Sign-patterns of Certain Infinite Products

TL;DR

This paper analyzes the sign patterns of Fourier coefficients of specific eta quotients using dissections of theta functions and product formulas, extending previous conjectures and characterizing sign distributions for various classes.

Contribution

It introduces a method to determine sign patterns of Fourier coefficients for certain eta quotients and extends conjectures by Bringmann et al. to broader classes.

Findings

Sign patterns characterized for \\frac{(q^i;q^i)_{\\infty}}{(q^p;q^p)_{\\infty}}

Extended conjectures on coefficient signs for eta quotients

Sign distribution analysis for additional eta quotient classes

Abstract

The signs of Fourier coefficients of certain eta quotients are determined by dissecting expansions for theta functions and by applying a general dissection formula for certain classes of quintuple products. A characterization is given for the coefficient sign patterns for \[ \frac{(q^i;q^i)_{\infty}}{(q^p;q^p)_{\infty}} \] for integers \( i > 1 \) and primes \( p > 3 \). The sign analysis for this quotient addresses and extends a conjecture of Bringmann et al. for the coefficients of \( (q^2;q^2)_{\infty}(q^5;q^5)_{\infty}^{-1} \). The sign distribution for additional classes of eta quotients is considered. This addresses multiple conjectures posed by Bringmann et al.