Abstract
We find several new congruences for \(\ell \)-regular partitions for \(\ell \in \{5,6,7,49\}\) and also find alternative proofs of the congruences for 10- and 20-regular partitions which were proved earlier by Carlson and Webb (Ramanujan J 33:329–337, 2014) by using the theory of modular forms. We use certain p-dissections of \((q;q)_{\infty }\), \(\psi (q)\), \((q;q)_{\infty }^3\) and \(\psi (q^2)(q;q)_{\infty }^2\).
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References
Baruah, N.D., Das, K.: Parity results for 7-regular and 23-regular partitions. Int. J. Number Theory 11, 2221–2238 (2015)
Berndt, B.C.: Ramanujan’s Notebooks. Part III. Springer, New York (1991)
Berndt, B.C.: Number Theory in the Spirit of Ramanujan. American Mathematical Society, Providence (2006)
Carlson, R., Webb, J.J.: Infinite families of congruences for \(k\)-regular partitions. Ramanujan J. 33, 329–337 (2014)
Cui, S.P., Gu, N.S.S.: Arithmetic properties of \(\ell \)-regular partitions. Adv. Appl. Math. 51, 507–523 (2013)
Dandurand, B., Penniston, D.: \(\ell \)-Divisibility of \(\ell \)-regular partition functions. Ramanujan J. 19, 63–70 (2009)
Furcy, D., Penniston, D.: Congruences for \(\ell \)-regular partitions modulo \(3\). Ramanujan J. 27, 101–108 (2012)
Garvan, F.G.: A simple proof of Watson’s partition congruences for powers of \(7\). J. Aust. Math. Soc. A 36, 316–334 (1984)
Hou, Q.-H., Sun, L.H., Zhang, L.: Quadratic forms and congruences for \(\ell \)-regular partitions modulo 3, 5 and 7. Adv. Appl. Math. 70, 32–44 (2015)
Lin, B.L.S.: An infinite family of congruences modulo 3-regular bipartitions. Ramanujan J. 2014. doi:10.1007/s11139-014-9610-7
Lin, B.L.S.: Arithmetic of the \(7\)-regular bipartition function modulo \(3\). Ramanujan J. 37, 469–478 (2015)
Webb, J.J.: Arithmetic of the 13-regular partition function modulo 3. Ramanujan J. 25, 49–56 (2011)
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The authors thank the anonymous referee for his/her insightful comments.
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The first author is supported by a fellowship of UGC-SAP (DRS-I), Mathematical Sciences.
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Ahmed, Z., Baruah, N.D. New congruences for \(\ell \)-regular partitions for \(\ell \in \{5,6,7,49\}\) . Ramanujan J 40, 649–668 (2016). https://doi.org/10.1007/s11139-015-9752-2
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DOI: https://doi.org/10.1007/s11139-015-9752-2