How to compute Selberg-like integrals?

TL;DR

This paper presents a general method for computing Selberg-like integrals using Jack polynomial expansions, providing closed-form coefficients for special cases and analyzing asymptotic behaviors.

Contribution

It introduces a novel approach based on Kaneko's formula and Gram-Schmidt orthogonalization for Selberg-Jack integrals, with explicit results for power-sum cases and asymptotic analysis.

Findings

Coefficients for power-sum integrals can be expressed in closed form.

The integral is a rational function in the number of variables.

Asymptotic behavior of integrals involving Jack polynomials is characterized.

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.