Congruences for sums involving $\left(\genfrac{}{}{0ex}{}{rk}{k}\right)$

Abstract

We primarily investigate congruences modulo $p$ for finite sums of the form ${\sum }_k\left(\genfrac{}{}{0ex}{}{rk}{k}\right)x^k/k$ over the ranges $0<k<p$ and $0<k<p/r$, where $p$ is a prime larger than the positive integer $r$. Here $x$ is an indeterminate, thus allowing specialization to numerical congruences where $x$ takes certain algebraic numbers as values. We employ two different approaches that have complementary strengths. In particular, we obtain congruences modulo $p^2$ for the sum ${\sum }_{0<k<p}\left(\genfrac{}{}{0ex}{}{rk}{k}\right)x^k$, expressed in terms of finite polylogarithms of certain quantities related to $x$.

24 pages; improved introduction in this version