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Given a stable polynomial $\phi(s)=a_0+a_1{s}+a_2{s^2}+a_3{s^3}+\cdots+a_n{s^n}=\phi^{e}(s)+s\phi^{o}(s)$ where $\phi^{e}(s)=a_0+a_2{s^2}+a_4{s^4}+\cdots,~\phi^{o}(s)=a_1+a_3{s^2}+a_5{s^4}+\cdots$, how to determine the stability of the polynomial $\phi^{e}(s)+\dfrac{d\phi^{e}(s)}{ds}=a_0+a_2{s^2}+2a_2{s}+a_4{s^4}+4a_4{s^3}+a_6{s^6}+6a_6{s^5}+\cdots$?

By stable polynomial I meant, all the roots of the polynomial are on the open left half plane.

I initially thought of checking interlacing property involving $\phi^{e}(s)$ and $\phi^{o}(s)$ but that didn't take me any further. Next from the characteristic equation $\dfrac{d\phi^{e}(s)}{ds}+\phi^{e}(s)=0$, we get $ln(\phi^{e}(s))=-s+c$,

or $\phi^{e}(s)=ke^{-s}=k(1-s+\dfrac{s^2}{2}-\dfrac{s^3}{6}-\cdots)$. Since the coefficients are not strictly positive or negative, $\phi^{e}(s)$ is not stable and hence $\phi^{e}(s)+\dfrac{d\phi^{e}(s)}{ds}=k(1-s)e^{-s}$ is also unstable.

I was actually supposed to show the stability of the polynomial $\phi^{e}(s)+\dfrac{d\phi^{e}(s)}{ds}$, but I don't know where I went wrong in the analysis. Any suggestions are greatly appreciated.

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