On the Frequency of Zeros of Solutions of Second Order Linear Differential Equations

Abstract

We consider the equation $\rm f^{\prime\prime}+{A}(z){f}=0$ with linearly independent solutions f₁,₂, where A(z) is a transcendental entire function of finite order. Conditions are given on A(z) which ensure that max{λ(f₁),λ(f₂)} = ∞, where λ(g) denotes the exponent of convergence of the zeros of g. We show as a special case of a further result that if P(z) is a non-constant, real, even polynomial with positive leading coefficient then every non-trivial solution of $\rm f^{\prime\prime}+{e}^P{f}=0$ satisfies λ(f) = ∞. Finally we consider the particular equation $\rm f^{\prime\prime}+({e}^Z-K){f}=0$ where K is a constant, which is of interest in that, depending on K, either every solution has λ(f) = ∞ or there exist two independent solutions f₁, f₂ each with λ(f_i) ≤ 1.