  Let A be a local ring with maximal ideal m. For an arbitrary ideal I of A, we
define the generalized Hilbert coefficients j_k(I) \in Z^{k+1} (k=0,1,...,dim
A). When the ideal I is m-primary, j_k(I)=(0,...,0,(-1)^k e_k(I)), where e_k(I)
is the classical k-th Hilbert coefficient of I. Using these coefficients, we
give a numerical characterization of the homogeneous components of the
S_2-ification of S=A[It,t^{-1}].
