Selberg integral

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

Selberg's integral formula[edit]

Selberg's formula implies Dixon's identity for well poised hypergeometric series, and some special cases of Dyson's conjecture.

Aomoto's integral formula[edit]

Aomoto (1987) proved a slightly more general integral formula:

Mehta's integral[edit]

Mehta's integral is

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

This was conjectured by Mehta & Dyson (1963), who were unaware of Selberg's earlier work.

Macdonald's integral[edit]

Macdonald (1982) conjectured the following extension of Mehta's integral to all finite root systems, Mehta's original case corresponding to the An−1 root system.

The product is over the roots r of the roots system and the numbers dj 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[edit]

  • 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, 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), 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