Selberg integral

When $Re(\backslash alpha)\; >\; 0,\; Re(\backslash beta)\; >\; 0,\; Re(\backslash gamma)\; >\; -\backslash min\; \backslash left(\backslash frac\; 1n\; ,\; \backslash frac\{Re(\backslash alpha)\}\{n-1\},\; \backslash frac\{Re(\backslash beta)\}\{n-1\}\backslash right)$, we have

:$\backslash begin\{align\}$

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 \le i < j \le 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{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 proved a slightly more general integral formula.{{cite journal|last=Aomoto|first=K|title=On the complex Selberg integral|journal=The Quarterly Journal of Mathematics|year=1987|volume=38|issue=4|pages=385–399|doi=10.1093/qmath/38.4.385|url=https://academic.oup.com/qjmath/article-abstract/38/4/385/1530985|url-access=limited}} With the same conditions as Selberg's formula,

:$$

\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 \le i < j \le n} |t_i - t_j |^{2 \gamma}\,dt_1 \cdots dt_n

:$=$

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 {{harvtxt|Andrews|Askey|Roy|1999}}.{{cite book|last1=Andrews|first1=George|last2=Askey|first2=Richard|last3=Roy|first3=Ranjan|title=Special functions|series=Encyclopedia of Mathematics and its Applications|publisher=Cambridge University Press|volume=71|year=1999|isbn=978-0-521-62321-6|mr=1688958|chapter=The Selberg integral and its applications}}

When $Re(\backslash gamma)\; >\; -1/n$,

:$$

\frac{1}{(2\pi)^{n/2}}\int_{-\infty}^{\infty} \cdots \int_{-\infty}^{\infty} \prod_{i=1}^n e^{-t_i^2/2}

\prod_{1 \le i < j \le 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 $\backslash alpha\; =\; \backslash beta$, and change of variables with $t\_i\; =\; \backslash frac\{1+t\text{'}\_i/\backslash sqrt\{2\backslash alpha\}\}\{2\}$, then taking $\backslash alpha\; \backslash to\; \backslash infty$.

This was conjectured by {{harvtxt|Mehta|Dyson|1963}}, who were unaware of Selberg's earlier work.{{cite journal|last1=Mehta|first1=Madan Lal|last2=Dyson|first2=Freeman J.|title=Statistical theory of the energy levels of complex systems. V|journal=Journal of Mathematical Physics|year=1963|volume=4|issue=5|pages=713–719|url=https://pubs.aip.org/aip/jmp/article-abstract/4/5/713/230167/Statistical-Theory-of-the-Energy-Levels-of-Complex|url-access=limited|doi=10.1063/1.1704009|mr=0151232|bibcode=1963JMP.....4..713M}}

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

{{cite book | last1=Mehta | first1=Madan Lal | title=Random matrices | publisher=Elsevier/Academic Press, Amsterdam | edition=3rd | series=Pure and Applied Mathematics (Amsterdam) | isbn=978-0-12-088409-4 | mr=2129906 | year=2004 | volume=142}}

In particular, when $\backslash gamma\; =\; 1$, the term on the right is $\backslash prod\_\{j=1\}^n\; j!$.

{{harvtxt|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.{{cite journal | last1=Macdonald | first1=I. G. | title=Some conjectures for root systems | doi=10.1137/0513070 | mr=674768 | year=1982 | journal=SIAM Journal on Mathematical Analysis | issn=0036-1410 | volume=13 | issue=6 | pages=988–1007}}

:$\backslash frac\{1\}\{(2\backslash pi)^\{n/2\}\}\backslash int\backslash cdots\backslash int\; \backslash left|\backslash prod\_r\backslash frac\{2(x,r)\}\{(r,r)\}\backslash right|^\{\backslash gamma\}e^\{-(x\_1^2+\backslash cdots+x\_n^2)/2\}dx\_1\backslash cdots\; dx\_n$

=\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.

{{harvtxt|Opdam|1989}} gave a uniform proof for all crystallographic reflection groups.{{cite journal|last=Opdam|first=E.M.|year=1989|title=Some applications of hypergeometric shift operators|issue=1|journal=Invent. Math.|volume=98|doi=10.1007/BF01388841|mr=1010152|pages=275–282|bibcode=1989InMat..98....1O|s2cid=54571505 |url=http://dare.uva.nl/personal/pure/en/publications/some-applications-of-hypergeometric-shift-operators(4d1bc98d-e707-47eb-aaec-164e5488d0bc).html}} Several years later he proved it in full generality, making use of computer-aided calculations by Garvan.{{cite journal|last=Opdam|first=E.M.|year=1993|title=Dunkl operators, Bessel functions and the discriminant of a finite Coxeter group|journal=Compositio Mathematica |volume=85|issue=3|pages=333–373|zbl=0778.33009|mr=1214452|url= http://www.numdam.org/item?id=CM_1993__85_3_333_0}}

{{Reflist}}

Category:Special functions