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CONGRUENCES FOR SUMS OF MACMAHON’S q-CATALAN POLYNOMIALS
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- Journal:
- Bulletin of the Australian Mathematical Society , First View
- Published online by Cambridge University Press:
- 08 October 2024, pp. 1-12
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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. Paris 358 (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.
