The zeros of certain family of self-reciprocal polynomials

Bull. Korean Math. Soc. 2007 Vol. 44, No. 3, 461-473 Printed September 1, 2007

Seon-Hong Kim Sookmyung Women's University

Abstract : For integral self-reciprocal polynomials $P(z)$ and $Q(z)$ with all zeros lying on the unit circle, does there exist integral self-reciprocal polynomial $G_r(z)$ depending on $r$ such that for any $r$, $0 \leq r \leq 1$, all zeros of $G_r(z)$ lie on the unit circle and $G_0(z)=P(z)$, $G_1(z)=Q(z)? $ We study this question by providing examples. An example answers some interesting questions. Another example relates to the study of convex combination of two polynomials. From this example, we deduce the study of the sum of certain two products of finite geometric series.

Keywords : self-reciprocal polynomials, convex combination, zeros, unit circle