Some new congruences and identities for $SOME(n)$, $DSOME(n)$, $\overline{SOME}(n)$ functions and analogues

Abstract

Andrews and Dastidar (\textit{Ramanujan J. 69, Article Number 26, (2026)} ) introduced the $SOME(n)$ and $DSOME(n)$ functions that calculate the sum of all odd parts minus the sum of all even parts of ordinary partitions and distinct partitions, respectively of a positive integer $n$, and proved their generating functions and some congruences modulo 4 and 5. Recently, Gireesh and Hemanthkumar introduced an overpartition analogue of $SOME(n)$ function, denoted by $\overline{SOME}(n)$ and proved some congruences modulo 3, 5 and powers of 2. In this paper, we prove some new identities and congruences for $SOME(n)$, $DSOME(n)$, and $\overline{SOME}(n)$ functions, including monotonicity results. We also define a general analogue of $SOME(n)$ function, denoted by $S_{\mathcal P}(n)$, which calculates the sum of all odd parts minus the sum of all even parts in any arbitrary family of partitions $\mathcal P(n)$ of a positive integer $n$, and prove some divisibility properties. Additionally, we define a colour partition analogue of $SOME(n)$ function and prove divisibility properties.