# 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…

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## References

SHOWING 1-10 OF 224 REFERENCES

A Short Proof of Selberg’s Generalized Beta Formula

- Mathematics
- 1991

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

- Mathematics
- 1997

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

- Mathematics
- 1996

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

- MathematicsAdv. Appl. Math.
- 2002

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

- Mathematics
- 1990

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

- Mathematics
- 1989

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

- Mathematics
- 1999

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…

A Proof of Some q-Analogues of Selberg’s Integral for $k=1$

- Mathematics
- 1988

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

- Mathematics
- 2007

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…