Eigenvalue statistics for product complex Wishart matrices

The eigenvalue statistics for complex $N \times N$ Wishart matrices $X_{r,s}^\dagger X_{r,s}$, where $ X_{r,s}$ is equal to the product of $r$ complex Gaussian matrices, and the inverse of $s$ complex Gaussian matrices, are considered. In the case $r=s$ the exact form of the global density is computed. The averaged characteristic polynomial for the corresponding generalized eigenvalue problem is calculated in terms of a particular generalized hypergeometric function ${}_{s+1} F_r$. For finite $N$ the eigenvalue probability density function is computed, and is shown to be an example of a biorthogonal ensemble. A double contour integral form of the corresponding correlation kernel is derived, which allows the hard edge scaled limit to be computed. The limiting kernel is given in terms of certain Meijer G-functions, and is identical to that found in the recent work of Kuijlaars and Zhang in the case $s=0$. Properties of the kernel and corresponding correlation functions are discussed.