why are the polynomials in this series all solvable by radicals?

Consider n+1 identical masses m in linear series with n identical
springs of spring constant k connecting adjacent masses. Taking the
lengths of the springs as n variables, you can set up n independent
second order linear differential equations and analyze the solutions.

What you want are eigenvalues, to determine characteristic vibrational
frequencies. After some simplification and change of variable, you get
polynomials according to the following recurrence relation:

p(1) = x
p(2) = x^2-1
p(n) = x * p(n-1) - p(n-2) for n >= 3

What is remarkable about this series is that every polynomial I've
checked up to degree fifty is factorable, and every factor I've checked
is solvable by radicals. (Some factors in some high degrees were too
computationally difficult.)

Furthermore, many of these solutions were expressible as linear
combinarions of pth roots of 1, or had such linear combinarions under
radicals.

I wonder why this recurrence relation invariably (at least as far as
I've checked) leads to such polynomials.

I tried varying the recurrence relation, and obtained some others with
similar behavior. However, other variations produce >= 5th degree
factors which are invariably not solvable by radicals, or produce
irreducible polynomials whose solution I did not attempt.

So there is some quality of recurrence relations, rather than some
feature of the original physical problem, which explains this?

.

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