On the generalised Selberg integral of Richards and Zheng

TL;DR

This paper shows that the generalized Selberg integral by Richards and Zheng is a special case of Kadell's integral over Jack polynomials and demonstrates how Okounkov and Olshanski's formula can be used to prove Kadell's result.

Contribution

It connects Richards and Zheng's generalization to Kadell's integral over Jack polynomials and applies Okounkov and Olshanski's formula for a new proof.

Findings

Richards and Zheng's Selberg generalization is a special case of Kadell's integral.

Okounkov and Olshanski's integral formula can be used to prove Kadell's integral.

The approach unifies different generalizations of the Selberg integral.

Abstract

In a recent paper Richards and Zheng compute the determinant of a matrix whose entries are given by beta-type integrals, thereby generalising an earlier result by Dixon and Varchenko. They then use their result to obtain a generalisation of the famous Selberg integral. In this note we point out that the Selberg-generalisation of Richards and Zheng is a special case of an integral over Jack polynomials due to Kadell. We then show how an integral formula for Jack polynomials of Okounkov and Olshanski may be applied to prove Kadell's integral along the lines of Richards and Zheng.