On a conjecture of Andrews and Bachraoui

TL;DR

The paper advances the Andrews–Bachraoui conjecture by proving that the generating function $F_{k,1}(q)$ has nonnegative coefficients for $k=5$ to $10$, extending prior results for $k\le4$. It reveals a deep link to Ramanujan's third-order mock theta function via $\lim_{k\to\infty} F_{k,1}(q)= q\omega(q)$ and a precise difference $q\omega(q)-F_{k,1}(q)= q^{2k+1}E_k(q)$ with $E_k(q)\in\mathbb{Z}[[q]]$ and $\operatorname{coeff}_{[q^0]}E_k(q)=1$, while showing $\operatorname{coeff}_{[q^m]}(q\omega(q)-F_{k,1}(q))=0$ for $1\le m\le 2k$. The work develops explicit $q$-series representations of $F_{k,1}(q)$, employs transformations such as Heine and Rogers–Fine, and uses computational verification to handle $k=8$–$10$; it also introduces a framework via $G_{k,n}(q)$ and poses conjectures involving $q$-binomial quotients that could yield a full proof. Overall, the results deepen connections between partition generating functions and mock theta functions, with potential implications for $q$-series identities and combinatorial interpretations.

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 $ω(q)$ and to quotients of certain $q$-binomial coefficients.

Theorems & Definitions (18)

• Conjecture 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Corollary 1.6
• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 3.1