Selberg integral

In mathematics, the Selberg integral is a generalization of Euler beta function to n dimensions introduced by Atle Selberg.[1][2]

Selberg's integral formula

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's integral formula

Aomoto proved a slightly more general integral formula.[3] 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 Andrews, Askey & Roy (1999).[4]

Mehta's integral

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.[5]

It is the partition function for a gas of point charges moving on a line that are attracted to the origin.[6]

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.[7]

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

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.[8] Several years later he proved it in full generality, making use of computer-aided calculations by Garvan.[9]

References

1. Selberg, Atle (1944). "Remarks on a multiple integral". Norsk Mat. Tidsskr. 26: 71–78. MR 0018287.
2. 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. S2CID 14185100.
3. Aomoto, K (1987). "On the complex Selberg integral". The Quarterly Journal of Mathematics. 38 (4): 385–399. doi:10.1093/qmath/38.4.385.
4. Andrews, George; Askey, Richard; Roy, Ranjan (1999). "The Selberg integral and its applications". Special functions. Encyclopedia of Mathematics and its Applications. Vol. 71. Cambridge University Press. ISBN 978-0-521-62321-6. MR 1688958.
5. 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. MR 0151232.
6. 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.
7. 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.
8. 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. S2CID 54571505.
9. 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.