Sign Changes of Coefficients of Powers of the Infinite Borwein Product

TL;DR

This paper studies the sign behavior of coefficients in powers of the infinite Borwein product, providing asymptotic formulas, characterizations, and conjectures about their periodic sign changes across various parameters.

Contribution

It introduces asymptotic formulas and characterizations for the coefficients' signs, and confirms a conjecture on their periodicity for specific parameter cases.

Findings

Coefficients exhibit ultimately periodic sign patterns.

Asymptotic formulas are derived for coefficients.

Conjecture on sign periodicity is confirmed for certain cases.

Abstract

We denote by $c_t^{(m)}(n)$ the coefficient of $q^n$ in the series expansion of $(q;q)_\infty^m(q^t;q^t)_\infty^{-m}$, which is the $m$-th power of the infinite Borwein product. Let $t$ and $m$ be positive integers with $m(t-1)\leq 24$. We provide asymptotic formula for $c_t^{(m)}(n)$, and give characterizations of $n$ for which $c_t^{(m)}(n)$ is positive, negative or zero. We show that $c_t^{(m)}(n)$ is ultimately periodic in sign and conjecture that this is still true for other positive integer values of $t$ and $m$. Furthermore, we confirm this conjecture in the cases $(t,m)=(2,m),(p,1),(p,3)$ for arbitrary positive integer $m$ and prime $p$.