Entire functions with an arithmetic sequence of exponents

Abstract

For a given entire function $f(z)={\sum }_{j=0}^{\infty }{a}_j{z}^j$, we study the zero distribution of $f_r(z)={\sum }_{j\equiv r(\mathrm{mod}m)}{a}_j{z}^j$ where $m\in \mathbb{N}$ and $0\le r<m$. We find conditions under which the zeros of $f_r(z)$ lie on $m$ radial rays defined by $\Im z^m=0$ and $\Re z^m\le 0$.