with respect to the torus scalar product (Macdonald, chapter VI, § 10). It is likely that our requirement that p − r + 1 is even in cases where β = 2 /α is odd can be lifted without affecting relationship (1). This would require a different proof of the proposition, which is left for future research. For α = 2 and α = 2 /β with even β , formula (9) has been independently estab- lished by Wang (2012). He uses the formula to study the asymptotic distribution of the largest eigenvalue of the real, complex and quaternionic Wishart matrices perturbed by matrices of rank one. Wang’s proof is similar to the first proof of Mo (2011) (see Forrester, 2011, for an alternative proof). For α = 2 , formula (9) has also been independently established by Onatski et al (2012). Their proof is based on the properties of the so-called Lauricella function. In contrast to (9), the general relationship (1) contains special functions on both left- and right-hand sides. However, for α = 1 , it is possible to further simplify the right-hand side of (1) using Harish-Chandra/Itzykson-Zuber formula (see Harish-Chandra, 1957, and Itzykson and Zuber, 1980) where V$_{r}$ ( A ) = ∏ r $_{j>i}$( a$_{j}$ − a$_{i}$ ) and V$_{r}$ ( Z ) = ∏ r $_{j>i}$( z$_{j}$ − z$_{i}$ ) are the Vandermonde determinants associated with the diagonal elements a$_{1}$, ..., a$_{r}$ of A and the diagonal elements z$_{1}$, ..., z$_{r}$ of Z , respectively. Using (10) in (1), noting that one of the terms in the definition of ω $^{(1)}$( A, B, Z ) equals V$_{r}$ ( Z ) 2 , and applying Andreief’s identity (Andreief, 1883) $_{0}$F (1) 0 ( A , Z ) = ∏ r − 1 j =1 j ! V$_{r}$ ( A ) V$_{r}$ ( Z ) det 1 ≤ i,j ≤ $_{r}$( e a$_{i}$$^{z$_{j}$}$) , (10) det 1 ≤ i,j ≤ r (∫ f$_{i}$ ( x ) g$_{j}$ ( x )d µ ( x ) ) = 1 r ! ∫ ... ∫ det 1 ≤ i,j ≤ $_{r}$( f$_{i}$ ( x$_{j}$ )) det 1 ≤ i,j ≤ $_{r}$( g$_{i}$ ( x$_{j}$ )) ∏ j d µ ( x$_{j}$ ) , we obtain the following Corollary. 12