Congruences for sums of MacMahon's $q$-Catalan polynomials

TL;DR

This paper investigates congruences related to sums of MacMahon's $q$-Catalan polynomials, extending previous results to cases where $n$ is divisible by 3 and exploring connections to Dirichlet character sums.

Contribution

It extends known congruences for $q$-Catalan polynomial sums to the case $n ot ot ext{equiv} ext{±}1 mod 3$, filling a gap in the literature.

Findings

Established congruences for sums when $n$ is divisible by 3.

Linked $q$-Catalan sum identities to Dirichlet character sums.

Extended the scope of previous congruence results.

Abstract

One variant of the $q$-Catalan polynomials is defined in terms of Gaussian polynomials by $\mathcal{C}_k(q)=\genfrac{[}{]}{0pt}{}{2k}{k}_q-q\genfrac{[}{]}{0pt}{}{2k}{k+1}_q$. Liu studied congruences of the form $\sum_{k=0}^{n-1} q^k\mathcal{C}_k$ modulo the cyclotomic polynomial $\Phi_n(q)^2$, provided that $n\equiv\pm 1\pmod3$. Apparently the case $n\equiv 0\pmod3$ has been missing from the literature. It is our primary purpose to fill this gap by the current work. In addition, we discuss certain fascinating link to Dirichlet character sum identities.