On a conjecture of Andrews and Bachraoui

TL;DR

This paper proves the non-negativity of coefficients for a specific generating function related to two-color partitions for certain parameters, and explores its connection to Ramanujan's mock theta functions and q-binomial quotients.

Contribution

It extends the proof of non-negativity of the generating function's coefficients to a broader range of parameters and links it to classical mock theta functions.

Findings

Proves non-negativity for 5 ≤ k ≤ 10

Establishes connection to Ramanujan's third order mock theta function

Relates the generating function to quotients of q-binomial coefficients

Abstract

Recently, Andrews and Bachraoui considered a generating function $F_{k,m}(q)$ associated with certain two-color partitions, and conjectured that this function has non-negative coefficients for $m=1$. They showed this property for $1 \leq k \leq 4$. In this note, we prove that $F_{k,1}(q)$ has non-negative coefficients for $5 \leq k \leq 10$. Moreover, we show that, as $k\to\infty$, $F_{k,1}(q)$ is related to Ramanujan's third order mock theta function $\omega(q)$ and to quotients of certain $q$-binomial coefficients.