Generating functions for ratios of sequences

Question

Suppose that I have two sequences $(a_0,a_1,a_2,\dots)$ and $(b_0,b_1,b_2,\dots)$ and the corresponding generating functions $A(x)=a_0+a_1x+a_2x^2+\dots$ and $B(x)=b_0+b_1x+b_2x^2+\dots$. Letting $c_i=\frac{a_i}{b_i}$, is there a way to retrieve the generating function for the sequence $c_0,c_1,c_2,\dots$?