The importance of the Selberg integral

@article{Forrester2007TheIO,
  title={The importance of the Selberg integral},
  author={Peter J. Forrester and S. Ole Warnaar},
  journal={Bulletin of the American Mathematical Society},
  year={2007},
  volume={45},
  pages={489-534}
}
It has been remarked that a fair measure of the impact of Atle Selberg’s work is the number of mathematical terms that bear his name. One of these is the Selberg integral, an n-dimensional generalization of the Euler beta integral. We trace its sudden rise to prominence, initiated by a question to Selberg from Enrico Bombieri, more than thirty years after its initial publication. In quick succession the Selberg integral was used to prove an outstanding conjecture in random matrix theory and… 
How to compute Selberg-like integrals?
AbstractIn this paper, we describe a general method for computing Selberg-like integralsbased on a formula, due to Kaneko, for Selberg-Jack integrals. The general principleconsists in expanding the
Multivariate Jacobi Polynomials and the Selberg Integral. II
The problem of harmonic analysis for infinite-dimensional classical groups and symmetric spaces leads to a family of probability measures with infinite-dimensional support. In the present paper, we
On a Selberg–Schur Integral
A generalization of Selberg’s beta integral involving Schur polynomials associated with partitions with entries not greater than 2 is explicitly computed. The complex version of this integral is
The sl3 Selberg integral
Selberg Integral Involving the Product of Multivariable Special Functions
AbstractThe Selberg integral was an integral first evaluated by Selberg in 1944. The aim of the present paper is to estimate generalized Selberg integral. It involves the product of the general class
The distribution of values of zeta and L-functions
This article concerns the distribution of values of the Riemann zeta-function, and related L-functions. We begin with a brief discussion of L-values at the edge of the critical strip, which give
A new q-Selberg integral, Schur functions, and Young books
Recently, Kim and Oh expressed the Selberg integral in terms of the number of Young books which are a generalization of standard Young tableaux of shifted staircase shape. In this paper the
The q-Dixon – Anderson integral and multi-dimensional 1 ψ 1 summations
The Dixon–Anderson integral is a multi-dimensional integral evaluation fundamental to the theory of the Selberg integral. The 1ψ1 summation is a bilateral generalization of the q-binomial theorem. It
Multiplicities of classical varieties
The j ‐multiplicity plays an important role in the intersection theory of Stückrad–Vogel cycles, while recent developments confirm the connections between the ϵ ‐multiplicity and equisingularity
Multivariate Jacobi polynomials and the Selberg integral
This work is motivated by the problem of harmonic analysis on “big” groups and can be viewed as a continuation of the first author’s paper in Functional Anal. Appl. 37 (2003), no. 4, 281–301. Our
...
1
2
3
4
5
...

References

SHOWING 1-10 OF 224 REFERENCES
A Short Proof of Selberg’s Generalized Beta Formula
in a quite nonobvious way. After an initial period in the shadows, Selberg's formula has come to play an important role in mathematics not only because of the interesting applications which have been
The Selberg–Jack Symmetric Functions
Abstract K. Aomoto has recently given a simple proof of an extension of A. Selberg's integral. We prove the following generalization of Aomoto's theorem. For eachk⩾0, there exists a family {skλ(t)}
$q$-Selberg integrals and Macdonald polynomials
We consider a Jackson integral with special integrand (g-Selberg integral) and give an explicit formula of a system of ^-difference equations satisfied by it. We also define a kind of hypergeometric
Determinants of Period Matrices and an Application to Selberg's Multidimensional Beta Integral
In work on critical values of linear functions and hyperplane arrangements, A. Varchenko (Izv. Akad. Nauk SSSR Ser. Mat. 53 (1989), 1206-1235; 54 (1990), 146-158) defined certain period matrices
A generalization of Selberg’s beta integral
We evaluate several infinite families of multidimensional integrals which are generalizations or analogs of Euler's classical beta integral. We first evaluate a ^-analog of Selberg's beta integral.
Some Macdonald-Mehta Integrals by Brute Force
Bombieri and Selberg showed how Mehta’s [6; p. 42] integral could be evaluated using Selberg’s [7] integral. Macdonald [5; §§5,6] conjectured two different generalizations of Mehta’s integral
Zeroes of zeta functions and symmetry
Hilbert and Polya suggested that there might be a natural spectral interpretation of the zeroes of the Riemann Zeta function. While at the time there was little evidence for this, today the evidence
On the generalised Selberg integral of Richards and Zheng
A Proof of Some q-Analogues of Selberg’s Integral for $k=1$
Selberg has given an important multiple beta type integral. We conjecture that for all $k \geqq 0$, there exists a family $\{ s_{n,\boldsymbol{\lambda} }^k ({\bf t})\} $ of homogeneous symmetric
A Selberg integral for the Lie algebra An
A new q-binomial theorem for Macdonald polynomials is employed to prove an An analogue of the celebrated Selberg integral. This confirms the $ \mathfrak{g} ={\rm{A}}_{n}$ case of a conjecture by
...
1
2
3
4
5
...