Selberg integral

Short description: Mathematical function

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

Selberg's integral formula

When [math]\displaystyle{ Re(\alpha) \gt 0, Re(\beta) \gt 0, Re(\gamma) \gt -\min \left(\frac 1n , \frac{Re(\alpha)}{n-1}, \frac{Re(\beta)}{n-1}\right) }[/math], we have

[math]\displaystyle{ \begin{align} 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 \le i \lt j \le 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{align} }[/math]

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,

[math]\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 \le i \lt j \le n} |t_i - t_j |^{2 \gamma}\,dt_1 \cdots dt_n }[/math]

[math]\displaystyle{ = S_n(\alpha,\beta,\gamma) \prod_{j=1}^k\frac{\alpha+(n-j)\gamma}{\alpha+\beta+(2n-j-1)\gamma}. }[/math]

A proof is found in Chapter 8 of (Andrews Askey).

Mehta's integral

When [math]\displaystyle{ Re(\gamma) \gt -1/n }[/math],

[math]\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 \le i \lt j \le n} |t_i - t_j |^{2 \gamma}\,dt_1 \cdots dt_n = \prod_{j=1}^n\frac{\Gamma(1+j\gamma)}{\Gamma(1+\gamma)}. }[/math]

It is a corollary of Selberg, by setting [math]\displaystyle{ \alpha = \beta }[/math], and change of variables with [math]\displaystyle{ t_i = \frac{1+t'_i/\sqrt{2\alpha}}{2} }[/math], then taking [math]\displaystyle{ \alpha \to \infty }[/math].

This was conjectured by (Mehta Dyson), 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

(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.

[math]\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)} }[/math]

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