Concerning an approximation of Copson

Abstract

The determination of the coefficients is quite complicated (and, for the coefficient of n-4, incorrect). In the present paper we obtain Copson's series by a simpler method which yields an asymptotic expansion for Sn(z) valid for every complex number z except z= 1. For our principle result we require the following three lemmas concerning the function Sn(z) defined by (1) and the function Tn(z) defined by