Selberg integral explained

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

Re(\alpha) > 0 , Re(\beta) > 0 , Re(\gamma) > -min \left( (1 n)/(,) (Re(\alpha))/(n -1) , (Re(\beta))/(n -1) \right) , we have\begin{align} S _{n} (\alpha, \beta, \gamma) & = \int ^{1} _{0} … \int ^{1} _{0} \prod ^{n} _{i =1} t ^{\alpha -1} _{i} ( 1- t ^{\beta -1} _{i)} \prod _{1} |t _{i} - t _{j} | ^{2} dt _{1} … dt _{n} \\ & = \prod ^{n -1} _{j =0} (\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.^[3] With the same conditions as Selberg's formula,

\int ^{1} _{0} … \int ^{1} _{0} \left(\prod ^{k} _{i =1} t _{i\right)\prod} ^{n} ^{i =1} t ^{\alpha -1} _{i} ( 1- t ^{\beta -1} _{i)} \prod _{1} |t _{i} - t _{j} | ^{2} dt _{1} … dt _{n} = S _{n(\alpha,\beta,\gamma)} \prod ^{k (\alpha + (n - j)\gamma)/(\alpha + \beta + ( 2 n - j -1 )\gamma)} _{j =1} . A proof is found in Chapter 8 of .^[4]

Mehta's integral

When

Re(\gamma) > -1 /n , (1)/(( 2 \pi) ^{n/2}) \int ^{inf ty} _{-inf ty} … \int ^{inf ty} _{-inf ty} \prod ^{n} _{i =1} e ^{- t ^{2 / 2} _{i}} \prod _{1} |t _{i} - t _{j} | ^{2} dt _{1} … dt _{n} = \prod ^{n (\Gamma( 1+ j\gamma))/(\Gamma( 1+ \gamma))} _{j =1} . It is a corollary of Selberg, by setting \alpha = \beta , and change of variables with t _{i} = (1+ t' _{i/\sqrt{2\alpha})/() }, then taking \alpha \to inf ty . This was conjectured by, 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

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]

(1)/(( 2 \pi) ^{n/2}) \int … \int \left|\prod _{r (2 (x,r))/((r,r))} \right| ^{\gamma} e ^{- (x ^{2 )/ 2} _{n}} dx _{1 …} dx _{n} = \prod ^{n (\Gamma( 1+ d _{j\gamma)})/(\Gamma( 1+ \gamma))} _{j =1} 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.^[8] Several years later he proved it in full generality, making use of computer-aided calculations by Garvan.^[9]

Notes and References

1. Atle. Selberg. Remarks on a multiple integral. Norsk Mat. Tidsskr.. 1944. 26. 71–78. 0018287. subscription.
2. The importance of the Selberg integral. Peter J.. Forrester. S. Ole . Warnaar. Bull. Amer. Math. Soc. . 45 . 2008. 489–534. 10.1090/S0273-0979-08-01221-4. 4 . 0710.3981. 14185100.
3. Aomoto. K. On the complex Selberg integral. The Quarterly Journal of Mathematics. 1987. 38. 4. 385–399. 10.1093/qmath/38.4.385. limited.
4. Book: Andrews. George. Askey. Richard. Roy. Ranjan. Special functions. Encyclopedia of Mathematics and its Applications. Cambridge University Press. 71. 1999. 978-0-521-62321-6. 1688958. The Selberg integral and its applications.
5. Mehta. Madan Lal. Dyson. Freeman J.. Statistical theory of the energy levels of complex systems. V. Journal of Mathematical Physics. 1963. 4. 5. 713–719. limited. 10.1063/1.1704009. 0151232. 1963JMP.....4..713M.
6. Book: Mehta . Madan Lal . Random matrices . Elsevier/Academic Press, Amsterdam . 3rd . Pure and Applied Mathematics (Amsterdam) . 978-0-12-088409-4 . 2129906 . 2004 . 142.
7. Macdonald . I. G. . Some conjectures for root systems . 10.1137/0513070 . 674768 . 1982 . SIAM Journal on Mathematical Analysis . 0036-1410 . 13 . 6 . 988–1007.
8. Opdam. E.M.. 1989. Some applications of hypergeometric shift operators. 1. Invent. Math.. 98. 10.1007/BF01388841. 1010152. 275 - 282. 1989InMat..98....1O. 54571505 .
9. Opdam. E.M.. 1993. Dunkl operators, Bessel functions and the discriminant of a finite Coxeter group. Compositio Mathematica . 85. 3. 333 - 373. 0778.33009. 1214452.