New Borwein-type conjectures

TL;DR

The paper addresses sign-pattern phenomena in Borwein-type $q$-polynomials and extends prior work by formulating new conjectures modulo $3$ and $5$ using the framework $P_{n,k}^i(q)=\dfrac{(q;q)_{kn}^i}{(q^k;q^k)_n^i}$. It proposes precise positivity/negativity regimes for coefficient sequences $c_m^{(i)}(n)$ and $d_m^{(i)}(n)$, including asymptotic thresholds $\alpha^{(i)}(n)$ and $\tilde{\alpha}^{(i)}(n),\tilde{\beta}^{(i)}(n)$ with limiting values tabulated for various $(k,i)$, and confirms these via Maple computations. The work builds on Wang–Krattenthaler’s asymptotic methods and duality relations to predict sign behavior, extending Borwein-type conjectures to new moduli and highlighting potential partition-theoretic interpretations. Overall, it charts a path for proving sign-patterns in high-degree $q$-series through computational verification and asymptotic analysis, with implications for positivity phenomena in partition theory.

Abstract

Motivated by recent research of Wang and Krattenthaler, we use Maple to propose five new ``Borwein-type'' conjectures modulo $3$ and two new ``Borwein-type'' conjectures modulo $5$.

Theorems & Definitions (10)

• Conjecture 1.1: First Borwein Conjecture
• Conjecture 1.2: Second Borwein Conjecture
• Conjecture 1.3: Third Borwein Conjecture
• Conjecture 1.4: Cubic Borwein Conjecture
• Conjecture 1.5: Modulus $7$ Borwein Conjecture
• Conjecture 2.1
• Remark 1
• Conjecture 2.2
• Remark 2
• Remark 3