I'm considering quasi-polynomials of the form
$$ P(z) = z^n + a_1z^{n-1}+...+a_n + K_1e^{-z\tau}(z^m + b_1z^{m-1}+...+b_m) + K_2e^{-2z\tau}(z^j + c_1z^{j-1}+...+c_j), $$
where all the constants are real. For the case where $K_2=0$, there is this article: https://community.ams.org/journals/bull/1964-70-02/S0002-9904-1964-11129-0/S0002-9904-1964-11129-0.pdf by Allan M. Krall that decsribed the location of the zeros of $P$.
I kept searching for a similar result for the case where $K_2\neq 0$, but I did not manage yet to find a reference on this. Do you know any reference for this problem?