In mathematics, the Selberg integral is a generalization of Euler beta function to n dimensions introduced by Atle Selberg. It has applications in statistical mechanics, multivariable orthogonal polynomials, random matrix theory, Calogero–Moser–Sutherland model, and Knizhnik–Zamolodchikov equations.
Selberg's integral formula When R e ( α ) > 0 , R e ( β ) > 0 , R e ( γ ) > − min ( 1 n , R e ( α ) n − 1 , R e ( β ) n − 1 ) {\displaystyle Re(\alpha )>0,Re(\beta )>0,Re(\gamma )>-\min \left({\frac {1}{n}},{\frac {Re(\alpha )}{n-1}},{\frac {Re(\beta )}{n-1}}\right)} , we have
S n ( α , β , γ ) = ∫ 0 1 ⋯ ∫ 0 1 ∏ i = 1 n t i α − 1 ( 1 − t i ) β − 1 ∏ 1 ≤ i < j ≤ n | t i − t j | 2 γ d t 1 ⋯ d t n = ∏ j = 0 n − 1 Γ ( α + j γ ) Γ ( β + j γ ) Γ ( 1 + ( j + 1 ) γ ) Γ ( α + β + ( n + j − 1 ) γ ) Γ ( 1 + γ ) {\displaystyle {\begin{aligned}S_{n}(\alpha ,\beta ,\gamma )&=\int _{0}^{1}\cdots \int _{0}^{1}\prod _{i=1}^{n}t_{i}^{\alpha -1}(1-t_{i})^{\beta -1}\prod _{1\leq i<j\leq n}|t_{i}-t_{j}|^{2\gamma }\,dt_{1}\cdots dt_{n}\\&=\prod _{j=0}^{n-1}{\frac {\Gamma (\alpha +j\gamma )\Gamma (\beta +j\gamma )\Gamma (1+(j+1)\gamma )}{\Gamma (\alpha +\beta +(n+j-1)\gamma )\Gamma (1+\gamma )}}\end{aligned}}}
Selberg's formula implies Dixon's identity for well poised hypergeometric series, and some special cases of Dyson's conjecture. This is a corollary of Aomoto.
Aomoto's integral formula Aomoto proved a slightly more general integral formula. With the same conditions as Selberg's formula,
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