Abstract
We consider the zeros of transcendental entire solutions f of the functional equation
where \(c\in {\rm C},0<|c|<1,\) and Q and the a j are polynomials. Under a suitable hypothesis concerning the associated Newton-Puiseux diagram we show that the zeros of f are asymptotic to certain geometric progressions. More precisely, with this hypothesis there exist positive integers M and N such that the zero set can written in the form \(\lbrace z_{n,u}:\mu \in \lbrace 1,2,\dots, M\rbrace, n\in {\rm N}\rbrace\) where for each μ in \(\lbrace 1,2,\dots, M\rbrace \) there exists A μ in ℂ {0} with \(z_{n,\mu}\sim A_{\mu}\ c^{-Nn}\ {\rm as}\ n\rightarrow \infty\). The proof is achieved by showing that f behaves asymptotically like a product of θ-functions. The hypothesis on the Newton-Puiseux diagram is satisfied, e.g., if for each positive σ and each real τ the line \(\lbrace (x,y)\in {\rm R}^2:y=\sigma x+\tau \rbrace\) contains at most two points of the form (j, deg(a j)). In particular, this is the case if all a j are linear, in which case the above conclusion follows with M = 1 which means that the zeros are asymptotic to only one geometric progression. The hypothesis on the Newton-Puiseux diagram is also satisfied if m = 1. If m = 1 and Q ≡ 0, however, we have a much simpler and more precise result. We illustrate our results by a number of examples. In particular, we show that if the hypothesis on the Newton-Puiseux diagram is not satisfied, then the zeros of the solutions need not be asymptotic to a finite number of geometric progessions.
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Dedicated to the memory of Professor Dieter Gaier, collaborator and true friend.
This research was initiated while the first author was visiting Imperial College, London. He hanks the Royal Society and the London Mathematical Society for financial support, and the econd author and the Department of Mathematics at Imperial College for the hospitality. Support f his research by the German—Israeli Foundation for Scientific Research and Development, rant G.I.F., G-643-117.6/1999, and by INTAS-99-00089 is also gratefully acknowledged.
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Bergweiler, W., Haymana, W.K. Zeros of Solutions of a Functional Equation. Comput. Methods Funct. Theory 3, 55–78 (2004). https://doi.org/10.1007/BF03321025
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DOI: https://doi.org/10.1007/BF03321025