We study Fredholm determinants related to a family of kernels that describe the edge eigenvalue behavior in unitary random matrix models with critical edge points. The kernels are natural higher-order analogues of the Airy kernel and are built out of functions associated with the Painlevé I hierarchy. The Fredholm determinants related to those kernels are higher-order generalizations of the Tracy-Widom distribution. We give an explicit expression for the determinants in terms of a distinguished smooth solution to the Painlevé II hierarchy. In addition, we compute large gap asymptotics for the Fredholm determinants.
