  In a recent study of large non-null sample covariance matrices, a new
sequence of functions generalizing the GUE Tracy-Widom distribution of random
matrix theory was obtained. This paper derives Painlev\'e formulas of these
functions and use them to prove that they are indeed distribution functions.
Applications of these new distribution functions to last passage percolation,
queues in tandem and totally asymmetric simple exclusion process are also
discussed. As a part of the proof, a representation of orthogonal polynomials
on the unit circle in terms of an operator on a discrete set is presented.
