^7)_\infty}{(q)_\infty}\)
\(\sum_{n_1,n_{2}\geq0}\frac{q^{n_{1}^2+2n_1n_2+2n_{2}^{2}+n_1+2n_2}}{(q)_{n_1}(q)_{n_{2}}}=\prod_{r\neq 0,\pm 1 \pmod {7}}\frac{1}{1-q^r}=\frac{(q;q^7)_\infty (q^6; q^7)_\infty(q^7;q^7)_\infty}{(q)_\infty}\)
\(\sum_{n_1,n_{2}\geq0}\frac{q^{n_{1}^2+2n_1n_2+2n_{2}^{2}+n_2}}{(q)_{n_1}(q)_{n_{2}}}=\prod_{r\neq 0,\pm 2 \pmod {7}}\frac{1}{1-q^r}=\frac{(q^2;q^7)_\infty (q^5; q^7)_\infty(q^7;q^7)_\infty}{(q)_\infty}\)
얻어지는 이차형식== \(n_{1}^{2}\) \((n_{1}+n_{2})^{2}+n_{2}^{2}\) \((n_{1}+n_{2}+n_{3})^{2}+(n_{2}+n_{3})^{2}+n_{3}^{2}\) \((n_{1}+n_{2}+n_{3}+n_{4})^{2}+(n_{2}+n_{3}+n_{4})^{2}+(n_{3}+n_{4})^{2}+n_{4}^{2}\) 행렬은 \(\text{A=}\left( \begin{array}{ccccc} 2 & 2 & 2 & 2 & 2 \\ 2 & 4 & 4 & 4 & 4 \\ 2 & 4 & 6 & 6 & 6 \\ 2 & 4 & 6 & 8 & 8 \\ 2 & 4 & 6 & 8 & 10 \end{array} \right)\)
역사==
메모==
관련된 항목들==
수학용어번역==
사전 형태의 자료==
관련논문==
- The Rogers–Selberg recursions, the Gordon–Andrews identities and intertwining operators
- Stefano Capparelli, James Lepowsky, Antun Milas, 2004
- Some formulas related to dilogarithms, the zeta function and the Andrews–Gordon identities
- B. Richmond and G. Szekeres, 1981
- A general theory of identities of the Rogers-Ramanujan type
- George E. Andrews, Bull. Amer. Math. Soc. Volume 80, Number 6 (1974), 1033-1052.
- On the General Rogers-Ramanujan Theorem.
- Andrews, G. E. Providence, RI: Amer. Math. Soc., 1974.
- An Analytic Generalization of the Rogers-Ramanujan Identities for Odd Moduli
- George E. Andrews, PNAS October 1, 1974 vol. 71 no. 10 4082-4085
- A Combinatorial Generalization of the Rogers-Ramanujan Identities
- Gordon, B. Amer. J. Math. 83, 393-399, 1961.
- Stefano Capparelli, James Lepowsky, Antun Milas, 2004
- B. Richmond and G. Szekeres, 1981
- George E. Andrews, Bull. Amer. Math. Soc. Volume 80, Number 6 (1974), 1033-1052.
- Andrews, G. E. Providence, RI: Amer. Math. Soc., 1974.
- George E. Andrews, PNAS October 1, 1974 vol. 71 no. 10 4082-4085
- Gordon, B. Amer. J. Math. 83, 393-399, 1961.