Selberg integral

<h2 id="selbergs-integral-formula">Selberg's integral formula</h2>
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 <a href="/facts/Dixon%2527s_identity/KpPCNErF">Dixon's identity</a> for well poised hypergeometric series, and some special cases of <a href="/facts/Dyson%2527s_conjecture/wHkzGBBk">Dyson's conjecture</a>. This is a corollary of Aomoto.

<h2 id="aomotos-integral-formula">Aomoto's integral formula</h2>
Aomoto proved a slightly more general integral formula.<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a> 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).<a class="footnote-ref" id="fnref:4" href="#fn:4">4</a>

<h2 id="mehtas-integral">Mehta's integral</h2>
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.<a class="footnote-ref" id="fnref:5" href="#fn:5">5</a>
It is the partition function for a gas of point charges moving on a line that are attracted to the origin.<a class="footnote-ref" id="fnref:6" href="#fn:6">6</a>
In particular, when 
 
 
 
 γ
 =
 1
 
 
 {\displaystyle \gamma =1}
 
, the term on the right is 
 
 
 
 
 ∏
 
 j
 =
 1
 
 
 n
 
 
 j
 !
 
 
 {\displaystyle \prod _{j=1}^{n}j!}
 
.

<h2 id="macdonalds-integral">Macdonald's integral</h2>
Macdonald (1982) conjectured the following extension of Mehta's integral to all <a href="/facts/Root_system/TFaEdwMU">finite root systems</a>, Mehta's original case corresponding to the An−1 root system.<a class="footnote-ref" id="fnref:7" href="#fn:7">7</a>

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.<a class="footnote-ref" id="fnref:8" href="#fn:8">8</a> Several years later he proved it in full generality, making use of computer-aided calculations by Garvan.<a class="footnote-ref" id="fnref:9" href="#fn:9">9</a>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Selberg, Atle (1944). "Remarks on a multiple integral". Norsk Mat. Tidsskr. 26: 71–78. MR 0018287. <a href="https://cds.cern.ch/record/411367" target="_blank">https://cds.cern.ch/record/411367</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn: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. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn: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. <a href="https://academic.oup.com/qjmath/article-abstract/38/4/385/1530985" target="_blank">https://academic.oup.com/qjmath/article-abstract/38/4/385/1530985</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn: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. <a href="978-0-521-62321-6" target="_blank">978-0-521-62321-6</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn: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. <a href="https://pubs.aip.org/aip/jmp/article-abstract/4/5/713/230167/Statistical-Theory-of-the-Energy-Levels-of-Complex" target="_blank">https://pubs.aip.org/aip/jmp/article-abstract/4/5/713/230167/Statistical-Theory-of-the-Energy-Levels-of-Complex</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn: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. <a href="978-0-12-088409-4" target="_blank">978-0-12-088409-4</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn: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. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
<li id="fn: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. <a href="http://dare.uva.nl/personal/pure/en/publications/some-applications-of-hypergeometric-shift-operators(4d1bc98d-e707-47eb-aaec-164e5488d0bc).html" target="_blank">http://dare.uva.nl/personal/pure/en/publications/some-applications-of-hypergeometric-shift-operators(4d1bc98d-e707-47eb-aaec-164e5488d0bc).html</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></li>
<li id="fn: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. <a href="http://www.numdam.org/item?id=CM_1993__85_3_333_0" target="_blank">http://www.numdam.org/item?id=CM_1993__85_3_333_0</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></li>
</ol>

Selberg integral open-in-new

Selberg integral