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 (1987) proved a slightly more general integral formula. 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.[1]

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.

It is the partition function for a gas of point charges moving on a line that are attracted to the origin (Mehta 2004).

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.

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

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. Several years later he proved it in full generality (Opdam (1993)), making use of computer-aided calculations by Garvan.

• Andrews, George E.; Askey, Richard; Roy, Ranjan (1999), Special functions, Encyclopedia of Mathematics and its Applications, vol. 71, Cambridge University Press, ISBN 978-0-521-62321-6, MR 1688958 (Chapter 8)
• Aomoto, K (1987), "On the complex Selberg integral", The Quarterly Journal of Mathematics, 38 (4): 385–399, doi:10.1093/qmath/38.4.385
• Forrester, Peter J.; Warnaar, S. Ole (2008), "The importance of the Selberg integral", Bull. Amer. Math. Soc., 45 (4): 489–534, arXiv:0710.3981, doi:10.1090/S0273-0979-08-01221-4
• Macdonald, I. G. (1982), "Some conjectures for root systems", SIAM Journal on Mathematical Analysis, 13 (6): 988–1007, doi:10.1137/0513070, ISSN 0036-1410, MR 0674768
• Mehta, Madan Lal (2004), Random matrices, Pure and Applied Mathematics (Amsterdam), vol. 142 (3rd ed.), Elsevier/Academic Press, Amsterdam, ISBN 978-0-12-088409-4, MR 2129906
• Mehta, Madan Lal; Dyson, Freeman J. (1963), "Statistical theory of the energy levels of complex systems. V", Journal of Mathematical Physics, 4 (5): 713–719, Bibcode:1963JMP.....4..713M, doi:10.1063/1.1704009, ISSN 0022-2488, MR 0151232
• Opdam, E.M. (1989), "Some applications of hypergeometric shift operators", Invent. Math., 98 (1): 275–282, Bibcode:1989InMat..98....1O, doi:10.1007/BF01388841, MR 1010152
• Opdam, E.M. (1993), "Dunkl operators, Bessel functions and the discriminant of a finite Coxeter group", Compositio Mathematica, 85 (3): 333–373, MR 1214452, Zbl 0778.33009
• Selberg, Atle (1944), "Remarks on a multiple integral", Norsk Mat. Tidsskr., 26: 71–78, MR 0018287