Polynomials with weighted sum

Abstract

In this paper, we study the equation z(n) = Sigma(k=0)(n-1) alpha(k)Z(k), where Sigma(k=0)(n-1) alpha(k) = >= 0 for each k. We show that, given p > 1, there exist C(1/p)-polynomials with the degree of weighted sum n - 1. However, we obtain sufficient conditions for nonexistence of certain lacunary C(1/p)-polynomials. In case of the degree of weighted sum n - 2, we see that, by giving an example, our sufficient condition is best possible in a certain sense.