Arithmetic properties of the DSOME function
Abstract
Recently, Andrews and Ghosh Dastidar [Ramanujan J. 69 (2026), art. 26] studied two interesting functions $\mathrm {SOME}(n)$ and $\mathrm {DSOME}(n)$, where $\mathrm {SOME}(n)$ is the sum of all the odd parts in the partitions of $n$ minus the sum of all the even parts, and $\mathrm {DSOME}(n)$ is the sum of all the odd parts in the partitions of $n$ into distinct parts minus the sum of all the even parts. They expressed the generating functions of $\mathrm {SOME}(n)$ and $\mathrm {DSOME}(n)$ in terms of $q$-series and found several interesting congruences modulo 4 and 5. In this paper, we express the generating function of $\mathrm {DSOME}(n)$ in closed form, which allows us to find some new congruences and internal congruences modulo 4 and 8 for $\mathrm {DSOME}(n)$.
