I have the following function
$$f(x)=\frac{\sum_{n=0}^{+\infty}a_n \, x^{-n}}{\sum_{n=0}^{+\infty}\left(b_n \, x^n + c_n \, x^{-n}\right)}$$
Is there a way, maybe using long division to find the following equivalence:
$$\sum_{n=0}^{+\infty}d_n \, x^n=\frac{\sum_{n=0}^{+\infty}a_n \, x^{-n}}{\sum_{n=0}^{+\infty}\left(b_n \, x^n + c_n \, x^{-n}\right)}$$
or maybe with going from $-\infty$ to $\infty$, in order to find iteratively the values of $d_n$ ? I have to add that $a_n$, $b_n$ and $c_n$ are numerically known, but their explicit expression is unattainable. You can see that the biggest problem is that I have negative exponents at the numerator and the denominator.