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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Zeros of successive derivatives of entire functions of the form $ h(z){\rm exp}(-e\sp{z})$


Author: Albert Edrei
Journal: Trans. Amer. Math. Soc. 259 (1980), 207-226
MSC: Primary 30D20; Secondary 30D15
DOI: https://doi.org/10.1090/S0002-9947-1980-0561833-6
MathSciNet review: 561833
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Abstract: Consider $ f(z)\, = \,h(z)\exp ( - {e^z})\,(z\, = \,x\, + \,iy)$ where $ h(z)$ is a real entire function of finite order having no zeros in some strip $ \{ x\, + \,iy:\,\left\vert {y - \pi } \right\vert\, < \,{\eta _1},\,x\, > \,{x_0}\} \,(0\, < \,{\eta _1})$. The author studies the power series (1) $ f(\tau \, + \,z)\, = \,\Sigma _{n = 0}^\infty \,{a_n}{z^n}$ ($ \tau $ real) and the number $ N({\tau _1},\,{\tau _2};\,n)$ of real zeros of $ {f^{(n)}}(z)$ which lie in the interval $ [{\tau _1},\,{\tau _2}]$. He proves (2) $ N({\tau _1},{\tau _2};\,n)\, \sim \,({\tau _2} - {\tau _1})n/{(\log \,n)^2}\,(n \to \infty )$. With regard to the expansion (1) he determines a positive, strictly increasing, unbounded sequence $ \{ {v_k}\} _{k = 1}^\infty $ such that $ ({v_{k + 1}} - {v_k})/\log \,{v_k} \to 1\,(k \to \infty )$ and having the following properties: (i) if $ {v_k}\, < \,n\, < \,{v_{k + 1}}$, then $ {a_n}\, \ne \,0$ and all the $ {a_n}$ have the same sign; (ii) if in addition $ {v_{k + 1}}\, < \,m\, < \,{v_{k + 2}}$, tnen $ {a_m}{a_n}\, < 0$. It is possible to deduce from (2) the complete characterization of the final set (in the sense of Pólya) of $ \exp ( - {e^z})$.


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DOI: https://doi.org/10.1090/S0002-9947-1980-0561833-6
Article copyright: © Copyright 1980 American Mathematical Society