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One helpful property of functions generating Pólya frequency sequences

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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 \(\{\frac{j}{k} \pi \leqslant \mathrm{Arg}\,\, z\leqslant \frac{j+1}{k} \pi \}\), \(j=0,\ldots , 2k-1\). 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 \(a_n\geqslant 0\). This includes the “disturbed exponential” function corresponding to \(a_n= q^{n(n-1)/2}/n!\) when \(0<q\leqslant 1\), as well as the partial theta function corresponding to \(a_n= q^{n(n-1)/2}\) when \(0<q\leqslant q_*\approx 0.7457224107\).

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Notes

  1. Functions of this form are the kth root transforms of \(z^pR^k(z)\). In the particular case when R(z) and \(R'(z)\) are holomorphic and non-zero at \(z=0\), the function \(zR^k(z)\) is univalent in some disk centred at the origin. Then, \(zR(z^{k})\) will be a univalent function with k-fold symmetry in this disk in the sense that the image of \(zR(z^{k})\) will be k-fold rotationally symmetric (see e.g. [8, Sect. 2.1] for the details). The term “functions with k-fold symmetry” could be good under the narrower conditions imposed; however, we study a more general case assuming no such regularity at the origin and allowing any integer p satisfying \(\gcd (|p|,k)=1\).

  2. As soon as the \(\alpha \)-set of the function \(z^pR(z^k)\) actually contains the point \(z_{i+2k}\): Theorem 20 asserts nothing about existence of \(\alpha \)-points, nor does Theorem 22.

  3. The definition of genus can be found in e.g. [7, p. 92] or [19, p. 9]. These books also introduce further basic notions of the theory of entire functions.

  4. The polynomial is called (Hurwitz) stable if all of its roots have negative real parts.

  5. The zeros of two functions are called interlacing if between each two consecutive zeros of the first function there is exactly one zero of the second function and vice versa. The Hermite-Biehler theorem assumes the interlacing property to be strict, which means that the functions have no common zeros.

  6. Many similar facts are well known. For example, considering functions \(\Phi (\zeta )\mathrel {\mathop :}=\phi (e^{-\zeta })\) gives the problem from [31] but in a strip. However, we place this lemma here since we need the relation between \(|z_1|\) and \(|z_2|\) satisfying (4) rather than the univalence itself.

  7. Here \(\mathrm{dist}(0,\gamma _i):= \inf _{z\in \gamma _i}|z|\) is the distance between the origin and the component \(\gamma _i\), \(i=1,2,\ldots \).

  8. In fact we have more: \(\partial \mathrm{Im}\psi (z) / \partial \nu =0\) implies that the gradient of \(\mathrm{Im}\psi \) on \(\gamma _i\) is tangential to \(\gamma _i\).

  9. This limit exists since the function \(\phi \) increases in \({\mathfrak {I}}\).

  10. This fact follows from the integral representation (2). It can be also deduced from the expression (7) when \(\mathcal {R}\)-functions have the form considered in this lemma.

  11. On the one hand, the condition that \(\overline{\alpha }=\alpha e_{-2p{\widetilde{s}}}\) for some integer \({\widetilde{s}}=0,\ldots ,k-1\) coincides with \(\overline{\alpha e_{-p{\widetilde{s}}}}=\alpha e_{-p{\widetilde{s}}}\) and therefore to \(\mathrm{Im}\alpha e_{-p{\widetilde{s}}}=0\). On the other hand, changing the order gives \(\overline{\alpha }\in \big \{\alpha e_{-2pm}\big \}_{m=0}^{k-1} =\big \{\alpha {}e_{2p(k-m)}\big \}_{m=0}^{k-1} =\big \{\alpha e_{2pm}\big \}_{m=0}^{k-1}\), which is \(\overline{\alpha }=\alpha e_{2ps}\) for some integer \(s=0,\ldots ,k-1\); the last expression can be written as \(\overline{\alpha e_{ps}}=\alpha e_{ps}\), or equivalently \(\mathrm{Im}\alpha e_{ps}=0\).

  12. The right-hand side follows from \(\alpha e_{2q+2ps} =\alpha e_{ps}\cdot e_{2q+ps} =\overline{\alpha e_{ps}}\cdot \overline{e_{-2q-ps}} =\overline{\alpha e_{-2q}} =\overline{\alpha }e_{2q}\).

  13. If \(R(w)\in \Omega \) for some \(w\in \mathbb {C}_{+}\) satisfying \(|w|<|w_0|\), then \(z_0\) cannot be the \(\alpha \)-point of F(z) minimal in absolute value; see the proof of Theorem 23 for the details.

  14. Recall that \(\left\lceil \frac{p}{2}\right\rceil \) stands for the minimal integer greater than or equal to \(\frac{p}{2}\). Here \(\left| \left\lceil \frac{p}{2}\right\rceil \right| \leqslant \left| \frac{p}{2}\right| \) since \(p<0\).

  15. The definition and properties of multiplier sequences can be found in e.g. [23], [22, Chapter II] and [19, Chapter VIII, Sect. 3]. The fact that \(\big (q^{n(n-1)/2}\big )_{n=0}^{\infty }\) is a multiplier sequence (of the first kind) for \(0<q\leqslant 1\) was first shown by Laguerre [17, p. 35]. The more modern proof follows from Satz 10.1 of [22, p. 42] applied to the function \(\Phi (z)\mathrel {\mathop :}=\exp \big (\frac{1}{2}z(z-1)\cdot \ln q\big )\).

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Acknowledgments

The author is grateful to the people who gave helpful comments concerning this study, especially the members (former and current) of his working group at the TU-Berlin. The anonymous referees suggested significant improvements to the paper, for which the author is grateful. I also thank the colleagues from Potsdam (Germany) and Ufa (Russian Federation) for useful discussions.

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  1. Institut für Mathematik, TU-Berlin, Sekretariat MA 4-2, Straße des 17. Juni 136, 10623, Berlin, Germany

    Alexander Dyachenko

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Correspondence to Alexander Dyachenko.

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Communicated by George Csordas.

This work was financially supported by the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007–2013)/ERC Grant Agreement No. 259173.

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Dyachenko, A. One helpful property of functions generating Pólya frequency sequences. Comput. Methods Funct. Theory 16, 529–566 (2016). https://doi.org/10.1007/s40315-016-0160-4

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