Congruences for sums involving $\binom{rk}{k}$

TL;DR

This paper studies congruences modulo a prime p for sums involving binomial coefficients of the form inom{rk}{k}nd explores their properties using two approaches, deriving congruences modulo p^2 expressed via finite polylogarithms.

Contribution

It introduces new methods to establish congruences for sums of binomial coefficients involving a parameter r, including results modulo p^2 expressed through finite polylogarithms.

Findings

Derived congruences modulo p for sums inom{rk}{k}or primes p > r.

Established congruences modulo p^2 involving finite polylogarithms.

Provided a framework for specializing x to algebraic numbers in these congruences.

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 $x$.