In mathematics, the Selberg integral is a generalization of Euler beta function to n dimensions introduced by Atle Selberg.
, 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 proved a slightly more general integral formula.
[3] With the same conditions as Selberg's formula, A proof is found in Chapter 8 of Andrews, Askey & Roy (1999).
, It is a corollary of Selberg, by setting
This was conjectured by Mehta & Dyson (1963), who were unaware of Selberg's earlier work.
[5] It is the partition function for a gas of point charges moving on a line that are attracted to the origin.
Macdonald (1982) conjectured the following extension of Mehta's integral to all finite root systems, Mehta's original case corresponding to the An−1 root system.
[7] 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.
Opdam (1989) gave a uniform proof for all crystallographic reflection groups.
[8] Several years later he proved it in full generality, making use of computer-aided calculations by Garvan.