Proof of a conjecture of Banerjee,Bringmann and Bachraoui on infinite families of congruences

TL;DR

This paper proves a conjecture on infinite families of congruences for a limiting sequence, building on recent work relating generating functions to modular forms and mock theta functions.

Contribution

It confirms a conjecture by Banerjee, Bringmann, and Bachraoui on infinite congruence families, extending their previous results.

Findings

Confirmed the conjecture on infinite families of congruences modulo 4 and 8.

Connected the conjecture to modular forms and mock theta functions.

Extended previous results on congruences for restricted two-color partitions.

Abstract

Recently, Andrews and Bachraoui investigated congruences for certain restricted two-color partitions. They made two conjectures for Ramanujan type congruences and a vanishing identity for the limiting sequence. Very recently, Banerjee, Bringmann and Bachraoui confirmed these three conjectures by relating the corresponding generating function to modular forms and mock theta functions. At the end of their paper, they posed a conjecture on infinite families of congruences modulo 4 and 8 for the limiting sequence. The Banerjee-Bringmann-Bachraoui's conjecture implies the two conjectures given by Andrews and Bachraoui. In this note, we settle Banerjee-Bringmann-Bachraoui's conjecture on infinite famlies of congruences based on Banerjee-Bringmann-Bachraoui's results and an identity due to Waston.