Heterogeneous hypergeometric functions with two matrix arguments and the exact distribution of the largest eigenvalue of a singular beta-Wishart matrix

This paper discusses certain properties of heterogeneous hypergeometric functions with two matrix arguments. These are newly defined but have already appeared in statistical literature and useful when dealing the derivation distributions for eigenvalues singular beta-Wishart matrices. The joint density function distribution largest eigenvalue can be expressed terms functions. Exact computation is conducted real complex cases.