8 - The Selberg Integral and Its Applications

Summary

Dirichlet's straightforward though useful multidimensional generalization of the beta integral was presented in Chapter 1. In the 1940s, more than 100 years after Dirichlet's work, Selberg found a more interesting generalized beta integral in which the integrand contains a power of the discriminant of the n variables of integration. Recently, Aomoto evaluated a yet slightly more general integral. An important feature of this evaluation is that it provides a simpler proof of Selberg's formula, reminiscent of Euler's evaluation of the beta integral by means of a functional equation. The depth of Selberg's integral formula may be seen in the fact that in two dimensions it implies Dixon's identity for a well-poised ₃F₂. Bressoud observed that Aomoto's extension implies identities for nearly poised ₃F₂.

After presenting Aomoto's proof, we give another proof of Selberg's formula due to Anderson. This proof is similar to Jacobi's or Poisson's evaluation of Euler's beta integral in that it depends on the computation of a multidimensional integral in two different ways. The basis for Anderson's proof is Dirichlet's multidimensional integral mentioned above. A very significant aspect of Anderson's method is that it applies to the finite-field analog of Selberg's integral as well. We give a brief treatment of this analog at the end of the chapter.