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In this work we study the solutions of the equation $z^pR(z^k)=\alpha$ withnonzero complex $\alpha$, integer $p,k$ and $R(z)$ generating a (possiblydoubly infinite) totally positive sequence. It is shown that the zeros of$z^pR(z^k)-\alpha$ are simple (or at most double in the case of real$\alpha^k$) and split evenly among the sectors $\{\frac jk\pi\le\operatorname{Arg} z\le\frac {j+1}k \pi\}$, $j=0,\dots, 2k-1$. Ourapproach rests on the fact that $z(\ln z^{p/k}R(z) )'$ is an $\mathcalR$-function (i.e. maps the upper half of the complex plane into itself).  This result guarantees the same localization to zeros of entire functions$f(z^k)+z^p g(z^k)$ and $g(z^k)+z^{p}f(z^k)$ provided that $f(z)$ and $g(-z)$have genus $0$ and only negative zeros. As an application, we deduce thatfunctions of the form $\sum_{n=0}^\infty (\pm i)^{n(n-1)/2}a_n z^{n}$ havesimple zeros distinct in absolute value under a certain condition on thecoefficients $a_n\ge 0$. This includes the "disturbed exponential" functioncorresponding to $a_n= q^{n(n-1)/2}/n!$ when $0<q\le 1$, as well as the partialtheta function corresponding to $a_n= q^{n(n-1)/2}$ when $0<q\le q_*\approx0.7457224107$.

One helpful property of functions generating P\'olya frequency sequences