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Ranks, cranks for overpartitions and Appell–Lerch sums

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Abstract

The definitions of the rank and crank for overpartitions were given by Bringmann, Lovejoy and Osburn. Let \(\overline{N}(s,l;n)\) (resp. \(\overline{M}(s,l;n)\), \(\overline{M2}(s,l;n)\)) denote the number of overpartitions of n with rank (resp. the first residual crank, the second residual crank) congruent to s modulo l. The rank differences of overpartitions modulo 3, 5, 6, 7 and 10 were determined. In this paper, we establish the generating functions for \(\overline{N}(s,l;n)\), \(\overline{M}(s,l;n)\) and \(\overline{M2}(s,l;n)\) with \(l=4, 8\) by utilizing Appell–Lerch sums and theta function identities. Moreover, in light of these generating functions, we obtain some equalities and inequalities on ranks and cranks of overpartitions modulo 4 and 8.

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Authors and Affiliations

  1. Department of Mathematics, Jiangsu University, Zhenjiang, 212013, Jiangsu, P.R. China

    Min Bian, Houqing Fang, Xiao Qian Huang & Olivia X. M. Yao

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  1. Min Bian
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  2. Houqing Fang
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  3. Xiao Qian Huang
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  4. Olivia X. M. Yao
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Correspondence to Houqing Fang.

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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 (Grant No. BK20180044).

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Bian, M., Fang, H., Huang, X.Q. et al. Ranks, cranks for overpartitions and Appell–Lerch sums. Ramanujan J 57, 823–844 (2022). https://doi.org/10.1007/s11139-021-00403-0

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  • DOI: https://doi.org/10.1007/s11139-021-00403-0

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