On the Generalised Selberg Integral of Richards and Zheng

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. Recently, Richards and Zheng established the following theorem [8]. Theorem 1 (Richards & Zheng). Let r be a nonnegative integer, x1, . . . , xn ∈ R and α1, . . . , αn ∈ C such that Re(αi) > 0 for all 1 ≤ i ≤ n. If aij = ∫ xi+1 xi y n ∏ l=1 (y − xl) αl−1 dy then det 1≤i,j≤n−1 (aij) = ∏ 1≤i<j≤n (xj − xi) n ∏ i,j=1 i6=j (xj − xi) αi−1