An application of sums of triple products of binomials

TL;DR

This paper investigates the modular properties of sums involving triple products of binomials, providing new closed-form formulas and revealing an alternating behavior modulo prime numbers.

Contribution

It introduces a novel approach to analyze sums of triple binomial products modulo primes, including deriving explicit closed-form expressions for related sums.

Findings

Sums exhibit alternating behavior modulo prime p

Closed-form expressions for sums of signed triple binomials

Enhanced understanding of binomial sum structures modulo primes

Abstract

We prove that a certain family of sums of products of three binomials has alternating behavior modulo a prime $p$. To accomplish this we rewrite these sums as signed sums of products of three binomials, the better to handle $p$, and we give closed-form expressions for two related sums of signed products of three binomials.