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

, 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 Chapter8 of .

Mehta's integral

,

.

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.

.

Macdonald's integral

conjectured the following extension of Mehta's integral to all finite root systems, Mehta's original case corresponding to the An1 root system.

The product is over the roots r of the roots system and the numbers dj 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.

References

Further reading