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Proofs of some conjectures on the reciprocals of the Ramanujan–Gordon identities

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Abstract

Recently, Lin and Wang introduced two special partition functions \(RG_1(n)\) and \(RG_2(n)\), the generating functions of which are the reciprocals of two identities due to Ramanujan and Gordon. They established several congruences modulo 5 and 7 for \(RG_1(n)\) and \(RG_2(n)\) and posed four conjectures on congruences modulo 25 for \(RG_1(n)\) and \(RG_2(n)\) at the end of their paper. In this paper, we confirm the four conjectures given by Lin and Wang by using Ramanujan’s modular equation of fifth degree. Moreover, we also obtain new congruences modulo 25 for \(RG_1(n)\) and \(RG_2(n)\) based on Newman’s identities. For example, we deduce that for any integer \(n\ge 0\),

$$\begin{aligned} RG_1\left( \frac{23375n(3n+1)}{2}+974\right)&\equiv RG_1\left( \frac{23375n(3n+5)}{2}+24349\right) \\&\equiv 0\pmod {25}. \end{aligned}$$

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Acknowledgements

All authors would like to acknowledge the helpful comments and suggestions made by the referee.

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Authors and Affiliations

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

    Min Bian, Ernest X. W. Xia & Fanggang Xue

  2. Center for Applied Mathematics, Tianjin University, Tianjin, 300072, People’s Republic of China

    Dazhao Tang

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  1. Min Bian
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  2. Dazhao Tang
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  3. Ernest X. W. Xia
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  4. Fanggang Xue
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Correspondence to Ernest X. W. Xia.

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Min Bian, Ernest X. W. Xia and Fanggang Xue were supported by the National Natural Science Foundation of China (Nos. 11971203 and 11571143) and the Nature Funds for Distinguished Young Scientists of Jiangsu Province (No. BK20180044). Dazhao Tang was supported by the Fundamental Research Funds for the Central Universities (No. 2018CDXYST0024) and the Postdoctoral Science Foundation of China (No. 2019M661005).

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Bian, M., Tang, D., Xia, E.X.W. et al. Proofs of some conjectures on the reciprocals of the Ramanujan–Gordon identities. Ramanujan J 55, 497–515 (2021). https://doi.org/10.1007/s11139-019-00247-9

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  • DOI: https://doi.org/10.1007/s11139-019-00247-9

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