Is there anything that can be said about the roots of the polynomial $f_n(x)$ if $f_n(x) = xf_{n-1}(x) + f_{n-2}(x)$ where these are polynomials of degree $n, n-1,$ and $n-2$, respectively. In addition, the roots of both $f_{n-1}(x)$ and $f_{n-2}(x)$ are known to be the maximum number of complex conjugate pairs (at most one purely real root), and all roots have positive real part.

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    $\begingroup$ These are the Fibonacci polynomials. See the paper "Roots of Fibonacci polynomials" mentioned in that page. $\endgroup$ – lhf Mar 24 '15 at 3:08
  • $\begingroup$ ... depending on $f_0$ and $f_1$. $\endgroup$ – lhf Mar 24 '15 at 3:15
  • $\begingroup$ @lhf Unfortunately, I'm not dealing with Fibonacci polynomials. For me, $ f_1(x) = x - 1$, and $f_2(x) = x^2 - x + 1$ $\endgroup$ – Pistol Pete Mar 24 '15 at 3:51

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