Periodicity of signs of Fourier coefficients of eta-quotients

We study the periodicity of signs of Fourier coefficients of the function, ∏ d | α f ( − q d ) r d , where α is a positive integer, f ( − q ) = ∏ n = 1 ∞ ( 1 − q n ) , and r d ∈ 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.