Congruences for Overpartition Pairs with Restricted Odd Differences

Jump To References Section

Authors

  • M. S. Mahadeva Naika
     ,IN
  • T. Harishkumar
     ,IN

DOI:

https://doi.org/10.18311/jims/2022/26254

Keywords:

Congruences, Overpartitions, Restricted Odd Differences.
11P83, 05A17

Abstract

Let b-(k) (n) denote the number of overpartition pairs of n where (i) consecutive parts di?er by a multiple of k + 1 unless the larger of the two is overlined, and (ii) the smallest part is overlined unless it is divisible by k+1. We prove many in?nite families of congruences modulo powers of 2 and 3 for b-(2) (n) and congruences modulo 4 and 5 for b-(4) (n). For example, for all n ? 0 and ?,? ? 0,

 

b-(4)(4·34? ·52?(5n + i) + 34? ·52?)? 0 (mod 5),

where i = 3,4.

Downloads

Download data is not yet available.

Metrics

Metrics Loading ...

Downloads

Published

2022-08-23

How to Cite

Naika, M. S. M., & Harishkumar, T. (2022). Congruences for Overpartition Pairs with Restricted Odd Differences. The Journal of the Indian Mathematical Society, 89(3-4), 353–371. https://doi.org/10.18311/jims/2022/26254

Issue

Volume 89, Issue 3-4, July-December 2022

Section

Articles

License

Copyright (c) 2022 M. S. Mahadeva Naika, T. Harishkumar

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

Crossref
Scopus
Google Scholar
Europe PMC
Received 2020-10-13
Accepted 2021-12-27
Published 2022-08-23

 

References

C. Adiga, B. C. Berndt, S. Bhargava and G. N. Watson, Chapter 16 of Ramanujan’s Second Notebook:Theta functions and q-series, Mem. Amer. Math. Soc., 315 (1985), 1–91. DOI: https://doi.org/10.1090/memo/0315

N. D. Baruah and K. K. Ojah, Partitions with designated summands in which all parts are odd, Integers, 15 (2015), #A9.

B. C. Berndt, Ramanujan’s Notebooks Part III, Springer-Verlag, New York, 1991. DOI: https://doi.org/10.1007/978-1-4612-0965-2

K. Bringmann, J. Dousse, J. Lovejoy and K. Mahlburg, Overpartitions with restricted odd di?erences, Electron. J. Combin., 22 (3) (2015), #P.3.17. DOI: https://doi.org/10.37236/5248

S. Chern and L. J. Hao, Congruences for two restricted overpartitions, Proc. Indian Acad. Sci. (Math. Sci.), (2019), 129:31. DOI: https://doi.org/10.1007/s12044-019-0474-z

M. D. Hirschhorn, The Power of q, Springer International Publishing, Switzerland, 2017.

M. D. Hirschhorn and J. A. Sellers, Elementary proofs of parity results for 5-regular partitions, Bull. Aust. Math. Soc., 81 (2010), 58–63. DOI: https://doi.org/10.1017/S0004972709000525

M. D. Hirschhorn and J. A. Sellers, Arithmetic properties of partitions with odd distinct, Ramanujan J., 22 (2010), 273–284. DOI: https://doi.org/10.1007/s11139-010-9225-6

M. D. Hirschhorn and J. A. Sellers, Congruences for overpartitions with restricted odd di?erences, Ramanujan J., 53 (2020), 167–180. DOI: https://doi.org/10.1007/s11139-019-00156-x

M. S. Mahadeva Naika and D. S. Gireesh, Congruences for overpartitions with restricted odd di?erences, Afr. Mat., 30 (2019), 1–21. DOI: https://doi.org/10.1007/s13370-018-0624-y

S. Ramanujan, Collected Papers, Cambridge University Press, 1927; reprinted by Chelsea, New York, 1962; reprinted by the American Mathematical Society, RI, 2000.

P. C. Toh, Ramanujan type identities and congruences for partition pairs, Discrete Math., 312 (2012), 1244–1250. DOI: https://doi.org/10.1016/j.disc.2011.10.013