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 where 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 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 11 $_{0}$F ( α ) 0 ( A, B ) = Γ ( p/α ) a 1 − p/α 1 1 2 π i ∮ K e a$_{1}$z$_{1}$ p ∏ s =1 ( z$_{1}$ − b$_{s}$ ) − 1 α d z$_{1}$. (9) ω ( α $^{)}$( A, B, Z ) = ( − 1) r ( r − 1) / (2 α ) r ∏ j =1 [ Γ (( p +1 − j ) /α ) Γ (1 /α ) Γ (( r +1 − j ) /α ) ] × r ∏ j>i ( z$_{j}$ − z$_{i}$ ) 2 /α r ∏ j =1 [ a 1 − ( p − r +1) /α j p ∏ s =1 ( z$_{j}$ − b$_{s}$ ) − 1 /α ] $_{0}$F ( α ) 0 ( A, B ) = 1 r ! (2 π i) r ∮ K ... ∮ K $_{0}$F ( α ) 0 ( A , Z ) ω ( α $^{)}$( A, B, Z ) r ∏ j =1 d z$_{j}$, (1)