In a recent study of large nonnull sample covariance matrices, a new sequence of functions generalizing the Gaussian unitary ensemble (GUE) Tracy-Widom distribution of random matrix theory was obtained. This article derives Painlevé formulas of these functions and uses 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