Advertisement

Arabian Journal of Mathematics

, Volume 7, Issue 2, pp 61–75 | Cite as

Congruences modulo 8 for \((2,\, k)\)-regular overpartitions for odd \(k > 1\)

  • Chandrashekar Adiga
  • M. S. Mahadeva Naika
  • D. Ranganatha
  • C. Shivashankar
Open Access
Article
  • 202 Downloads

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 \(k\equiv 1\pmod {8}\), \(\overline{p}_{2,\,\, k}(2^{1+\beta }(16n+14))\equiv ~0\pmod {8}\).

Mathematics Subject Classification

05A15 05A17 11P83 

Notes

References

  1. 1.
    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)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Adiga, C.; Ranganatha, D.: Congruences modulo powers of 2 for \(\ell \)-regular overpartitions. J. Ramanujan Math. Soc. 32, 147–163 (2017)MathSciNetGoogle Scholar
  3. 3.
    Alanazi, A.M.; Munagi, A.O.; Sellers, J.A.: An infinite family of congruences for \(\ell \)-regular overpartitions. Integers 16, #A37 (2016)Google Scholar
  4. 4.
    Chen, S.C.: On the number of overpartitions into odd parts. Discrete Math. 325, 32–37 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Corteel, S.; Lovejoy, J.: Overpartitions. Trans. Am. Math. Soc. 356, 1623–1635 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Cui, S.-P.; Gu, N.S.S.: Arithmetic properties of \(\ell \)-regular partition. Adv. Appl. Math. 51, 507–523 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Hirschhorn, M.D.; Sellers, J.A.: Arithmetic properties of overpartitions into odd parts. Ann. Comb. 10, 353–367 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Lin, B.L.S.: Arithmetic properties of overpartitions pairs into odd parts. Electron. J. Comb. 19(2), P17 (2012)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Lovejoy, J.: Gordon’s theorem for overpartitions. J. Comb. Theory Ser. A 103, 393–401 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Naika, M.S.M.; Gireesh, D.S.: Congruences for Andrews’s singular overpartitions. J. Number Theory 165, 109–130 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Naika, M.S.M.; Shivashankar, C.: Arithmetic propertities of \(\ell \)-regular overpartition pairs. Turk. J. Math. 41(3), 756–774 (2017)CrossRefGoogle Scholar
  12. 12.
    Ranganatha, D.: On some new congruences for \(\ell \)-regular overpartitions. Palest. J. Math. 7(1), 345–362 (2018)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Shen, E.Y.Y.: Arithmetic properties of \(\ell \)-regular overpartitions. Int. J. Number Theory 12(3), 841–852 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Xia, E.X.W.; Yao, O.X.M.: Analogues of Ramanujan’s partition identities. Ramanujan J. 31, 373–396 (2013)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© The Author(s) 2017

Open AccessThis 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.

Authors and Affiliations

  1. 1.Department of Studies in MathematicsUniversity of MysoreMysoreIndia
  2. 2.Department of MathematicsBangalore UniversityBangaloreIndia
  3. 3.Department of MathematicsSiddaganga Institute of TechnologyTumakuruIndia

Personalised recommendations