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where β is a positive integer. Then, assuming that p − r + 1 is an even integer in cases where β is odd, and without this additional assumption in cases where β is even, we have 

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where 

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The proposition reduces $_{0}$F ( α ) 0 ( A, B ), a hypergeometric function with poten- tially high-dimensional matrix arguments, to a repeated contour integral of $_{0}$F ( α ) 0 ( A , Z ), a hypergeometric function with matrix arguments of possibly much lower dimen- sions. In the special case where r = 1 , $_{0}$F ( α ) 0 ( A , Z ) = e a$_{1}$z$_{1}$ and (1) becomes 

<img src='content_image/314465.jpg'>

For α = 2 , this formula has been established by Mo (2011), who used it to an- alyze the asymptotic behavior of the largest eigenvalue of a rank-one perturbation of a real Wishart matrix. He gives two proofs of the formula. One of the proofs uses Jack polynomial expansions and requires that p be an even integer (consistent with our requirement that p − r + 1 is even). The other proof, which Mo (2011) calls geometric, allows for odd p . 

Similar to the first proof of Mo, our proof of Lemma 1 uses Jack polynomial expansions. In contrast to that proof, we do not rely on the simplification of the Jack polynomials for top-order partitions, but use Jack polynomials’ orthogonality 

