Congruences for partial sums of the generating series for $\left(\genfrac{}{}{0ex}{}{3k}{k}\right)$

Abstract

We produce congruences modulo a prime $p>3$ for sums ${\sum }_k\left(\genfrac{}{}{0ex}{}{3k}{k}\right)x^k$ over ranges $0\le k<q$ and $0\le k<q/3$, where $q$ is a power of $p$. Here $x$ equals either $c^2/(1-c{)}^3$, or $4s^2/(27(s^2-1))$, where $c$ and $s$ are indeterminates. In the former case we deal more generally with shifted binomial coefficients $\left(\genfrac{}{}{0ex}{}{3k+e}{k}\right)$. Our method derives such congruences directly from closed forms for the corresponding series.

arXiv comment18 pages