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.[3] With the same conditions as Selberg's formula,
It is a corollary of Selberg, by setting , and change of variables with , then taking .
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.[6]
Macdonald's integral
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.[9]
^Andrews, George; Askey, Richard; Roy, Ranjan (1999). "The Selberg integral and its applications". Special functions. Encyclopedia of Mathematics and its Applications. Vol. 71.
Cambridge University Press.
ISBN978-0-521-62321-6.
MR1688958.
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.[3] With the same conditions as Selberg's formula,
It is a corollary of Selberg, by setting , and change of variables with , then taking .
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.[6]
Macdonald's integral
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.[9]
^Andrews, George; Askey, Richard; Roy, Ranjan (1999). "The Selberg integral and its applications". Special functions. Encyclopedia of Mathematics and its Applications. Vol. 71.
Cambridge University Press.
ISBN978-0-521-62321-6.
MR1688958.