Selberg integral

When ${\displaystyle Re(\alpha )>0,Re(\beta )>0,Re(\gamma )>-\min \left({\frac {1}{n}},{\frac {Re(\alpha )}{n-1}},{\frac {Re(\beta )}{n-1}}\right)}$, we have

${\displaystyle {\begin{aligned}S_{n}(\alpha ,\beta ,\gamma )&=\int _{0}^{1}\cdots \int _{0}^{1}\prod _{i=1}^{n}t_{i}^{\alpha -1}(1-t_{i})^{\beta -1}\prod _{1\leq i<j\leq n}|t_{i}-t_{j}|^{2\gamma }\,dt_{1}\cdots dt_{n}\\&=\prod _{j=0}^{n-1}{\frac {\Gamma (\alpha +j\gamma )\Gamma (\beta +j\gamma )\Gamma (1+(j+1)\gamma )}{\Gamma (\alpha +\beta +(n+j-1)\gamma )\Gamma (1+\gamma )}}\end{aligned}}}$

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,

${\displaystyle \int _{0}^{1}\cdots \int _{0}^{1}\left(\prod _{i=1}^{k}t_{i}\right)\prod _{i=1}^{n}t_{i}^{\alpha -1}(1-t_{i})^{\beta -1}\prod _{1\leq i<j\leq n}|t_{i}-t_{j}|^{2\gamma }\,dt_{1}\cdots dt_{n}}$

${\displaystyle =S_{n}(\alpha ,\beta ,\gamma )\prod _{j=1}^{k}{\frac {\alpha +(n-j)\gamma }{\alpha +\beta +(2n-j-1)\gamma }}.}$

A proof is found in Chapter 8 of Andrews, Askey & Roy (1999).^[4]

When ${\displaystyle Re(\gamma )>-1/n}$,

${\displaystyle {\frac {1}{(2\pi )^{n/2}}}\int _{-\infty }^{\infty }\cdots \int _{-\infty }^{\infty }\prod _{i=1}^{n}e^{-t_{i}^{2}/2}\prod _{1\leq i<j\leq n}|t_{i}-t_{j}|^{2\gamma }\,dt_{1}\cdots dt_{n}=\prod _{j=1}^{n}{\frac {\Gamma (1+j\gamma )}{\Gamma (1+\gamma )}}.}$

It is a corollary of Selberg, by setting ${\displaystyle \alpha =\beta }$, and change of variables with ${\displaystyle t_{i}={\frac {1+t'_{i}/{\sqrt {2\alpha }}}{2}}}$, then taking ${\displaystyle \alpha \to \infty }$.

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]

In particular, when ${\displaystyle \gamma =1}$, the term on the right is ${\displaystyle \prod _{j=1}^{n}j!}$.

Macdonald (1982) 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.^[7]

${\displaystyle {\frac {1}{(2\pi )^{n/2}}}\int \cdots \int \left|\prod _{r}{\frac {2(x,r)}{(r,r)}}\right|^{\gamma }e^{-(x_{1}^{2}+\cdots +x_{n}^{2})/2}dx_{1}\cdots dx_{n}=\prod _{j=1}^{n}{\frac {\Gamma (1+d_{j}\gamma )}{\Gamma (1+\gamma )}}}$

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. 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]