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- W3105332159 abstract "In this work we study solutions of the equation $$z^pR(z^k)=alpha $$ with non-zero 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 $$mathrm{Im},,alpha ^k=0$$ ) and split evenly among the sectors $${frac{j}{k} pi leqslant mathrm{Arg},, zleqslant frac{j+1}{k} pi }$$ , $$j=0,ldots , 2k-1$$ . 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 $$begin{aligned} f(z^k)+z^p g(z^k) quad text {and}quad g(z^k)+z^{p}f(z^k) end{aligned}$$ 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 $$a_ngeqslant 0$$ . This includes the “disturbed exponential” function corresponding to $$a_n= q^{n(n-1)/2}/n!$$ when $$0<qleqslant 1$$ , as well as the partial theta function corresponding to $$a_n= q^{n(n-1)/2}$$ when $$0<qleqslant q_*approx 0.7457224107$$ ." @default.
- W3105332159 created "2020-11-23" @default.
- W3105332159 creator A5085020785 @default.
- W3105332159 date "2016-04-20" @default.
- W3105332159 modified "2023-09-30" @default.
- W3105332159 title "One helpful property of functions generating Pólya frequency sequences" @default.
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- W3105332159 doi "https://doi.org/10.1007/s40315-016-0160-4" @default.
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