Exponential polynomials in the oscillation theory

Abstract

Supposing that $A(z)$ is an exponential polynomial of the form $$A(z)=H_0(z)+H_1(z)e^{{\zeta }_1{z}^n}+\cdots +H_m(z)e^{{\zeta }_m{z}^n},$$ where $H_j$'s are entire and of order $<n$, it is demonstrated that the function $H_0(z)$ and the geometric location of the leading coefficients ${\zeta }_1,\dots ,{\zeta }_m$ play a key role in the oscillation of solutions of the differential equation $f^{\prime \prime }+A(z)f=0$. The key tools consist of value distribution properties of exponential polynomials, and elementary properties of the Phragmén-Lindelöf indicator function. In addition to results in the whole complex plane, results on sectorial oscillation are proved.

29 pages