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$ (R,{\frak m},k) $\end{document} be a Noetherian local ring of prime characteristic p and d its Krull dimension. It is known that for an \documentclass[12pt]{minimal}
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$ \frak m $\end{document}-primary ideal I of R and a finitely generated R-module N the limit \documentclass[12pt]{minimal}
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$\lim \limits_{n\to \infty } l_R(N/I^{[n]}N)/p^{dn}$\end{document} exists where I [n] denotes the ideal of R generated by \documentclass[12pt]{minimal}
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$ x^{p^n} $\end{document}, \documentclass[12pt]{minimal}
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$ x \in I $\end{document}, and l R (M) the length of an R-module M.¶ We will show that the ordinary generating function¶¶\documentclass[12pt]{minimal}
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$ \sum _{n = 0}^\infty l_R (N / I^{[n]} N) t^n \in {\Bbb Q} [[t]] $\end{document}¶¶of the Hilbert-Kunz function \documentclass[12pt]{minimal}
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$ {\Bbb N} \to {\Bbb N}, n \mapsto l_R (N/I^{[n]}N) $\end{document} is rational, i.e., an element of \documentclass[12pt]{minimal}
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$ {\Bbb Q} (t) $\end{document}, if R (1) is a finite R-module, N a maximal Cohen-Macaulay module and R is of finite Cohen-Macaulay type, i.e., the number of isomorphism classes of finite, indecomposable maximal Cohen-Macaulay modules over R is finite. From this result, we deduce that \documentclass[12pt]{minimal}
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$ \lim _ {n\to \infty } l_R (N / I^{[n]} N) / p^{dn} \in {\Bbb Q} $\end{document}. Here R (1) denotes R considered as an R-algebra via the Frobenius map \documentclass[12pt]{minimal}
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$ R \to R, x \mapsto x^{p} $\end{document}. Actually we will consider a somewhat more general situation using the Frobenius functor.
