What is the asymptotic solution to the roots of $x^n+a_1 \epsilon x^{n-1} +\cdots+ a_{n-1}\epsilon^{n-1}x+a_n \epsilon^n$?

Question

The polynomial I'm working with is:

$$ \lambda^n +\frac{\epsilon}{p \ 1!}\lambda^{n-1} +\frac{\epsilon^2}{p^2 2!}\lambda^{n-1} +\dots +\frac{\epsilon^{n-1}}{p^{n-1} (n-1)!}\lambda +\frac{\epsilon^{n}}{p^n (n!)} =0 $$

I've been scouring the internet for a method for finding it's solution, and the closest I've found is this. In the paper they only deal with polynomials with known coefficients, and a large part of their method is graphical.

Can anyone help me untagle the paper, or point in the direction of another resource which might help?

Answer 1

The polynomial above can be transformed into the exponential sum function:

$$ e_n = \sum_{k = 0}^n \frac{x^k}{k!} $$

The roots of this polynomial are known to approach the curve in the unit disk $|x e^{1-x}|=1$ for large $n$. No perturbation theory necessary.