Zero Distribution of Polynomials Generated by a Power of a Cubic Polynomial

Abstract

For each $\alpha>0$ and $A(z),B(z)\in\mathbb{C}[z]$, we study the zero distribution of the sequence of polynomials $\left\{ P_{m}^{(\alpha)}(z)\right\} _{m=0}^{\infty}$ generated by $(1+B(z)t+A(z)t^{3})^{-\alpha}$. We show that for large $m$, the zeros of $P_{m}^{(\alpha)}(z)$ lie on an explicit curve on the complex plane.