  We study Fredholm determinants related to a family of kernels which 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 Painlev\'e 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 Painlev\'e II
hierarchy. In addition we compute large gap asymptotics for the Fredholm
determinants.
