The Selberg integral is a higher-dimensional generalization of the integral representation of Euler’s beta function due Alte Selberg. Many further “Selberg-type” generalizations appear in the study of multidimensional generalizations of hypergeometric functions, arrangements of hyperplanes, Knizhnik-Zamolodchikov equation, representation theory of quantum and affine Lie algebras and of vertex operator algebras, random matrix theory etc. There is also an ellitic generalization, see elliptic Selberg integral.
where $p\gt 0$ is a positive integer, $Re x\gt 0$, $Re y\gt 0$ , $Re z \gt max\{-p^{-1},- Re x/(p-1), - Re y/(p-1)\}$. Notice that the discriminant $\prod_{i\lt j} (u_j-u_i)$ is the value of the standard Vandermonde determinant.
J.-G. Luque, J.-Y. Thibon, Hankel hyperdeterminants and Selberg integrals, J. Phys. A36 (2003), 5267–5292, MR1985318 (2004d:15011)
K. Aomoto, Jacobi polynomials associated with Selberg integrals, SIAM J. Math. Anal. 18 (1987) 545–549; On the complex Selberg integral, Q. J. Math. Oxford 38 (1987) 385–399.
R. S. Askey, Some basic hypergeometric extensions of integrals of Selberg and Andrews, SIAM J. Math. 11 (1980) 938–951.
Revised on November 7, 2012 22:54:56
by Urs Schreiber
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