On the divisibility of sums of even powers of $q$-binomial coefficients

TL;DR

This paper proves a divisibility conjecture related to sums of even powers of $q$-binomial coefficients, advancing understanding in $q$-series and combinatorial identities.

Contribution

It provides a proof of a recent conjecture on divisibility of sums involving $q$-binomial coefficients, using $q$-harmonic series congruences.

Findings

Confirmed the divisibility conjecture for sums of even powers of $q$-binomial coefficients.

Established new congruences involving $q$-harmonic series.

Enhanced methods for analyzing $q$-series divisibility properties.

Abstract

We prove the divisibility conjecture on sums of even powers of $q$-binomial coefficients, which was recently proposed by Guo, Schlosser and Zudilin. Our proof relies on two $q$-harmonic series congruences due to Shi and Pan.