Selberg integral

Selberg's integral formula

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

& = \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}

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, : \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 := 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 .

Mehta's integral

When Re(\gamma) -1/n, : \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 It is a corollary of Selberg, by setting \alpha = \beta, and change of variables with t_i = \frac{1+t'_i/\sqrt{2\alpha}}{2}, then taking \alpha \to \infty.

This was conjectured by , 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.

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

Macdonald's integral

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. :\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. 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.

References

References

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2. (2008). "The importance of the Selberg integral". Bull. Amer. Math. Soc..
3. Aomoto, K. (1987). "On the complex Selberg integral". The Quarterly Journal of Mathematics.
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7. Opdam, E.M.. (1989). "Some applications of hypergeometric shift operators". Invent. Math..
8. Opdam, E.M.. (1993). "Dunkl operators, Bessel functions and the discriminant of a finite Coxeter group". Compositio Mathematica.