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CONGRUENCES FOR SUMS OF MACMAHON’S q-CATALAN POLYNOMIALS

Part of: Elementary number theory Combinatorics Sequences and sets

Published online by Cambridge University Press:  08 October 2024

TEWODROS AMDEBERHAN Open the ORCID record for TEWODROS AMDEBERHAN [Opens in a new window]  and
ROBERTO TAURASO Open the ORCID record for ROBERTO TAURASO [Opens in a new window]
TEWODROS AMDEBERHAN
Affiliation:
Department of Mathematics, Tulane University, New Orleans, LA 70118, USA e-mail: tamdeber@tulane.edu
ROBERTO TAURASO*
Affiliation:
Dipartimento di Matematica, Università di Roma ‘Tor Vergata’, 00133 Roma, Italy
*
e-mail: tauraso@mat.uniroma2.it
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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. 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.

Keywords

q-Catalan numberq-analogue congruenceq-binomial coefficientcyclotomic polynomialroots of unity

MSC classification

Primary: 11B65: Binomial coefficients; factorials; $q$-identities
Secondary: 05A10: Factorials, binomial coefficients, combinatorial functions 05A19: Combinatorial identities, bijective combinatorics 11A07: Congruences; primitive roots; residue systems

Information

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 111 , Issue 3 , June 2025 , pp. 416 - 427
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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References

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