Selberg integral

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In mathematics the Selberg integral is a generalization of Euler beta function to n dimensions introduced by Atle Selberg (1944).

Selberg's integral formula[edit]

Selberg's formula implies Dixon's identity for well poised hypergeometric series, and some special cases of Dyson's conjecture.

Aomoto's integral formula[edit]

Aomoto (1987) proved a slightly more general integral formula:

Mehta's integral[edit]

Mehta's integral is

It is the partition function for a gas of point charges moving on a line that are attracted to the origin (Mehta 2004). Its value can be deduced from that of the Selberg integral, and is

This was conjectured by Mehta & Dyson (1963), who were unaware of Selberg's earlier work.

Macdonald's integral[edit]

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.

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. Several years later he proved it in full generality (Opdam (1993)), making use of computer-aided calculations by Garvan.