CONGRUENCES FOR SUMS OF MACMAHON’S q-CATALAN POLYNOMIALS

Abstract

Abstract One variant of the q-Catalan polynomials is defined in terms of Gaussian polynomials by $$ \begin{align*}\mathcal{C}_k(q)=\bigg[\begin{matrix}{2k}\\ {k}\end{matrix}\bigg]_q-q \bigg[\begin{matrix}{2k}\\ {k+1} \end{matrix}\bigg]_q.\end{align*} $$ Liu [‘On a congruence involving q-Catalan numbers’, C. R. Math. Acad. Sci. Paris358 (2020), 211–215] 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 $n\equiv \pm 1\pmod 3$ . Apparently, the case $n\equiv 0\pmod 3$ has been missing from the literature. Our primary purpose is to fill this gap. In addition, we discuss a certain fascinating link to Dirichlet character sum identities.