Frequently, in Physics, we need to determine the smallest root of a polynomial when its degree tends to infinity. Namely, let's suppose we have a polynomial of degree $d$,
$\displaystyle P_d(x) = \sum_{i=0}^d c_i x^i$
where the coefficients $c_i$ are known. I am particularly interested in polynomials that satisfy 3-term recurrence relation, e.g.,
$\displaystyle P_{d+1}(x) = 2x P_{d}(x)-P_{d-1}(x)$.
In this case, $P(x)$ is the Chebyshev polynomial of second kind. We can in fact find an explicit solution for its roots:
$\displaystyle x_k = \cos\left(\frac{k \pi}{d+1}\right),\quad k=1,\dots,d$.
The smallest root corresponds to $k=d$, and we can expand,
$\displaystyle \cos\left(\frac{d \pi}{d+1}\right) = -1 + \frac{\pi^2}{2d^2} + O(1/d)^3$
Then we can easily see that in the limit $d\rightarrow \infty$ the smallest root of P tends to -1.
This is a simple example where we can find the explicit expression for the roots and then take the limit. But, in general, one can have polynomials for which we do not know the explicit expression for the roots (e.g., polynomials generated by 3-term relation of third order), but we do know the expansion $\sum_{i=0}^d c_i x^i$. So:
Question 1: Is there a way to compute the smallest root of $P(x)$ when $d\rightarrow\infty$ in terms of the coefficients $c_i$? One can suppose that all roots of $P(x)$ are real and distinct.
Searching this forum I found this post:
Root of a polynomial $p(x)$ and asymptotic behaviour of the Taylor series of $\frac{1}{p(x)}$
that could lead to a nice answer. It says:
$\displaystyle x_{\text{smallest}} = \lim_{k\rightarrow\infty}\frac{a_k}{a_{k+1}}$
where $a_k$ is the coefficient of the Taylor expansion of $\displaystyle 1/P(x) = \sum_{k=0}^\infty a_k x^k$. In principle, if one could have a explicit expression of $a_k$, one could compute the behavior of $z_{\text{smallest}}$ when $d\rightarrow \infty$. Computing $a_k$ means that we have to compute the $k$-th derivative of $1/P(x)$.
Question 2: Is there a way to compute the coefficients of the Taylor expansion of $1/P(x)$ in terms of $c_i$?.
Thanks!