If I have an $n^{th}$ order polynomial set equal to zero, is there some way I can invert it to get a series expansion in terms of $n$?
In particular I'm interested in solving the equation
$$ U_n(x) - (1+ \mu)U_{n-1}(x) + \mu U_{n-2}(x) = 0$$
where $U_n(x)$ is a Chebyshev polynomial of the second kind and $1-\delta \leq \mu \leq 1$. In particular I expect the solution to scale something like $x \approx \alpha n^{-2} + ...$ for some $\alpha = \alpha(\mu)$.
Other suggestions for solutions are welcome!