  In this work we study the solutions of the equation $z^pR(z^k)=\alpha$ with
nonzero complex $\alpha$, integer $p,k$ and $R(z)$ generating a (possibly
doubly 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 $\{\frac jk
\pi\le\operatorname{Arg} z\le\frac {j+1}k \pi\}$, $j=0,\dots, 2k-1$. Our
approach rests on the fact that $z(\ln z^{p/k}R(z) )'$ is an $\mathcal
R$-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 that
functions of the form $\sum_{n=0}^\infty (\pm i)^{n(n-1)/2}a_n z^{n}$ have
simple zeros distinct in absolute value under a certain condition on the
coefficients $a_n\ge 0$. This includes the "disturbed exponential" function
corresponding to $a_n= q^{n(n-1)/2}/n!$ when $0<q\le 1$, as well as the partial
theta function corresponding to $a_n= q^{n(n-1)/2}$ when $0<q\le q_*\approx
0.7457224107$.
