Factors of sums involving $q$-binomial coefficients and powers of $q$-integers

Victor J. W. Guo, Su-Dan Wang

We show that, for all positive integers $n_1,\dots ,n_m$, $n_{m+1}=n_1$, and any non-negative integers $j$ and $r$ with $j⩽m$, the expression $$ \frac{1}{[n_1]}{n_1+n_{m…