DOI QR코드

DOI QR Code

ON 3k-REGULAR CUBIC PARTITIONS

  • Received : 2021.08.27
  • Accepted : 2022.02.04
  • Published : 2022.07.01

Abstract

Recently, Gireesh, Shivashankar, and Naika [11] found some infinite classes of congruences for the 3- and the 9-regular cubic partitions modulo powers of 3. We extend their study to all the 3k-regular cubic partitions. We also find new families of congruences.

Keywords

Acknowledgement

The authors are thankful to the anonymous reviewer for his/her helpful comments on the paper.

References

  1. Z. Ahmed and N. D. Baruah, New congruences for ℓ-regular partitions for ℓ ∈ {5, 6, 7, 49}, Ramanujan J. 40 (2016), no. 3, 649-668. https://doi.org/10.1007/s11139-015-9752-2
  2. N. D. Baruah and H. Das, Generating functions and congruences for 9-regular and 27-regular partitions in 3 colours, Hardy-Ramanujan J. 44 (2021), 101-115.
  3. N. D. Baruah and K. Nath, Infinite families of arithmetic identities and congruences for bipartitions with 3-cores, J. Number Theory 149 (2015), 92-104. https://doi.org/10.1016/j.jnt.2014.10.010
  4. B. C. Berndt, Ramanujan's notebooks. Part III, Springer-Verlag, New York, 1991. https://doi.org/10.1007/978-1-4612-0965-2
  5. R. Carlson and J. J. Webb, Infinite families of infinite families of congruences for kregular partitions, Ramanujan J. 33 (2014), no. 3, 329-337. https://doi.org/10.1007/s11139-013-9523-x
  6. H.-C. Chan, Ramanujan's cubic continued fraction and an analog of his "most beautiful identity", Int. J. Number Theory 6 (2010), no. 3, 673-680. https://doi.org/10.1142/S1793042110003150
  7. H.-C. Chan, Ramanujan's cubic continued fraction and Ramanujan type congruences for a certain partition function, Int. J. Number Theory 6 (2010), no. 4, 819-834. https://doi.org/10.1142/S1793042110003241
  8. H.-C. Chan, Distribution of a certain partition function modulo powers of primes, Acta Math. Sin. (Engl. Ser.) 27 (2011), no. 4, 625-634. https://doi.org/10.1007/s10114-011-8620-2
  9. H. H. Chan and P. C. Toh, New analogues of Ramanujan's partition identities, J. Number Theory 130 (2010), no. 9, 1898-1913. https://doi.org/10.1016/j.jnt.2010.02.017
  10. B. Dandurand and D. Penniston, ℓ-divisibility of ℓ-regular partition functions, Ramanujan J. 19 (2009), no. 1, 63-70. https://doi.org/10.1007/s11139-007-9042-8
  11. D. S. Gireesh, C. Shivashankar, and M. S. M. Naika, On 3- and 9-regular cubic partitions, J. Integer Seq. 23 (2020), no. 7, Art. 20.7.2, 12 pp.
  12. M. D. Hirschhorn, Cubic partitions modulo powers of 5, Ramanujan J. 51 (2020), no. 1, 67-84. https://doi.org/10.1007/s11139-018-0074-z
  13. M. Hirschhorn, F. Garvan, and J. Borwein, Cubic analogues of the Jacobian theta function θ (z, q), Canad. J. Math. 45 (1993), no. 4, 673-694. https://doi.org/10.4153/CJM1993-038-2
  14. Q.-H. Hou, L. H. Sun, and L. Zhang, Quadratic forms and congruences for ℓ-regular partitions modulo 3, 5 and 7, Adv. in Appl. Math. 70 (2015), 32-44. https://doi.org/10.1016/j.aam.2015.06.005
  15. B. L. S. Lin, Arithmetic of the 7-regular bipartition function modulo 3, Ramanujan J. 37 (2015), no. 3, 469-478. https://doi.org/10.1007/s11139-013-9542-7
  16. S. Ramanujan, Congruence properties of partitions, Proc. London Math. Soc. 18 (1920), 19.
  17. S. Ramanujan, Congruence properties of partitions, Math. Z. 19 (1921), no. 1-2, 147-153. https://doi.org/10.1007/BF01378341
  18. S. Ramanujan, The lost notebook and other unpublished papers, Narosa, New Delhi, 1988.
  19. S. Ramanujan, Some properties of p(n), the number of partitions of n, Proc. Cambridge Philos. Soc. 19 (1919), 207-210.
  20. J. J. Webb, Arithmetic of the 13-regular partition function modulo 3, Ramanujan J. 25 (2011), no. 1, 49-56. https://doi.org/10.1007/s11139-010-9227-4
  21. X. Xiong, The number of cubic partitions modulo powers of 5, Sci. Sin. Math. 41 (2011), no. 1, 1-15. https://doi.org/10.1360/012010-517