nLab Selberg integral

Idea

The Selberg integral is a higher-dimensional generalization of the integral representation of Euler’s beta function due Alte Selberg. Many further “Selberg-type” generalizations appear in the study of multidimensional generalizations of hypergeometric functions, arrangements of hyperplanes, Knizhnik-Zamolodchikov equation, representation theory of quantum and affine Lie algebras and of vertex operator algebras, random matrix theory etc. There is also an elliptic generalization, see elliptic Selberg integral.

$$\int_0^1 \cdots \int_0^1 (u_1\cdots u_p)^{x-1} [(1-u_1)\cdots (1-u_p)]^{y-1} \left( \prod_{i\lt j} (u_j-u_i)\right)^{2 z} d u_1 \ldots d u_p = \prod_{s = 1}^p \frac{\Gamma(1+s z) \Gamma(x+(s-1)z)\Gamma(y+(s-1)z)}{ \Gamma(1+z)\Gamma(x+y+(p+s-2)z)}$$

where $p\gt 0$ is a positive integer, $Re x\gt 0$, $Re y\gt 0$ , $Re z \gt max\{-p^{-1},- Re x/(p-1), - Re y/(p-1)\}$. Notice that the discriminant $\prod_{i\lt j} (u_j-u_i)$ is the value of the standard Vandermonde determinant.

References

• wikipedia Selberg integral
• Alte Selberg, Remarks on a multiple integral, Norsk Matematisk Tidsskrift 26: 71–78, MR0018287
• Atle Selberg – utdypning, biography in Norweigian
• Nils A. Baas, Atle Selberg (1917-2007) pdf
• Peter J. Forrester, S. Ole Warnaar, The importance of the Selberg integral, Bull. Amer. Math. Soc. (N.S.) 45 (2008), 489-534, pdf, arxiv/0710.3981
• G. E. Andrews, R. Askey, R. Roy, Special functions, Enc. of Math. and its Appl. (1999)
• I. M. Gelfand, M. M. Kapranov, A. Zelevinsky, Discriminants, resultants and multidimensional determinants, Birkhäuser 1994, 523 pp.
• J.-G. Luque, J.-Y. Thibon, Hankel hyperdeterminants and Selberg integrals, J. Phys. A36 (2003), 5267–5292, MR1985318 (2004d:15011)
• K. Aomoto, Jacobi polynomials associated with Selberg integrals, SIAM J. Math. Anal. 18 (1987) 545–549; On the complex Selberg integral, Q. J. Math. Oxford 38 (1987) 385–399.
• R. S. Askey, Some basic hypergeometric extensions of integrals of Selberg and Andrews, SIAM J. Math. 11 (1980) 938–951.