Congruences for sums involving $\binom{rk}{k}$
Sandro Mattarei, Roberto Tauraso
TL;DR
The paper addresses congruences modulo primes for finite sums of the form \(\sum_{k} \binom{rk}{k} x^k/k\). It develops two complementary methods: a p-analytic approach expressing the sums in terms of finite polylogarithms evaluated at algebraic roots \(c_i\) of \(x(c-1)^r+c^{r-1}=0\), and a p^2-refinement via symmetric-function identities that yields explicit formulas in terms of \pounds_2(c_i) and \pounds_2(1-c_i). The key contributions include a general modulo-p formula for the full and short ranges, a stronger modulo-p^2 result for the full range, and extensive numerical applications that connect these congruences to Fermat quotients, Euler and Lucas numbers, and known conjectures. Together, these results extend generating-function congruences to arbitrary \(r\) and provide practical tools for evaluating these finite sums modulo powers of primes.
Abstract
We primarily investigate congruences modulo $p$ for finite sums of the form $\sum_k\binom{rk}{k}x^k/k$ 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 $\sum_{0<k<p}\binom{rk}{k}x^k$, expressed in terms of finite polylogarithms of certain quantities related to $x$.