On the frequency of complex zeros of solutions of certain differential equations

Abstract

In this paper, we investigate the frequency of zeros of solutions of linear differential equations of the form w^(k)+∑\limits_{\jmath}=1^k−1Q_jw^(j)+(Q₀+Re^P)w=0, where k{≥}2, and where Q₀, …, Q_k−1, R and P are arbitrary polynomials with R{¬≡}0 and P non-constant. All solutions f{¬≡}0 of such an equation are entire functions of infinite order of growth, but there are examples of such equations which can possess a solution whose zero-sequence has a finite exponent of convergence. In this paper, we show that unless a special relation exists between the polynomials Q₀, …, Q_k−1, and P, all solutions of such an equation have an infinite exponent of convergence for their zero-sequences. This result extends earlier results for the equation, w^(k)+(Q₀+Re^P)w=0.