Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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NEW CONGRUENCES FOR ℓ-REGULAR OVERPARTITIONS

  • Jindal, Ankita (Indian Statistical Institute) ;
  • Meher, Nabin K. (Birla Institute of Technology and Science Pilani, Hyderbad Campus)
  • Received : 2022.01.07
  • Accepted : 2022.04.25
  • Published : 2022.09.01
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Abstract

For a positive integer ℓ, $\bar{A}_{\ell}(n)$ denotes the number of over-partitions of n into parts not divisible by ℓ. In this article, we find certain Ramanujan-type congruences for $\bar{A}_{r{\ell}}(n)$, when r ∈ {8, 9} and we deduce infinite families of congruences for them. Furthermore, we also obtain Ramanujan-type congruences for $\bar{A}_{13}(n)$ by using an algorithm developed by Radu and Sellers [15].

Keywords

Acknowledgement

The first author is supported by ISI Delhi Post doctoral fellowship. The second author is thankful to BITS Pilani, Hyderabad campus for providing warm hospitality, nice facilities for research and computing facility. The second author is supported by ISI Delhi Post doctoral fellowship.

References

  1. Z. Ahmed and N. D. Baruah, New congruences for Andrews' singular overpartitions, Int. J. Number Theory 11 (2015), no. 7, 2247-2264. https://doi.org/10.1142/S1793042115501018
  2. 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
  3. G. E. Andrews, Singular overpartitions, Int. J. Number Theory 11 (2015), no. 5, 1523-1533. https://doi.org/10.1142/S1793042115400059
  4. R. Barman and C. Ray, Congruences for ℓ-regular overpartitions and Andrews' singular overpartitions, Ramanujan J. 45 (2018), no. 2, 497-515. https://doi.org/10.1007/s11139-016-9860-7
  5. N. D. Baruah and K. K. Ojah, Analogues of Ramanujan's partition identities and congruences arising from his theta functions and modular equations, Ramanujan J. 28 (2012), no. 3, 385-407. https://doi.org/10.1007/s11139-011-9296-z
  6. B. C. Berndt, Ramanujan's Notebooks. Part III, Springer-Verlag, New York, 1991. https://doi.org/10.1007/978-1-4612-0965-2
  7. S.-C. Chen, M. D. Hirschhorn, and J. A. Sellers, Arithmetic properties of Andrews' singular overpartitions, Int. J. Number Theory 11 (2015), no. 5, 1463-1476. https://doi.org/10.1142/S1793042115400011
  8. S. Corteel and J. Lovejoy, Overpartitions, Trans. Amer. Math. Soc. 356 (2004), no. 4, 1623-1635. https://doi.org/10.1090/S0002-9947-03-03328-2
  9. S.-P. Cui and N. S. S. Gu, Arithmetic properties of ℓ-regular partitions, Adv. in Appl. Math. 51 (2013), no. 4, 507-523. https://doi.org/10.1016/j.aam.2013.06.002
  10. L. Euler, Introduction to Analysis of the Infinite. Book I, translated from the Latin and with an introduction by John D. Blanton, Springer-Verlag, New York, 1988. https://doi.org/10.1007/978-1-4612-1021-4
  11. M. D. Hirschhorn and J. A. Sellers, Arithmetic relations for overpartitions, J. Combin. Math. Combin. Comput. 53 (2005), 65-73.
  12. J. Lovejoy, Gordon's theorem for overpartitions, J. Combin. Theory Ser. A 103 (2003), no. 2, 393-401. https://doi.org/10.1016/S0097-3165(03)00116-X
  13. M. S. Mahadeva Naika and D. S. Gireesh, Congruences for Andrews' singular overpartitions, J. Number Theory 165 (2016), 109-130. https://doi.org/10.1016/j.jnt.2016.01.015
  14. S. Radu, An algorithmic approach to Ramanujan's congruences, Ramanujan J. 20 (2009), no. 2, 215-251. https://doi.org/10.1007/s11139-009-9174-0
  15. S. Radu and J. A. Sellers, Congruence properties modulo 5 and 7 for the pod function, Int. J. Number Theory 7 (2011), no. 8, 2249-2259. https://doi.org/10.1142/S1793042111005064
  16. C. Ray and K. Chakraborty, Certain eta-quotients and ℓ-regular overpartitions, Ramanujan J. 57 (2022), no. 2, 453-470. https://doi.org/10.1007/s11139-020-00322-6
  17. The Sage Developers, Sagemath, the Sage Mathematics Software System (Version 8.1). https://www.sagemath.org
  18. E. Y. Y. Shen, Arithmetic properties of l-regular overpartitions, Int. J. Number Theory 12 (2016), no. 3, 841-852. https://doi.org/10.1142/S1793042116500548
  19. L. Wang, Arithmetic properties of (k, ℓ)-regular bipartitions, Bull. Aust. Math. Soc. 95 (2017), no. 3, 353-364. https://doi.org/10.1017/S0004972716000964