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On congruence conjectures of Andrews and Bachraoui

Koustav Banerjee, Kathrin Bringmann, Mohamed El Bachraoui

Abstract

Andrews and the third author recently studied congruences for certain restricted two-color partitions. They made two conjectures for Ramanujan-type congruences and a vanishing identity for the limiting sequence. In this paper, we settle these conjectures by relating the corresponding generating function to modular forms and mock theta functions.

On congruence conjectures of Andrews and Bachraoui

Abstract

Andrews and the third author recently studied congruences for certain restricted two-color partitions. They made two conjectures for Ramanujan-type congruences and a vanishing identity for the limiting sequence. In this paper, we settle these conjectures by relating the corresponding generating function to modular forms and mock theta functions.

Paper Structure

This paper contains 8 sections, 15 theorems, 81 equations.

Table of Contents

  1. Introduction and Statements of results
  2. Preliminaries
  3. Proof of \ref{['lem1']}
  4. Proof of \ref{['ABConj1']}
  5. Proof of \ref{['ABConj1a']}
  6. Proof of \ref{['newthm']}
  7. Proof of \ref{['ABConj2']}
  8. Open questions

Key Result

Theorem 1.1

We have $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (28)

  • Theorem 1.1
  • Conjecture 1.2
  • Conjecture 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Conjecture 1.7
  • Theorem 1.8
  • Lemma 2.1
  • Lemma 2.2
  • ...and 18 more