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.