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ON THE DIVISIBILITY OF SUMS OF q-SUPER CATALAN NUMBERS

Part of: Elementary number theory Sequences and sets Combinatorics

Published online by Cambridge University Press:  15 May 2023

JI-CAI LIU Open the ORCID record for JI-CAI LIU [Opens in a new window]  and
YAN-NI LI
JI-CAI LIU*
Affiliation:
Department of Mathematics, Wenzhou University, Wenzhou 325035, PR China
YAN-NI LI
Affiliation:
Department of Mathematics, Wenzhou University, Wenzhou 325035, PR China e-mail: ynli2022@foxmail.com
*
e-mail: jcliu2016@gmail.com
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Abstract

The integrality of the numbers $A_{n,m}={(2n)!(2m)!}/{n!m!(n+m)!}$ was observed by Catalan as early as 1874 and Gessel named $A_{n,m}$ the super Catalan numbers. The positivity of the q-super Catalan numbers (q-analogue of the super Catalan numbers) was investigated by Warnaar and Zudilin [‘A q-rious positivity’, Aequationes Math. 81 (2011), 177–183]. We prove the divisibility of sums of q-super Catalan numbers, which establishes a q-analogue of Apagodu’s congruence involving super Catalan numbers.

Keywords

Catalan numberq-super Catalan numbercyclotomic polynomial

MSC classification

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

Information

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 109 , Issue 2 , April 2024 , pp. 215 - 224
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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Footnotes

The first author was supported by the National Natural Science Foundation of China (grant 12171370).

References

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