Arithmetic properties of DSOME function

TL;DR

This paper explores the arithmetic properties of the DSOME function, providing a closed-form generating function and discovering new congruences modulo 4 and 8, advancing understanding of partition-related functions.

Contribution

It derives a closed-form generating function for DSOME(n) and uncovers new congruences, extending previous work on partition functions.

Findings

New congruences modulo 4 and 8 for DSOME(n)

Closed-form generating function for DSOME(n)

Enhanced understanding of partition function properties

Abstract

Recently, Andrews and Ghosh Dastidar (Ramanujan J. \textbf{69}, Art. No. 26, 2026) studied two interesting functions $SOME(n)$ and $DSOME(n)$, where $SOME(n)$ is the sum of all the odd parts in the partitions of $n$ minus the sum of all even parts and $DSOME(n)$ is the sum of all the odd parts in the partitions of $n$ into distinct parts minus sum of all the even parts. They expressed the generating functions of $SOME(n)$ and $DSOME(n)$ in terms of $q$-series and found several interesting congruences modulo 4 and 5. In this paper, we express the generating function of $DSOME(n)$ in a closed form, which allows us to find some new congruences and internal congruences modulo 4 and 8 for $DSOME(n)$.