1. Introduction

In a recent paper [3], an extended Liouville–Green formula

formula here

was developed for solutions of the second-order differential equation

formula here

Here γM(x) ∼Q−¼(x), QM(x) ∼Q−½(x) and εM(x)→0 as x→∞, while M([ges ]2) is an integer and γM and QM can be defined in terms of Q and its derivatives up to order M−1. The general form of (1·1) had been obtained previously by Cassell [5], [6], [7] and Eastham [10], [11, section 2·4]. In particular, the proof of (1·1) in [10] and [11] depended on the formulation of (1·2) as a first-order system and then on a process of M repeated diagonalization of the coefficient matrices in a sequence of related differential systems.