Proof of a Conjecture on Overcolored Partition Restricted by Parity of the Parts

Abstract

In a recent paper, Thejitha and Fathima introduced the overcolored partition function ${\bar{a}}_{r,s}(n)$, which enumerates overpartitions in which even parts may appear in one of $r$ colors and odd parts in one of $s$ colors, for fixed integers $r,s\ge 1$. They also proposed several conjectures concerning families of congruences modulo powers of $2$ for specific arithmetic progressions of ${\bar{a}}_{r,s}(n)$. In this paper, we provide an elementary proof of this conjecture that relies only on classical $q$-series manipulations and properties of Ramanujan's theta function.