Derived functors and Hilbert polynomials over regular local rings

Let $(A,\mathfrak{m})$ be a regular local ring of dimension $d\ge 1$, $I$ an $\mathfrak{m}$-primary ideal. Let $N$ be a non-zero finitely generated $A$-module. Consider the functions $$t^I(N,n)=\sum _{i=0}^d\mathrm{\ell }({\text{Tor}}_i^A(N,A\mathrm{/}{I}^n))\text{ and }{e}^I(N,n)=\sum _{i=0}^d\mathrm{\ell }({\text{Ext}}_A^i(N,A\mathrm{/}{I}^n))$$ of polynomial type and let their degrees be $t^I(N)$ and $e^I(N)$. We prove that $t^I(N)=e^I(N)=\max \{\dim N,d-1\}$.