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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

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
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.

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Footnotes

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

References

Andrews, G. E., The Theory of Partitions (Addison-Wesley, Reading, MA, 1976).Google Scholar
Andrews, G. E. and Merca, M., ‘The truncated pentagonal number theorem’, J. Combin. Theory Ser. A 119 (2012), 1639–1643.10.1016/j.jcta.2012.05.001CrossRefGoogle Scholar
Fürlinger, J. and Hofbauer, J., ‘q-Catalan numbers’, J. Combin. Theory Ser. A 40 (1985), 248–264.CrossRefGoogle Scholar
Gu, C.-Y. and Guo, V. J. W., ‘q-Analogues of two supercongruences of Z.-W. Sun’, Czechoslovak Math. J. to appear.Google Scholar
Guo, V. J. W., ‘Common q-analogues of some different supercongruences’, Results Math. 74 (2019), Article ID 131.10.1007/s00025-019-1056-1CrossRefGoogle Scholar
Guo, V. J. W. and Zeng, J., ‘Multiple extensions of a finite Euler’s pentagonal number theorem and the Lucas formulas’, Discrete Math. 308 (2008), 4069–4078.10.1016/j.disc.2007.07.106CrossRefGoogle Scholar
Guo, V. J. W. and Zudilin, W., ‘A q-microscope for supercongruences’, Adv. Math. 346 (2019), 329–358.10.1016/j.aim.2019.02.008CrossRefGoogle Scholar
Hardy, G. H. and Wright, E. M., An Introduction to the Theory of Numbers (Oxford University Press, Oxford, 1938), 280–286.Google Scholar
Liu, J.-C., ‘Some finite generalizations of Euler’s pentagonal number theorem’, Czechoslovak Math. J. 142 (2017), 525–531.CrossRefGoogle Scholar
Liu, J.-C. and Petrov, F., ‘Congruences on sums of q-binomial coefficients’, Adv. Appl. Math. 116 (2020), Article ID 102003.10.1016/j.aam.2020.102003CrossRefGoogle Scholar
Shanks, D., ‘A short proof of an identity of Euler’, Proc. Amer. Math. Soc. 2 (1951), 747–749.10.1090/S0002-9939-1951-0043808-6CrossRefGoogle Scholar
Tauraso, R., ‘q-Analogs of some congruences involving Catalan numbers’, Adv. Appl. Math. 48 (2012), 603–614.10.1016/j.aam.2011.12.002CrossRefGoogle Scholar