Assume I am given points $(i, f(i))$ for $i = 1, \dots, n$ of an unknown polynomial $f$ of degree $n-1$. Further assume that the coefficients of the polynomial come from a field. In this case we know that $f$ is uniquely defined by the given points and we can evaluate $f$ on a new point $z$ without explicitly determining $f$'s coefficients.
That is, we can compute
$$f(z) = \sum_{i=1}^{n} \delta_i(z) \cdot f(i)$$ from the $f(i)$'s, where $\delta_i(z)$ is the Kronecker delta function that evaluates to $1$ if $i = z$ and $0$ on any other of the points.
Question: Is there some similar expression known for rational functions $h(x) = \frac{p(x)}{q(x)}$, where $p(x)$ and $q(x)$ are polynomials of degree $n$ with coefficients in a field? Given some amount of points $(i, h(i))$, can I extrapolate $h$ on a new point without explicitly interpolating $h$ and determining its coefficients?
If it helps one can assume that the field is $\mathbb{Z}_p$ for a prime $p$.