Selberg integral

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, we have
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,
A proof is found in Chapter 8 of.

Mehta's integral

When,
It is a corollary of Selberg, by setting, and change of variables with, then taking.
This was conjectured by, who were unaware of Selberg's earlier work.
It is the partition function for a gas of point charges moving on a line that are attracted to the origin.
In particular, when, the term on the right is.

Macdonald's integral

conjectured the following extension of Mehta's integral to all finite root systems, Mehta's original case corresponding to the A_n−1 root system.
The product is over the roots r of the roots system and the numbers d_j are the degrees of the generators of the ring of invariants of the reflection group.
gave a uniform proof for all crystallographic reflection groups. Several years later he proved it in full generality, making use of computer-aided calculations by Garvan.