Bulletin of the
Korean Mathematical Society BKMS

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

MSC numbers: Primary 30C15; Secondary 26C10