  We generalize the notion of Hilbert-Kunz multiplicity of a graded triple
$(M,R,I)$ in characteristic $p>0$ by proving that for any complex number $y$,
the limit
  $$\underset{n \to \infty}{\lim}(\frac{1}{p^n})^{\text{dim}(M)}\sum
\limits_{j= -\infty}^{\infty}\lambda \left(
(\frac{M}{I^{[p^n]}M})_j\right)e^{-iyj/p^n}$$ exists. We prove that the
limiting function in the complex variable $y$ is entire and name this function
the \textit{Frobenius-Poincar\'e function}. We establish various properties of
Frobenius-Poincar\'e functions including its relation with the tight closure of
the defining ideal $I$; and relate the study Frobenius-Poincar\'e functions to
the behaviour of graded Betti numbers of $\frac{R}{I^{[p^n]}} $ as $n$ varies.
Our description of Frobenius-Poincar\'e functions in dimension one and two and
other examples raises questions on the structure of Frobenius-Poincar\'e
functions in general.
