  We prove exact asymptotic expansions for the partial sums of the sequences of
central binomial coefficients and Catalan numbers, $\sum_{k=0}^n \binom{2k}{k}$
and $\sum_{k=0}^n C_n$. We also obtain closed forms for the polynomials
$\sum_{k=0}^{q-1}\binom{2k}{k}x^k$ and $\sum_{k=0}^{q-1}C_kx^k$ over the field
of $p$ elements, where $q$ is a power of the prime $p$.
