How to compute Selberg-like integrals?

Abstract: In this paper, we describe a general method for computing Selberg-like integrals based on a formula, due to Kaneko, for Selberg-Jack integrals. The general principle consists in expanding the integrand \emph{w.r.t.} the Jack basis, which is obtained by a Gram-Schmidt orthogonalization process. The resulting algorithm is not very efficient because of this decomposition. But for special cases, the coefficients admit a closed form. As an example, we study the case of the power-sums since for which the coefficients are obtained by manipulating generating series by means of transformations of alphabets. Furthermore, we prove that the integral is a rational function in the number of variables which allows us to study asymptotics. As an application, we investigate the asymptotic behavior when the integrand involves Jack polynomials and power sums.