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

## TL;DR

This paper investigates the properties of alternating sums involving powers of $q$-Narayana numbers, establishing divisibility results by $q$-Catalan numbers and proposing related conjectures.

## Contribution

The paper proves a divisibility property of alternating sums of powers of $q$-Narayana numbers by $q$-Catalan numbers, introducing new identities and conjectures.

## Key findings

- Proved divisibility of specific alternating sums by $q$-Catalan numbers.
- Established identities involving $q$-Narayana and $q$-Catalan numbers.
- Proposed several related conjectures for further research.

## Abstract

The $q$-Narayana numbers $N_q(n,k)$ and $q$-Catalan numbers $C_n(q)$ are respectively defined by $$ N_q(n,k)=\frac{1-q}{1-q^n}{n\brack k}{n\brack k-1}\quad\text{and}\quad C_n(q)=\frac{1-q}{1-q^{n+1}}{2n\brack n}, $$ where ${n\brack k}=\prod_{i=1}^{k}\frac{1-q^{n-i+1}}{1-q^i}$. 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 $0\leqslant j\leqslant 2r-1$. We also propose several related conjectures.