Divisibility of some binomial sums

TL;DR

This paper proves divisibility properties of certain binomial sums using q-congruence techniques, revealing new divisibility relations involving binomial coefficients and powers of integers.

Contribution

It introduces novel divisibility results for binomial sums employing q-congruence methods, expanding understanding of binomial sum divisibility.

Findings

Proves divisibility of specific binomial sums modulo certain binomial coefficients.

Establishes a general divisibility relation for sums involving binomial coefficients and powers.

Uses q-congruence techniques to derive new number-theoretic properties.

Abstract

With help of $q$-congruence, we prove the divisibility of some binomial sums. For example, for any integers $\rho,n\geq 2$, $$\sum_{k=0}^{n-1}(4k+1) \binom{2k}{k}^\rho \cdot (-4)^{\rho(n-1-k)} \equiv 0\pmod{2^{\rho-2}n\binom{2n}{n}}.$$