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

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 $\{ x + iy: \left | {y - \pi } \right | < {\eta _1}, x > {x_0}\} (0 < {\eta _1})$. 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 $\exp ( - {e^z})$.