Let M be a finite module and let I be an arbitrary ideal over a
Noetherian local ring. We define the generalized Hilbert function of I on M
using the 0th local cohomology functor. We show that our definition
re-conciliates with that of CiupercaΛ. By generalizing Singh's
formula (which holds in the case of Ξ»(M/IM)<β), we prove that the
generalized Hilbert coefficients j0β,...,jdβ2β are preserved under a
general hyperplane section, where d=dimM. We also keep track of the
behavior of jdβ1β. Then we apply these results to study the generalized
Hilbert function for ideals that have minimal j-multiplicity or almost
minimal j-multiplicity. We provide counterexamples to show that the
generalized Hilbert series of ideals having minimal or almost minimal
j-multiplicity does not have the `expected' shape described in the case where
Ξ»(M/IM)<β. Finally we give a sufficient condition such that the
generalized Hilbert series has the desired shape.Comment: arXiv admin note: text overlap with arXiv:1101.228