# Congruences for $q$-binomial coefficients

## TL;DR

This paper explores advanced congruences related to $q$-binomial coefficients, extending classical results and establishing new congruences for $q$-analogues of factorial ratios, with implications for number theory.

## Contribution

It introduces new $q$-congruences generalizing classical binomial coefficient congruences and proves related results for $q$-factorial ratios.

## Key findings

- Established $q$-analogues of classical binomial congruences
- Proved new congruences for $q$-factorial ratios
- Extended classical results to broader $q$-analogues

## Abstract

We discuss $q$-analogues of the classical congruence $\binom{ap}{bp}\equiv\binom{a}{b}\pmod{p^3}$, valid for primes $p>3$, as well as its generalisations. In particular, we prove related congruences for ($q$-analogues of) integral factorial ratios.