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Identities and inequalities for the \(M_2\)-rank of partitions without repeated odd parts modulo 8

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Abstract

In 2002, Berkovich and Garvan introduced the \(M_2\)-rank of partitions without repeated odd parts. Let \(N_2(a, M, n)\) denote the number of partitions of n without repeated odd parts in which \(M_2\)-rank is congruent to a mod M. Lovejoy and Osburn, and Mao found a number of nice results for \(M_2 \)-rank differences modulo 3, 5, 6, and 10. In this paper, by using some properties for Appell–Lerch sums, we establish the generating functions for \(N_2(a,8,n) \) with \(0\le a \le 7\). With these generating functions, we obtain some equalities and inequalities on \(M_2\)-rank modulo 8 of partitions without repeated odd parts. We also relate some differences of the \(M_2\)-rank to eighth-order mock theta functions.

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Acknowledgements

The authors cordially thank the anonymous referee for his/her helpful comments.

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  1. Department of Mathematics, Jiangsu University, Zhenjiang, 212013, Jiangsu, People’s Republic of China

    Ernest X. W. Xia & Xiang Zhao

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  1. Ernest X. W. Xia
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  2. Xiang Zhao
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Correspondence to Ernest X. W. Xia.

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This work was supported by the National Science Foundation of China (grant No. 11971203) and the Nature Funds for Distinguished Young Scientists of Jiangsu Province (No. BK20180044).

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Xia, E.X.W., Zhao, X. Identities and inequalities for the \(M_2\)-rank of partitions without repeated odd parts modulo 8. Ramanujan J 58, 1259–1284 (2022). https://doi.org/10.1007/s11139-021-00486-9

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  • DOI: https://doi.org/10.1007/s11139-021-00486-9

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