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

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.