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.
Zakir Ahmed, Rupam Barman, Chiranjit Ray
Barnagar College; Indian Institute of Technology Guwahati; Indian Institute of Technology Guwahati
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.
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