On Some Polynomials with Weighted Sums

Abstract

In this note, we study a generalization of a certain polynomial <TEX>$z^n-{\sum\limits_{k=0}^{n-1}}a_kz^k$</TEX>, where <TEX>$\sum\limits_{k=0}^{n-1}a_k=1$</TEX>, <TEX>$a_k{\geq}0$</TEX> for each k, whose all zeros except for z=1 lie on the circle of radius 1/2 with center at the origin.