Is it possible to express the inverse of a polynomial as a series?

Question

Is it possible to express the multiplicative inverse of a polynomial in descending powers of n i.e.
\begin{equation} \frac{1}{\left[\sum_{k=0}^\infty a_kt^{n-2k}\right]^2} \end{equation} as a series using binomial theorem or any other tool?

additional notes: $ a_0 \neq 0 $ and the polynomial has real and distinct roots.

Thank you very much!

Answer 1

This is basically the inversion for formal power series: If you have a power series $$ \sum_i c_i x^i $$ and want to write its inverse as $$ \sum_i b_i x^i = \frac{1}{\sum_i c_i x^i} $$ you see that it has to hold $$ \sum_i b_i x^i \sum_i c_ix^i \equiv 1. $$ Using the Cauchy product on the left gives some $$ \sum_i d_i x^i\equiv 1 $$ and comparing of the coefficients (i.e. noting that $d_0 = 1$, $d_i=0$ for $i\geq 1$) a gives equations for the $b_i$. Note that there is no guarantee that the series $\sum_i b_i x^i$ has positive radius of convergence…