The zeros of certain family of self-reciprocal polynomials

Abstract

For integral self-reciprocal polynomialsP(z) and Q(z) withall zeros lying on the unit circle, does there exist integral self-reciprocalpolynomial Gr(z) depending on r such that for any r, 0 r 1, all zerosof Gr(z) lie on the unit circle and G0(z) = P(z), G1(z) = Q(z)? Westudy this question by providing examples. An example answers someinteresting questions. Another example relates to the study of convexcombination of two polynomials. From this example, we deduce the studyof the sum of certain two products of nite geometric series.