I have some expressions of the form
$$\left ( \sum_{n=0}^\infty a_n x^n \right ) ^l - \sum_{n=0}^\infty b_n x^n = \sum_{n=0}^k c_n x^n $$
Where ("isolatedly") each series on the LHS converge for $|x|<1$, the RHS is a polynomial of finite degree and $l>0$ is an integer.
I would be interested in finding the roots of the LHS (values for $x$ where both summands are equal) in an empirical or computational way. I think a good approach could be to apply Newton-Raphson Method to the RHS to get those approximate values.
My problem is that, since the RHS is a polynomial, we know, by the Fundamental Theorem of Algebra, that it will have $k$ roots. However, some (most) of them will lie outside the region of convergence of the LHS.
Imagine then that I were able to compute all $k$ roots. What should I do with those for which $|x| \ge 1$? Do I simply discard them? Do these roots of the polynomial mean anything special? How can I interpret the results?
To clarify: I know that if I expanded the first series by using repetitive Cauchy Products and then subtracted from it the second series, I would end up with a series whose coefficients $c_m$ would equal $0$ for every $m>k$ (hence, the polynomial). However, neither of the original series independently do converge for $|x| \ge 1$.
Thank you.
The Jenkins-Traub algorithm, a black box polynomial root finding algorithm, finds the roots of smaller modulus first and roots of largest modulus last. It may be useful, if you want to stop finding roots at $|x| \ge 1$.
Regarding how to interpret the results for roots $|x|\ge1$: I would consider your RHS to be the analytic continuation of the LHS, since it is a polynomial with a finite number of terms that converges for all finite $x$.