  Some new formulas for the KP hierarchy are derived from the differential Fay
identity. They proved to be useful for the $k$-constrained hierarchies
providing a series of determinant identities for them. A differential equation
is introduced which is called ``universal" since it plays an important role for
all the $k$-constrained hierarchies. In the cases $k=1,2$ and 3 explicit
formulas are presented, in all the others recurrence relations are given which
enable one to obtain the identities.
