  In this paper we find the tight closure of powers of parameter ideals of
certain diagonal hypersurface rings. In many cases the associated graded ring
with respect to tight closure filtration turns out to be Cohen-Macaulay. This
helps us find the tight Hilbert polynomial in these diagonal hypersurfaces. We
determine the tight Hilbert polynomial in the following cases: (1) F-pure
diagonal hypersurfaces where number of variables is equal to the degree of
defining equation, (2) diagonal hypersurface rings where characteristic of the
ring is one less than the degree of defining equation and (3) quartic diagonal
hypersurface in four variables.
