Selberg integral

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

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Selberg's integral formula

\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 < j \le n} |t_i - t_j |^{2 \gamma}\,dt_1 \cdots dt_n = \\ & = \prod_{j = 0}^{n-1} \frac {\Gamma(\alpha %2B j \gamma) \Gamma(\beta %2B j \gamma) \Gamma (1 %2B (j%2B1)\gamma)} {\Gamma(\alpha %2B \beta %2B (n%2Bj-1)\gamma) \Gamma(1%2B\gamma)} \end{align}

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

Aomoto's integral formula

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

\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 < j \le n} |t_i - t_j |^{2 \gamma}\,dt_1 \cdots dt_n
= S_n(\alpha,\beta,\gamma) \prod_{j=1}^k\frac{\alpha%2B(n-j)\gamma}{\alpha%2B\beta%2B(2n-j-1)\gamma}.

Mehta's integral

Mehta's integral is

\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 < j \le n} |t_i - t_j |^{2 \gamma}\,dt_1 \cdots dt_n.

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

\prod_{j=1}^n\frac{\Gamma(1%2Bj\gamma)}{\Gamma(1%2B\gamma)}.

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

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.

\frac{1}{(2\pi)^{n/2}}\int\cdots\int \left|\prod_r\frac{2(x,r)}{(r,r)}\right|^{\gamma}e^{-(x_1^2%2B\cdots%2Bx_n^2)/2}dx_1\cdots dx_n =\prod_{j=1}^n\frac{\Gamma(1%2Bd_j\gamma)}{\Gamma(1%2B\gamma)}

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.

References