On an Overpartition Analogue of $SOME(n)$

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.

💡 Research Summary

The paper introduces an overpartition analogue of the weighted partition function SOME(n) originally defined by Andrews and Dastidar. For a non‑negative integer n, the new function (\overline{SOME}(n)) is defined as the total of all odd parts minus the total of all even parts taken over every overpartition of n (an overpartition allows the first occurrence of each distinct part to be overlined). The authors first recall the classical partition function p(n) and the known Ramanujan congruence (p(5n+4)\equiv0\pmod5), then present the generating function for overpartitions, \