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New congruences for \(\ell \)-regular overpartition pairs

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Abstract

Let \(\overline{B}_{\ell }(n)\) denote the number of \(\ell -\) regular overpartition pairs of n. In this paper, we find some Ramanujan like congruences and infinite families of congruences for \(\overline{B}_{\ell }(n)\) for \(\ell \in \{3,4,5,8\}\). For example, for \(n \ge 0\) and \(\alpha \ge 0\), \(\overline{B}_{3}(4^{\alpha +1}(6n+5))\equiv 0\pmod {9}\), \(\overline{B}_{3}(12n+10)\equiv 0\pmod {9}\), \(\overline{B}_{5}(4n+3)\equiv 0\pmod {8}\) and \(\overline{B}_{8}(16n+12)\equiv 0\pmod {16}\).

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References

  1. Berndt, B.C.: Ramanujan’s Notebooks. Springer-Verlag, New York, NY, USA, Part III (1991)

  2. Bringmann, K., Lovejoy, J.: Rank and congruences for overpartition pairs. Int. J. Number Theory 4, 303–322 (2008)

    Article  MathSciNet  Google Scholar 

  3. Chen, W.Y.C., Lin, B.L.S.: Arithmetic properties of overpartition pairs. Acta Arith. 151, 263–277 (2012)

    Article  MathSciNet  Google Scholar 

  4. Cui, S.P., Gu, N.S.S.: Arithmetic properties of \(\ell \)-regular partitions. Adv. Appl. Math. 51, 507–523 (2013)

    Article  MathSciNet  Google Scholar 

  5. Hirschhorn, M.D., Sellers, J.A.: Arithmetic relations for overpartitions. J. Combin. Math. Combin. Comput. 53, 65–73 (2005)

    MathSciNet  MATH  Google Scholar 

  6. 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)

    Article  Google Scholar 

  7. Hirschhorn, M.D., Sellers, J.A.: Elementary proofs of parity results for \(5\)-regular partitions. Bull. Aust. Math. Soc. 81, 58–63 (2010)

    Article  MathSciNet  Google Scholar 

  8. Hirschhorn, M.D., and Sellers J.A.: A congruence modulo 3 for partitions into distinct non-multiples of four. J. Integer Sequences, 17, Article 14.9.6 (2014)

  9. Keith, W.J.: Congruences for \(9\)-regular partitions modulo 3. Ramanujan J. 35, 157–164 (2014)

    Article  MathSciNet  Google Scholar 

  10. Kim, B.: Overpartition pairs modulo powers of 2. Discrete Math. 311, 835–840 (2011)

    Article  MathSciNet  Google Scholar 

  11. Lovejoy, J.: Overpartition pairs. Ann. Inst. Fourier. 56, 781–794 (2006)

    Article  MathSciNet  Google Scholar 

  12. Lovejoy, J.: Gordon’s theorem for overpartitions. J. Combin. Theory Ser. A, 103, 393-401 (2003)

  13. Mahadeva Naika, M.S., Shivashankar, C.: Arithmetic properties of \(\ell \)-regular overpartition pairs. Turk. J. Math., 41, 756-774 (2017)

  14. Penniston, D.: Arithmetic of \(\ell \)-regular partition functions. Int. J. Number Theory 4, 295–302 (2008)

    Article  MathSciNet  Google Scholar 

  15. Shen, E.Y.Y.: Arithmetic properties of \(\ell -\)regular overpartition. Int. J. Number Theory 12, 841–852 (2016)

    Article  MathSciNet  Google Scholar 

  16. Webb, J.J.: Arithmetic of the 13-regular partition function modulo 3. Ramanujan J. 25, 49–56 (2011)

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Vidyavardhaka College of Engineering, Gokulam III Stage, Mysuru, 570002, India

    C. Shivashankar

  2. Department of Mathematics, M. S. Ramaiah University of Applied Sciences Peenya, Bengaluru, Karnataka, 560 058, India

    D. S. Gireesh

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  1. C. Shivashankar
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  2. D. S. Gireesh
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Correspondence to C. Shivashankar.

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Shivashankar, C., Gireesh, D.S. New congruences for \(\ell \)-regular overpartition pairs. Afr. Mat. 33, 82 (2022). https://doi.org/10.1007/s13370-022-01018-4

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