Browse > Article
http://dx.doi.org/10.4134/JKMS.j220011

NEW CONGRUENCES FOR ℓ-REGULAR OVERPARTITIONS  

Jindal, Ankita (Indian Statistical Institute)
Meher, Nabin K. (Birla Institute of Technology and Science Pilani, Hyderbad Campus)
Publication Information
Journal of the Korean Mathematical Society / v.59, no.5, 2022 , pp. 945-962 More about this Journal
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
Partition functions; regular overpartitions; theta function; congruences;
Citations & Related Records
연도 인용수 순위
  • Reference
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   DOI
2 G. E. Andrews, Singular overpartitions, Int. J. Number Theory 11 (2015), no. 5, 1523-1533. https://doi.org/10.1142/S1793042115400059   DOI
3 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   DOI
4 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   DOI
5 B. C. Berndt, Ramanujan's Notebooks. Part III, Springer-Verlag, New York, 1991. https://doi.org/10.1007/978-1-4612-0965-2   DOI
6 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   DOI
7 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   DOI
8 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   DOI
9 M. D. Hirschhorn and J. A. Sellers, Arithmetic relations for overpartitions, J. Combin. Math. Combin. Comput. 53 (2005), 65-73.
10 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   DOI
11 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   DOI
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   DOI
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   DOI
14 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   DOI
15 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   DOI
16 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   DOI
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   DOI
19 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   DOI