Relative Hilbert co-efficients

Let $(A,\text{m})$ be a \CM \ local ring of dimension $d$ and let $I\subseteq J$ be two $\text{m}$-primary ideals with $I$ a reduction of $J$. For $i=0,\dots ,d$ let $e_i^J(A)$ ($e_i^I(A)$) be the $i^{th}$ Hilbert coefficient of $J$ ($I$) respectively. We call the number $c_i(I,J)=e_i^J(A)-e_i^I(A)$ the $i^{th}$ relative Hilbert coefficient of $J$ \wrt \ $I$. If $G_I(A)$ is \CM \ then $c_i(I,J)$ satisfy various constraints. We also show that vanishing of some $c_i(I,J)$ has strong implications on $\text{depth}{G}_{J^n}(A)$ for $n\gg 0$.