Abstract
In this paper, we study various arithmetic properties of the function \(\overline{p}_{2,\,\, k}(n)\), which denotes the number of \((2,\,\, k)\)-regular overpartitions of n with odd \(k > 1\). We prove several infinite families of congruences modulo 8 for \(\overline{p}_{2,\,\, k}(n)\). For example, we find that for all non-negative integers \(\beta , n\) and \(k\equiv 1\pmod {8}\), \(\overline{p}_{2,\,\, k}(2^{1+\beta }(16n+14))\equiv ~0\pmod {8}\).
Article PDF
Similar content being viewed by others
References
Adiga, C.; Berndt, B.C.; Bhargava, S.; Watson, G.N.: Chapter 16 of Ramanujan’s second notebook: theta functions and \(q\)-series. Mem. Am. Math. Soc. 315, 1–91 (1985)
Adiga, C.; Ranganatha, D.: Congruences modulo powers of 2 for \(\ell \)-regular overpartitions. J. Ramanujan Math. Soc. 32, 147–163 (2017)
Alanazi, A.M.; Munagi, A.O.; Sellers, J.A.: An infinite family of congruences for \(\ell \)-regular overpartitions. Integers 16, #A37 (2016)
Chen, S.C.: On the number of overpartitions into odd parts. Discrete Math. 325, 32–37 (2014)
Corteel, S.; Lovejoy, J.: Overpartitions. Trans. Am. Math. Soc. 356, 1623–1635 (2004)
Cui, S.-P.; Gu, N.S.S.: Arithmetic properties of \(\ell \)-regular partition. Adv. Appl. Math. 51, 507–523 (2013)
Hirschhorn, M.D.; Sellers, J.A.: Arithmetic properties of overpartitions into odd parts. Ann. Comb. 10, 353–367 (2006)
Lin, B.L.S.: Arithmetic properties of overpartitions pairs into odd parts. Electron. J. Comb. 19(2), P17 (2012)
Lovejoy, J.: Gordon’s theorem for overpartitions. J. Comb. Theory Ser. A 103, 393–401 (2003)
Naika, M.S.M.; Gireesh, D.S.: Congruences for Andrews’s singular overpartitions. J. Number Theory 165, 109–130 (2016)
Naika, M.S.M.; Shivashankar, C.: Arithmetic propertities of \(\ell \)-regular overpartition pairs. Turk. J. Math. 41(3), 756–774 (2017)
Ranganatha, D.: On some new congruences for \(\ell \)-regular overpartitions. Palest. J. Math. 7(1), 345–362 (2018)
Shen, E.Y.Y.: Arithmetic properties of \(\ell \)-regular overpartitions. Int. J. Number Theory 12(3), 841–852 (2016)
Xia, E.X.W.; Yao, O.X.M.: Analogues of Ramanujan’s partition identities. Ramanujan J. 31, 373–396 (2013)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Adiga, C., Naika, M.S.M., Ranganatha, D. et al. Congruences modulo 8 for \((2,\, k)\)-regular overpartitions for odd \(k > 1\) . Arab. J. Math. 7, 61–75 (2018). https://doi.org/10.1007/s40065-017-0195-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40065-017-0195-z