타원 초기하 적분 (elliptic hypergeometric integrals)

개요

정리 (스피리도노프 Spiridonov).

복소수 $t_1, \dots ,t_6,p,q$가 $|t_1|, \dots , |t_6|,|p|,|q| <1$이고, $\prod_{i=1}^6 t_i=pq$을 만족한다고 하자. 다음이 성립한다. \begin{equation} \label{betaint} \frac{(p;p)_\infty (q;q)_\infty}{2} \int_{\mathbb{T}} \frac{\prod_{i=1}^6 \Gamma(t_i z ;p,q)\Gamma(t_i z^{-1} ;p,q)}{\Gamma(z^{2};p,q) \Gamma(z^{-2};p,q)} \frac{dz}{2 \pi i z} = \prod_{1 \leq i < j \leq 6} \Gamma(t_i t_j;p,q), \end{equation} 여기서 $\mathbb{T}$는 단위원 (양의 방향)

• $p \rightarrow 0$일 때, 나스랄라-라만 적분을 얻는다

\begin{equation} \frac{(q;q)_\infty}{2} \int_{\mathbb{T}}\frac{(z \prod_{i=1}^5 t_i;q)_\infty (z^{-1} \prod_{i=1}^5 t_i;q)_\infty (z^2;q)_\infty (z^{-2};q)_\infty}{\prod_{i=1}^5 (t_i z)_\infty (t_i z^{-1})_\infty} \frac{dz}{2\pi i z} \ = \ \frac{\prod_{j=1}^5 (\frac{t_1 t_2 t_3 t_4 t_5}{t_j};q)_\infty}{\prod_{1 \leq i < j \leq 5} (t_i t_j;q)_\infty} \end{equation}

리뷰, 에세이, 강의노트

• Gahramanov, Ilmar. “Mathematical Structures behind Supersymmetric Dualities.” arXiv:1505.05656 [hep-Th, Physics:math-Ph], May 21, 2015. http://arxiv.org/abs/1505.05656.
• van Diejen, J. F., and V. P. Spiridonov. “Elliptic Beta Integrals and Mudular Hypergeometric Sums: An Overview.” Rocky Mountain Journal of Mathematics 32, no. 2 (June 2002): 639–56. doi:10.1216/rmjm/1030539690.