On the Absolute Convergence of Polynomial Series

Abstract

where { P,(z) } is a given sequence of polynomials in the complex variable z and I a, } is a given sequence of complex numbers. With such a series are associated two point-sets Co and Ca in the complex z-plane. Co is the set of points z for which the series converges and Ca the set for which it converges absolutely. Either Ca or both Co and Ca may be vacuous in special cases. In the present paper we shall assume that Pn(z) is a polynomial of degree n having all its n roots real and we shall be concerned with the corresponding set Ca exclusively. In a number of cases the geometric character of Ca is fairly well known. The following table gives a list of some important cases.* The table gives the interior of Ca rather than Ca itself. The latter may differ from its interior by points on the boundary or, in the third case, also by a finite number of isolated points.