integer. 2 The importance of finding “serviceable approximations” to $_{0}$F (2) 0 ( A, B ) has been recently emphasized by Johnstone (2007, p.322). Since in applications that rely on the spiked covariance matrix framework r is typically much smaller than p , analyzing $_{0}$F (2) 0 ( A , Z ) is much easier than analyzing $_{0}$F (2) 0 ( A, B ), and the established contour integral representation of the latter may be of the welcomed service to practitioners. 

For α = 2 , 1 and 1 / 2 , function $_{0}$F ( α ) 0 ( A, B ) has an integral representation ∫ G ( α $^{)}$( p $_{)}$e tr ( AGBG − 1 ) (d G ) , where G ( α $^{)}$( p ) is the orthogonal group O ( p ) for α = 2, the unitary group U ( p ) for α = 1, and the compact symplectic group S p ( p ) for α = 1 / 2 , and where (d G ) is the normalized Haar measure over G ( α $^{)}$( p ). Such integrals have various important applications in mathematics and physics, where they are referred to as Harish-Chandra/Itzykson-Zuber (HCIZ) integrals (Zinn-Justin and Zuber, 2003). The HCIZ integrals with rank-deficient A have been used in the analysis of spin glasses (Marinari et al, 1994), wireless communication systems (Muller et al, 2008), statistical tests for signal detection (Bianchi et al, 2010, and Onatski et al, 2012), distribution of the largest sample covariance eigenvalue (Mo, 2011, and Wang, 2012), and spiked Wishart β -ensembles (Forrester, 2011). Their asymptotic behavior as p → ∞ has been studied in Guionnet and Ma¨ıda (2005) and Collins and ´ Sniady (2007). We hope that the reduction of HCIZ integrals over large group G ( α $^{)}$( p ) to those over smaller group G ( α $^{)}$( r ) that follows from (1) will be useful in a wide spectrum of applications. 

The rest of this paper is organized as follows. In Section 2, we derive explicit formulae for the likelihood ratios. Section 3 establishes relationship (1). Section 4 uses (1) to derive contour integral representations for the likelihood ratios. Section 5 applies Laplace approximations to the contour integrals in the derived represen- 

