Congruences for Overpartition Pairs with Restricted Odd Differences

Authors

  • M. S. Mahadeva Naika
  • T. Harishkumar

DOI:

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

Keywords:

Congruences, Overpartitions, Restricted Odd Differences.

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.

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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.

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.

K. Bringmann, J. Dousse, J. Lovejoy and K. Mahlburg, Overpartitions with restricted odd di?erences, Electron. J. Combin., 22 (3) (2015), #P.3.17.

S. Chern and L. J. Hao, Congruences for two restricted overpartitions, Proc. Indian Acad. Sci. (Math. Sci.), (2019), 129:31.

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.

M. D. Hirschhorn and J. A. Sellers, Arithmetic properties of partitions with odd distinct, Ramanujan J., 22 (2010), 273–284.

M. D. Hirschhorn and J. A. Sellers, Congruences for overpartitions with restricted odd di?erences, Ramanujan J., 53 (2020), 167–180.

M. S. Mahadeva Naika and D. S. Gireesh, Congruences for overpartitions with restricted odd di?erences, Afr. Mat., 30 (2019), 1–21.

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.

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