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

Abstract

The $q$-Narayana numbers $N_q(n,k)$ and $q$-Catalan numbers $C_n(q)$ are respectively defined by $$N_q(n,k)=\frac{1-q}{1-q^n}\left[\genfrac{}{}{0ex}{}{n}{k}\right]\left[\genfrac{}{}{0ex}{}{n}{k-1}\right]\text{and}{C}_n(q)=\frac{1-q}{1-q^{n+1}}\left[\genfrac{}{}{0ex}{}{2n}{n}\right],$$ where $\left[\genfrac{}{}{0ex}{}{n}{k}\right]={\prod }_{i=1}^k\frac{1-q^{n-i+1}}{1-q^i}$. We prove that, for any positive integers $n$ and $r$, there holds \begin{align*} \sum_{k=-n}^{n}(-1)^{k}q^{jk^2+{k\choose 2}}N_q(2n+1,n+k+1)^r \equiv 0 \pmod{C_n(q)}, \end{align*} where $0⩽j⩽2r-1$. We also propose several related conjectures.

6 pages, accepted by Journal of Number Theory