Hurwitz Polynomial - Why does continued fraction expansion work?

Question

While checking whether the given polynomial is Hurwitz or not, we perform continued fraction expansion. We were taught how to perform this check(i.e) look at the sign of the coefficients of the quotients after performing the normal expansion steps.

I can understand why this method would work for a normal fraction like for example $\frac{158}{18}$. But, why separating a single polynomial into odd and even terms and dividing them(and invert and continue) yields all positive terms in the quotients for a Hurwitz polynomial?

Answer 1

The subject is beyond my knowledge. However, I came across the following proofs, I hope you find any value in either of them.

1-Paper-Elementary proof of the Routh-Hurwitz test by Gjerrit Meinsma

2-paper -A NETWORK PROOF OF A THEOREM ON HURWITZ POLYNOMIALS AND ITS GENERALIZATION - By T. R. BASHKOW and C. A. DESOER

3-Chapter 4 - Page 75 of:

Google Books:A Mathematical Introduction to Control Theory by Shlomo Engelberg