Periodicity of signs of Fourier coefficients of eta-quotients

We study the periodicity of signs of Fourier coefficients of the function,∏d|αf(−qd)rd, where α   is a positive integer, f(−q)=∏n=1∞(1−qn), and rd∈Zrd∈Z. As an application, we will prove the periodicity of signs for the crank differences conjectured by G.E. Andrews and R. Lewis and for the weighted counts of certain types of partitions.