Congruences modulo powers of 2 for Overpartition pairs into odd parts
J. Korean Math. Soc.
Published online June 12, 2019
Zakir Ahmed, Rupam Barman, and Chiranjit Ray
Barnagar College, 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 : 11P83; 05A15; 05A17
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