On non-oscillatory solutions of the generalized Hill equation

Abstract

Let $\varrho\neq -1$ be a complex number with modulus $|\varrho|=1$ and $K(x,y)=P_1(x)+P_2(y)$ for two polynomials $P_1$ and $P_2$. We consider the non-oscillatory solutions such that $\lambda(f)<\infty$ of the generalized Hill equation $f''-K(e^z,e^{-\varrho z})f=0$ ($\sharp$). When $\varrho=1$, the Hill equation $f''-K(e^z,e^{-z})f=0$ ($\dag$) is also written as $x^2u''-[K(x,x^{-1})-1/4]u=0$ ($\ddag$). We point out that there is a full correspondence between the non-oscillatory solutions of equation ($\dag$) and the Liouvillian solutions of equation ($\ddag$). Then this paper has two purposes. First, we show that if equation ($\sharp$) has a nonzero non-oscillatory solution, then $\varrho=1$. To this end, we solve entire solutions of a general Tumura--Clunie type differential equation. Second, for the particular Hill equation $f''-(e^{\mathbf{k}z}+b_{\mathbf{s}}e^{\mathbf{s}z}+b_0)f=0$, where $\mathbf{k}>\mathbf{s}\geq 1$ are integers and $b_{\mathbf{s}}\not=0$, we use Kovacic's algorithms to determine the non-oscillatory solutions with relatively few zeros.