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On \(\ell \)-regular bipartitions modulo \(\ell \)

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Abstract

Let \(B_\ell (n)\) denote the number of \(\ell \)-regular bipartitions of n. In this paper, we prove several infinite families of congruences satisfied by \(B_\ell (n)\) for \(\ell \in {\{5,7,13\}}\). For example, we show that for all \(\alpha >0\) and \(n\ge 0\),

$$\begin{aligned} B_5\left( 4^\alpha n+\frac{5\times 4^\alpha -2}{6}\right)\equiv & {} 0 \ (\text {mod}\ 5),\\ B_7\left( 5^{8\alpha }n+\displaystyle \frac{5^{8\alpha }-1}{2}\right)\equiv & {} 3^\alpha B_7(n)\ (\text {mod}\ 7) \end{aligned}$$

and

$$\begin{aligned} B_{13}\left( 5^{12\alpha }n+5^{12\alpha }-1\right) \equiv B_{13}(n)\ (\text {mod}\ 13). \end{aligned}$$

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Acknowledgements

The authors are extremely grateful to Professor Michael Hirschhorn who read our manuscript with great care, uncovered several errors and offered his valuable suggestions which have substantially improved our paper. The authors also thank the anonymous referee for his/her helpful suggestions and comments.

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Authors and Affiliations

  1. Department of Mathematics, Ramanujan School of Mathematics, Pondicherry University, Kalapet, Pondicherry, 605 014, India

    T. Kathiravan & S. N. Fathima

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  1. T. Kathiravan
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  2. S. N. Fathima
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Correspondence to S. N. Fathima.

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The first author research is supported by UGC-BSR, Research Fellowship, New Delhi, Government of India.

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Kathiravan, T., Fathima, S.N. On \(\ell \)-regular bipartitions modulo \(\ell \) . Ramanujan J 44, 549–558 (2017). https://doi.org/10.1007/s11139-016-9850-9

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