General expression for ratio of polynomials of equal degree and nonzero coefficients

Question

Let $\{a_k\}$ and $\{b_k\}$ be sequences of real and positive constants and $x\in(0,1)$. If we perform polynomial division, we should be able to write this ratio as a single sum. Is there a known closed expression for the coefficients of this sum? Specifically, is there is closed formula for the coefficients $c_k$ below? $$ \frac{\sum_{k=0}^na_kx^k}{\sum_{k=0}^nb_kx^k}=\sum_{j=0}^n c_kx^{-k} $$