Selberg integral

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

Selberg's integral formula

{\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.

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\leq i<j\leq n}}|t_{i}-t_{j}|^{{2\gamma }}\,dt_{1}\cdots dt_{n}

=S_{n}(\alpha ,\beta ,\gamma )\prod _{{j=1}}^{k}{\frac {\alpha +(n-j)\gamma }{\alpha +\beta +(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\leq i<j\leq 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+j\gamma )}{\Gamma (1+\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 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. 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

Andrews, George E.; Askey, Richard; Roy, Ranjan (1999), Special functions, Encyclopedia of Mathematics and its Applications, 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, 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 674768
Mehta, Madan Lal (2004), Random matrices, Pure and Applied Mathematics (Amsterdam), 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, 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, 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

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