Abstract
Let \(B_{l,m}(n)\) denote the number of (l, m)-regular bipartitions of n. Recently, many authors proved several infinite families of congruences modulo 3, 5 and 11 for \(B_{l,m}(n)\). In this paper, we use theta function identities to prove infinite families of congruences modulo m for (l, m)-regular bipartitions, where \(m\in \{7,3,11,13,17\}\).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
C. Adiga and D. Ranganatha, A simple proof of a conjecture of Dou on (3,7)-regular bipartitions modulo 3, Integers, 17, (2017) #A14.
G. E. Andrews, M. D. Hirschhorn and J. A. Sellers, Arithmetic properties of partitions with even parts distinct, Ramanujan J., 23, (2010) 169-181.
R. Carlson and J.J. Webb, Infinite families of congruences for k-regular partitions, Ramanujan J., 33, (2014) 329-337.
S.P. Cui and N.S.S. Gu, Arithmetic properties of the l-regular partitions, Adv. Appl. Math., 51, (2013) 507-523.
S.P. Cui and N.S.S. Gu, Congruences for 9-regular partitions modulo 3, Ramanujan J., 38, (2015) 503-512.
B. Dandurand and D. Penniston, l-divisibility of l-regular partition functions, Ramanujan J., 19, (2009) 63-70.
D.Q.J. Dou, Congruences for (3, 11)-regular bipartitions modulo 11, Ramanujan J., 40, (2016) 535-540.
D. Furcy and D. Penniston, Congruences for l-regular partition functions modulo 3, Ramanujan J., 27, (2012) 101-108.
B. Gordon and K. Ono, Divisibility of certain partition functions by powers of primes, Ramanujan J., 1, (1997) 25-34.
M.D. Hirschhorn, The Power of q. A Personal Journey, Developments in Mathematics, Vol. 49 (Springer, Cham, 2017), xxii+415 pp.
M.D. Hirschhorn and J.A. Sellers, Elementary proofs of parity results for 5-regular partitions, Bull. Aust. Math. Soc., 81, (2010) 58-63.
T. Kathiravan and S.N. Fathima, On l-regular bipartitions modulo l, Ramanujan J., 44, (2017) 549-558.
W.J. Keith, Congruences for 9-regular partitions modulo 3, Ramanujan J., 35, (2014) 157-164.
B.L.S. Lin, Arithmetic of the 7-regular bipartition function modulo 3, Ramanujan J., 37, (2015) 469-478.
B.L.S. Lin, An infinite family of congruences modulo 3 for 13-regular bipartitions, Ramanujan J., 39, (2016) 169-178.
L. Wang, (2017). Arithmetic properties of (k, l)-regular bipartitions, Bull. Aust. Math. Soc., 95, (2017) 353-364.
J.J. Webb, Arithmetic of the 13-regular partition function modulo 3, Ramanujan J., 25, (2011) 49-56.
E.X.W. **a, Congruences for some l-regular partitions modulo l, J. Number Theory, 152, (2015) 105-117.
E.X.W. **a and O.X.M. Yao, Parity results for 9-regular partitions, Ramanujan J., 34, (2014) 109-117.
E.X.W. **a and O.X.M. Yao, A proof of Keith’s conjecture for 9-regular partitions modulo 3, Int. J. Number Theory, 10, (2014) 669-674
E.X.W. **a and O.X.M. Yao, Arithmetic properties for (s, t)-regular bipartition functions, J. Number Theory, 171, (2017) 1-17.
O.X.M. Yao, New congruences modulo powers of 2 and 3 for 9-reguar partitions, J. Number Theory, 142, (2014) 89-101.
Acknowledgements
I extremely grateful to Professor Michael D. Hirschhorn, who read our manuscript with great care, uncovered several corrections and offered his valuable suggestions in this paper. I would like to thank Prof. K. Srinivas for some useful discussions which improve the presentation of the paper. This work is party supported by SERB MATRIX project No. MTR/2017/001006.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Sanoli Gun, Phd.
Rights and permissions
About this article
Cite this article
Kathiravan, T. Ramanujan-type congruences modulo m for (l, m)-regular bipartitions . Indian J Pure Appl Math 53, 375–391 (2022). https://doi.org/10.1007/s13226-021-00015-w
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s13226-021-00015-w