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

Imdadul Hussain, Suparno Ghoshal, Arijit Jana

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.

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 , which enumerates overpartitions in which even parts may appear in one of colors and odd parts in one of colors, for fixed integers . They also proposed several conjectures concerning families of congruences modulo powers of for specific arithmetic progressions of . In this paper, we provide an elementary proof of this conjecture that relies only on classical -series manipulations and properties of Ramanujan's theta function.
Paper Structure (2 sections, 1 theorem, 25 equations)

This paper contains 2 sections, 1 theorem, 25 equations.

Table of Contents

  1. Introduction
  2. Proof of Conjecture \ref{['conj']}

Key Result

Theorem 1.2

Conjecture conj is true.

Theorems & Definitions (2)

  • Conjecture 1.1
  • Theorem 1.2