# Congruences on sums of $q$-binomial coefficients

## TL;DR

This paper develops new $q$-congruences related to harmonic sums and central $q$-binomial coefficients, extending classical results into the $q$-analogue setting and confirming a recent conjecture by Guo.

## Contribution

It introduces a $q$-analogue of Sun--Zhao's harmonic sum congruence and proves a conjecture on sums of central $q$-binomial coefficients, expanding the theory of $q$-congruences.

## Key findings

- Established a $q$-analogue of Sun--Zhao's harmonic sum congruence.
- Proved a conjecture on sums of central $q$-binomial coefficients by Guo.
- Derived a $q$-analogue of a congruence by Apagodu and Zeilberger.

## Abstract

We establish a $q$-analogue of Sun--Zhao's congruence on harmonic sums. Based on this $q$-congruence and a $q$-series identity, we prove a congruence conjecture on sums of central $q$-binomial coefficients, which was recently proposed by Guo. We also deduce a $q$-analogue of a congruence due to Apagodu and Zeilberger from Guo's $q$-congruence.