Entire functions of finite order as solutions to certain complex linear differential equations

Abstract

When is an entire function of finite order a solution to a complex 2nd order homogeneous linear differential equation with polynomial coefficients? In this paper we will give two (equivalent) answers to this question. The starting point of both answers is the Hadamard product representation of a given entire function of finite order. While the first answer involves certain Stieltjes-like relations associated to the function, the second one requires the vanishing of all but finitely many suitable expressions constructed via the Gil’ sums of the zeros of the function. Applications of these results will also be given, most notably to the spectral theory of one-dimensional Schrödinger operators with polynomial potentials.