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Selberg's integral formula

When R e ( α ) > 0 , R e ( β ) > 0 , R e ( γ ) > − min ( 1 n , R e ( α ) n − 1 , R e ( β ) n − 1 ) {\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

S n ( α , β , γ ) = ∫ 0 1 ⋯ ∫ 0 1 ∏ i = 1 n t i α − 1 ( 1 − t i ) β − 1 ∏ 1 ≤ i < j ≤ n | t i − t j | 2 γ d t 1 ⋯ d t n = ∏ j = 0 n − 1 Γ ( α + j γ ) Γ ( β + j γ ) Γ ( 1 + ( j + 1 ) γ ) Γ ( α + β + ( n + j − 1 ) γ ) Γ ( 1 + γ ) {\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,

∫ 0 1 ⋯ ∫ 0 1 ( ∏ i = 1 k t i ) ∏ i = 1 n t i α − 1 ( 1 − t i ) β − 1 ∏ 1 ≤ i < j ≤ n | t i − t j | 2 γ d t 1 ⋯ d t n {\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}} = S n ( α , β , γ ) ∏ j = 1 k α + ( n − j ) γ α + β + ( 2 n − j − 1 ) γ . {\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 R e ( γ ) > − 1 / n {\displaystyle Re(\gamma )>-1/n} ,

1 ( 2 π ) n / 2 ∫ − ∞ ∞ ⋯ ∫ − ∞ ∞ ∏ i = 1 n e − t i 2 / 2 ∏ 1 ≤ i < j ≤ n | t i − t j | 2 γ d t 1 ⋯ d t n = ∏ j = 1 n Γ ( 1 + j γ ) Γ ( 1 + γ ) . {\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 α = β {\displaystyle \alpha =\beta } , and change of variables with t i = 1 + t i ′ / 2 α 2 {\displaystyle t_{i}={\frac {1+t'_{i}/{\sqrt {2\alpha }}}{2}}} , then taking α → ∞ {\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

In particular, when γ = 1 {\displaystyle \gamma =1} , the term on the right is ∏ j = 1 n j ! {\displaystyle \prod _{j=1}^{n}j!} .

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 An−1 root system.⁷

1 ( 2 π ) n / 2 ∫ ⋯ ∫ | ∏ r 2 ( x , r ) ( r , r ) | γ e − ( x 1 2 + ⋯ + x n 2 ) / 2 d x 1 ⋯ d x n = ∏ j = 1 n Γ ( 1 + d j γ ) Γ ( 1 + γ ) {\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 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.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. https://cds.cern.ch/record/411367 ↩

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. /wiki/ArXiv_(identifier) ↩

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. https://academic.oup.com/qjmath/article-abstract/38/4/385/1530985 ↩

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. 978-0-521-62321-6 ↩

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. https://pubs.aip.org/aip/jmp/article-abstract/4/5/713/230167/Statistical-Theory-of-the-Energy-Levels-of-Complex ↩

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. 978-0-12-088409-4 ↩

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. /wiki/Doi_(identifier) ↩

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. http://dare.uva.nl/personal/pure/en/publications/some-applications-of-hypergeometric-shift-operators(4d1bc98d-e707-47eb-aaec-164e5488d0bc).html ↩

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. http://www.numdam.org/item?id=CM_1993__85_3_333_0 ↩