  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.
