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

TL;DR

This paper proves a conjecture about overcolored partitions with parity restrictions, using classical q-series and theta function properties, confirming specific congruences modulo powers of 2.

Contribution

It provides an elementary proof of a conjecture on overcolored partition functions involving parity, using classical q-series techniques.

Findings

Confirmed conjectures on congruences modulo powers of 2

Established properties of overcolored partition functions

Demonstrated elementary proof techniques for partition congruences

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 $r,s \geq 1$. 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.