Oscillation results for {$n$}-th order linear differential equations with meromorphic periodic coefficients

Abstract

Consider $n$-th order linear differential equations with meromorphic periodic coefficients of the form $w^{(n)}+R_{n-1}(e^{z})w^{(n-1)}+\cdots+ R_{1}(e^{z})w'+R_{0}(e^{z})w = 0$, $n \ge 2$, where $R_{\nu}(t)$ $(0 \le \nu \le n-1)$ are rational functions of $t$. Under certain assumptions, we prove oscillation theorems concerning meromorphic solutions, which contain necessary conditions for the existence of a meromorphic solution with finite exponent of convergence of the zero-sequence. We also discuss meromorphic or entire solutions whose zero-sequences have an infinite exponent of convergence, and give a new zero-density estimate for such solutions.