A $d$-dimensional extension of a lemma of Huneke's and formulas for the Hilbert coefficients

A $d$-dimensional version is given of a $2$-dimensional result due to C. Huneke. His result produced a formula relating the length $\lambda (I^{n+1}/JI^{n})$ to the difference $P_{I}(n+1)-H_{I}(n+1)$, where $I$ is primary for the maximal ideal of a $2$-dimensional Cohen-Macaulay local ring $R$, $J$ is a minimal reduction of $I$, $H_{I}(n)=\lambda (R/I^{n})$, and $P_{I}(n)$ is the Hilbert-Samuel polynomial of $I$. We produce a formula that is valid for arbitrary dimension, and then use it to establish some formulas for the Hilbert coefficients of $I$. We also include a characterization, in terms of the Hilbert coefficients of $I$, of the condition $depth(G(I))\geq d-1$.