Abstract
In recent work, Bringmann et al. used q-difference equations to compute a two-variable q-hypergeometric generating function for the number of overpartitions where (i) the difference between two successive parts may be odd only if the larger of the two is overlined, and (ii) if the smallest part is odd then it is overlined, given by \(\overline{t}(n)\). They also established the two-variable generating function for the same overpartitions where (i) consecutive parts differ by a multiple of \((k+1)\) unless the larger of the two is overlined, and (ii) the smallest part is overlined unless it is divisible by \(k+1\), enumerated by \(\overline{t}^{(k)}(n)\). As an application they proved that \(\overline{t}(n)\equiv 0\pmod {3}\) if n is not a square. In this paper, we extend the study of congruence properties of \(\overline{t}(n)\), and we prove congruences modulo 3 and 6 for \(\overline{t}(n)\), congruences modulo 2 and 4 for \(\overline{t}^{(3)}(n)\) and \(\overline{t}^{(7)}(n)\), congruences modulo 4 and 5 for \(\overline{t}^{(4)}(n)\), and congruences modulo 3, 6 and 12 for \(\overline{t}^{(8)}(n)\).
Similar content being viewed by others
References
Ahmed, Z., Baruah, N.D.: New congruences for Andrews’ singular overpartitions. Int. J. Number Theory (2015). https://doi.org/10.1142/S1793042115501018
Andrews, G.E.: The Theory of Partitions. Cambridge Univ. Press, Cambridge (1998)
Andrews, G.E.: Singular overpartitions. Int. J. Number Theory 5(11), 1523–1533 (2015)
Baruah, N.D., Ojah, K.K.: Analogues of Ramanujan’s partition identities and congruences arising from his theta functions and modular equations. Ramanujan J. 28, 385–407 (2012)
Berndt, B.C.: Ramanujan’s Notebooks, Part III. Springer, New York (1991)
Bringmann, K., Dousse, J., Lovejoy, J., Mahlburg, K.: Overpartitions with restricted odd differences. Electron. J. Comb. 22(3), 3–17 (2015)
Chen, S.C., Hirschhorn, M.D., Sellers, J.A.: Arithmetic properties of Andrews’ singular overpartitions. Int. J. Number Theory 5(11), 1463–1476 (2015)
Chern, S., Hao, L.-J.: Congruences for two restricted overpartitions. Proc. Indian Acad. Sci. Math. Sci. (to appear)
Corteel, S., Lovejoy, J.: Overpartitions. Trans. Am. Math. Soc. 356, 1623–1635 (2004)
Cui, S.P., Gu, N.S.S.: Arithmetic properties of \(l\)-regular partitions. Adv. Appl. Math. 51, 507–523 (2013)
Hirschhorn, M.D., Garvan, F., Borwein, J.: Cubic analogs of the Jacobian cubic theta function \(\theta (z, q)\). Can. J. Math. 45, 673–694 (1993)
Hirschhorn, M.D., Sellers, J.A.: Elementary proofs of parity results for \(5\)-regular partitions. Bull. Aust. Math. Soc. 81, 58–63 (2010)
Mahadeva Naika, M.S., Gireesh, D.S.: Congruences for Andrew’s singular overpartitions. J. Number Theory 165, 109–130 (2016)
Ramanujan, S.: Some properties of \(p(n)\), the number of partitions of \(n\). Proc. Camb. Philos. Soc. 19, 207–210 (1919)
Ramanujan, S.: Collected Papers. Cambridge Univ. Press, Cambridge (1927)
Xia, E.X.W.: New congruences modulo powers of \(2\) for broken \(3\)-diamond partitions and \(7\)-core partitions. J. Number Theory 141, 119–135 (2014)
Xia, E.X.W., Yao, O.X.M.: New Ramanujan-like congruences modulo powers of 2 and 3 for overpartitions. J. Number Theory 133, 1932–1949 (2013)
Xia, E.X.W., Yao, O.X.M.: Some modular relations for the Göllnitz–Gordon functions by an even-odd method. J. Math. Anal. Appl. 387, 126–138 (2012)
Acknowledgements
The authors would like to thank the referees for helpfull suggestions and comments.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Naika, M.S.M., Gireesh, D.S. Congruences for overpartitions with restricted odd differences. Afr. Mat. 30, 1–21 (2019). https://doi.org/10.1007/s13370-018-0624-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s13370-018-0624-y