Sign-patterns of Certain Infinite Products

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 $$\frac{(q^i;q^i{)}_{\mathrm{\infty }}}{(q^p;q^p{)}_{\mathrm{\infty }}}$$ 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 $(q^2;q^2{)}_{\mathrm{\infty }}(q^5;q^5{)}_{\mathrm{\infty }}^{-1}$. The sign distribution for additional classes of eta quotients is considered. This addresses multiple conjectures posed by Bringmann et al.