Asymptotics of partial sums of central binomial coefficients and Catalan numbers

Abstract

We prove exact asymptotic expansions for the partial sums of the sequences of central binomial coefficients and Catalan numbers, ${\sum }_{k=0}^n\left(\genfrac{}{}{0ex}{}{2k}{k}\right)$ and ${\sum }_{k=0}^n{C}_n$. We also obtain closed forms for the polynomials ${\sum }_{k=0}^{q-1}\left(\genfrac{}{}{0ex}{}{2k}{k}\right)x^k$ and ${\sum }_{k=0}^{q-1}{C}_k{x}^k$ over the field of $p$ elements, where $q$ is a power of the prime $p$.

8 pages; a new section has been added, about congruences modulo a prime

References in corpus (2)

• On some new congruences for binomial coefficients
• New congruences for central binomial coefficients