On an Overpartition Analogue of $SOME(n)$

TL;DR

This paper introduces an overpartition analogue of the $SOME(n)$ function, derives its generating function, and establishes new congruences using classical $q$-series techniques.

Contribution

It defines $ar{SOME}(n)$, provides its generating function, and finds new congruences modulo 3, 5, and powers of 2, extending previous work on $SOME(n)$.

Findings

Derived the generating function for $ar{SOME}(n)$

Established congruences modulo 3, 5, and powers of 2

Extended the theory of partition functions using $q$-series

Abstract

Recently, Andrews and Dastidar introduced the partition function $SOME(n)$, defined as the sum of all the odd parts in the partitions of $n$ minus the sum of all the even parts in the partitions of $n$. They derived its generating function and established some congruences satisfied by \(SOME(n)\). In this paper, we introduce an overpartition analogue of $SOME(n)$, denoted by $\overline{SOME}(n)$, the sum of all the odd parts in the overpartitions of \(n\) minus the sum of all the even parts in the overpartitions of \(n\). We derive the generating function for $\overline{SOME}(n)$ and obtain congruences modulo \(3, \ 5\) and powers of \(2\). Our method is based on classical $q$-series identities and manipulations of infinite products and sums.