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Truncated versions of three identities of Euler and Gauss

Part of: Combinatorics Additive number theory; partitions

Published online by Cambridge University Press:  30 August 2022

Olivia X.M. Yao
Olivia X.M. Yao*
Affiliation:
School of Mathematical Sciences, Suzhou University of Science and Technology, Suzhou, Jiangsu Province 215009, P. R. China (yaoxiangmei@163.com)
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Abstract

In 2012, Andrews and Merca proved a truncated theorem on Euler's pentagonal number theorem. Motivated by the works of Andrews and Merca, Guo and Zeng deduced truncated versions for two other classical theta series identities of Gauss. Very recently, Xia et al. proved new truncated theorems of the three classical theta series identities by taking different truncated series than the ones chosen by Andrews–Merca and Guo–Zeng. In this paper, we provide a unified treatment to establish new truncated versions for the three identities of Euler and Gauss based on a Bailey pair due to Lovejoy. These new truncated identities imply the results proved by Andrews–Merca, Wang–Yee, and Xia–Yee–Zhao.

Keywords

partitionstheta functionstruncated sumspartition-theoretic interpretationsBailey pair

MSC classification

Primary: 11P81: Elementary theory of partitions
Secondary: 05A17: Partitions of integers
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 65 , Issue 3 , August 2022 , pp. 775 - 798
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Copyright
Copyright © The Author(s), 2022. Published by Cambridge University Press on Behalf of The Edinburgh Mathematical Society

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