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 is a positive integer, , , . Notice that the discriminant is the value of the standard Vandermonde determinant.