Growth of Hilbert coefficients of Syzygy modules

Let $(A,\mathfrak{m})$ be a complete intersection ring of dimension $d$ and let $I$ be an $\mathfrak{m}$-primary ideal. Let $M$ be a maximal \CM
$A$-module. For $i=0,1,\cdots ,d$, let $e_i^I(M)$ denote the $i^{th}$ Hilbert -coefficient of $M$ with respect to $I$. We prove that for $i=0,1,2$, the function $j\mapsto e_i^I(Sy{z}_j^A(M))$ is of quasi-polynomial type with period $2$. Let $G_I(M)$ be the associated graded module of $M$ with respect to $I$. If $G_I(A)$ is Cohen-Macaulay and $\dim A\le 2$ we also prove that the functions $j\mapsto depth{G}_I(Sy{z}_{2j+i}^A(M))$ are eventually constant for $i=0,1$. Let ${\xi }_I(M)={\lim }_{l\to \mathrm{\infty }}depth{G}_{I^l}(M)$. Finally we prove that if $\dim A=2$ and $G_I(A)$ is Cohen-Macaulay then the functions $j\mapsto {\xi }_I(Sy{z}_{2j+i}^A(M))$ are eventually constant for $i = 0, 1$.