맥도날드-메타 적분

개요

• \(G\)는 \(\mathbb{R}^n\)에 작용하는 유한반사군 (콕세터군)
• \(R\)은 루트시스템, \(R_+\)는 양의 루트
• \((\cdot,\cdot)\)는 \(\mathbb{R}^n\)에 정의된 \((\alpha,\alpha)=2,\,\alpha\in R\)을 만족하는 내적
• 다음이 성립한다

\[ \int_{\Bbb R''}\prod_{\alpha \in R_+} |(\alpha,x)|^{2 k}\, d\varphi(x)=\prod_{j=1}^n\frac{\Gamma(1+k d_j)}{\Gamma(1+k)}, \] 여기서 \(d_i\)는 콕세터군의 차수이고 \(\varphi\)는 \(\mathbb{R}^n\)에 정의된 가우스 측도 \[ d\varphi(x):=\frac{e^{-|x|^2/2}}{(2\pi)^{n/2}}\, d x_1\cdots d x_n \]

관련된 항목들

• 셀베르그 적분(Selberg integral)

매스매티카 파일 및 계산리소스

• https://docs.google.com/file/d/0B8XXo8Tve1cxSi1QSGF0bGdTaU0/edit
• http://mathpro.blogspot.com.au/2008/01/selberg-integral.html

리뷰, 에세이, 강의노트

• P. Etingof, The Macdonald-Mehta integral
• S. Ole Warnaar, The Selberg Integral, 2011
• S. Ole Warnaar, Beta Integrals

관련논문

• Brent, Richard P., Christian Krattenthaler, and S. Ole Warnaar. “Discrete Analogues of Macdonald-Mehta Integrals.” arXiv:1601.06536 [math-Ph], January 25, 2016. http://arxiv.org/abs/1601.06536.
• Cacciatori, S. L., F. Dalla Piazza, and A. Scotti. “Compact Lie Groups: Euler Constructions and Generalized Dyson Conjecture.” arXiv:1207.1262 [hep-Th, Physics:math-Ph], July 5, 2012. http://arxiv.org/abs/1207.1262.
• Brent, Richard P., Hideyuki Ohtsuka, Judy-anne H. Osborn, and Helmut Prodinger. “Some Binomial Sums Involving Absolute Values.” arXiv:1411.1477 [math, Stat], November 5, 2014. http://arxiv.org/abs/1411.1477.
• Opdam, E. M. “Dunkl Operators, Bessel Functions and the Discriminant of a Finite Coxeter Group.” Compositio Mathematica 85, no. 3 (1993): 333–73.
• Opdam, E. M. “Some Applications of Hypergeometric Shift Operators.” Inventiones Mathematicae 98, no. 1 (February 1989): 1–18. doi:10.1007/BF01388841.
• Garvan, Frank G. “Some Macdonald-Mehta Integrals by Brute Force.” In \(q\)-Series and Partitions (Minneapolis, MN, 1988), 18:77–98. IMA Vol. Math. Appl. Springer, New York, 1989. http://www.ams.org/mathscinet-getitem?mr=1019845.