Journal of the
Korean Mathematical Society
JKMS

ISSN(Print) 0304-9914 ISSN(Online) 2234-3008

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J. Korean Math. Soc. 2020; 57(2): 471-487

Online first article July 17, 2019      Printed March 1, 2020

https://doi.org/10.4134/JKMS.j190143

Copyright © The Korean Mathematical Society.

Congruences modulo powers of 2 for overpartition pairs into odd parts

Zakir Ahmed, Rupam Barman, Chiranjit Ray

Barnagar College; Indian Institute of Technology Guwahati; Indian Institute of Technology Guwahati

Abstract

We find congruences modulo $32$, $64$ and $128$ for the partition function $\overline{pp}_o(n)$, the number of overpartition pairs of $n$ into odd parts, with the aid of Ramamnujan's theta function identities and some known identities of $t_k(n)$, for $k=6, 7$, where $t_k(n)$ denotes the number of representations of $n$ as a sum of $k$ triangular numbers. We also find two Ramanujan-like congruences for $\overline{pp}_o(n)$ modulo $128$.

Keywords: Partition, $p$-dissection, theta function, triangular numbers, congruence

MSC numbers: Primary 11P83; Secondary 05A15, 05A17

Supported by: The first author acknowledges the financial support of SERB, Department of Science and Technology, Government of India.
The third author acknowledges the financial support of Department of Atomic Energy, Government of India for supporting a part of this work under NBHM Fellowship.