On the explicit determination of certain solutions of periodic differential equations

Abstract

We consider the class of equations of the form (∗)w′′ + A(z)w = 0, where A(z) is a nonconstant entire periodic function (where, for convenience, we take the period to be 27φi), and where we assume that A(z) is a rational function of ez . The solutions of such equations are entire functions of infinite order of growth, but such equations can possess one or two linearly independent solutions each of whose zero-sequences has a finite exponent of convergence. In this paper, we develop a method which involves using only the simple technique of undetermined coefficients, which can test any equation (∗) for the existence of a solution whose zero-sequence has a finite exponent of convergence. The method will also produce explicitly any such solution.