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A TRUNCATED IDENTITY OF EULER AND RELATED $q$-CONGRUENCES

Part of: Elementary number theory Combinatorics

Published online by Cambridge University Press:  08 April 2020

JI-CAI LIU Open the ORCID record for JI-CAI LIU [Opens in a new window]  and
ZHONG-YU HUANG
JI-CAI LIU*
Affiliation:
Department of Mathematics,Wenzhou University, Wenzhou325035, PR China email jcliu2016@gmail.com
ZHONG-YU HUANG
Affiliation:
Department of Mathematics,Wenzhou University, Wenzhou325035, PR China email 2357207357@qq.com
*
✉jcliu2016@gmail.com
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Abstract

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We discuss a truncated identity of Euler and present a combinatorial proof of it. We also derive two finite identities as corollaries. As an application, we establish two related $q$-congruences for sums of $q$-Catalan numbers, one of which has been proved by Tauraso [‘$q$-Analogs of some congruences involving Catalan numbers’, Adv. Appl. Math. 48 (2012), 603–614] by a different method.

Keywords

pentagonal number theoremq-congruencesq-Catalan numbers

MSC classification

Primary: 05A19: Combinatorial identities, bijective combinatorics
Secondary: 11A07: Congruences; primitive roots; residue systems

Information

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 102 , Issue 3 , December 2020 , pp. 353 - 359
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Copyright
© 2020 Australian Mathematical Publishing Association Inc.

Footnotes

The first author was supported by the National Natural Science Foundation of China (grant 11801417).

References

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