Factors of alternating sums of powers of $q$-Narayana numbers

Victor J. W. Guo, Qiang-Qiang Jiang

The $q$-Narayana numbers $N_q(n,k)$ and $q$-Catalan numbers $C_n(q)$ are respectively defined by $$ N_q(n,k)=\frac{1-q}{1-q^n}{n\brack k}{n\brack k-1}\quad\text{and}\quad C_n(q)=\f…