Entire functions with an arithmetic sequence of exponents

For a given entire function $f(z)={\sum }_{j=0}^{\mathrm{\infty }}{a}_j{z}^j$, we study the zero distribution of $f_r(z)={\sum }_{j\equiv r(\mathrm{m}\mathrm{o}\mathrm{d}m)}{a}_j{z}^j$ 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 $\mathrm{\Im }{z}^m=0$ and $\mathrm{\Re }{z}^m\le 0$.