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

Consider n-th order linear differential equations with meromorphic periodic coefficients of the form w⁽ⁿ⁾ + R_n-1(e^z)w^(n-1) + ⋯ + R₁(e^z)w′ + R₀(e^z)w = 0, n ≥ 2, where R_ν(t) (0 ≤ ν ≤ 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.